A generalization of the alcove model and its applications

Abstract

The alcove model of the first author and Postnikov uniformly describes highest weight crystals of semisimple Lie algebras. We construct a generalization, called the quantum alcove model. In joint work of the first author with Naito, Sagaki, Schilling, and Shimozono, this was shown to uniformly describe tensor products of column shape Kirillov–Reshetikhin crystals in all untwisted affine types; moreover, an efficient formula for the corresponding energy function is available. In the second part of this paper, we specialize the quantum alcove model to types $A$ and $C$. We give explicit affine crystal isomorphisms from the specialized quantum alcove model to the corresponding tensor products of column shape Kirillov–Reshetikhin crystals, which are realized in terms of Kashiwara–Nakashima columns.

1 Introduction

Kashiwara’s crystals [11] are colored directed graphs encoding the structure of certain bases (called crystal bases) for certain representations of quantum groups $U_q({\mathfrak g})$ as $q$ goes to zero. The first author and Postnikov [23, 24] defined the so-called alcove model for highest weight crystals associated to a semisimple Lie algebra $\mathfrak g$ (in fact, the model was defined more generally, for symmetrizable Kac–Moody algebras $\mathfrak g$). A related model is the one of Gaussent–Littelmann, based on LS-galleries [7]. Both models are discrete counterparts of the celebrated Littelmann path model [26, 27].

In this paper, we construct a generalization of the alcove model, which we call the quantum alcove model, as it is based on enumerating paths in the so-called quantum Bruhat graph of the corresponding finite Weyl group. This graph originates in the quantum cohomology theory for flag varieties [6], and was first studied in [2]. The path enumeration is determined by the choice of a certain sequence of alcoves (an alcove path or, equivalently, a $\lambda$-chain of roots), like in the classical alcove model. If we restrict to paths in the Hasse diagram of the Bruhat order, we recover the classical alcove model. The mentioned paths in the quantum Bruhat graph first appeared in [18], where they index the terms in the specialization $t=0$ of the Ram–Yip formula [30] for Macdonald polynomials $P_{\lambda }(X;q,t)$. We construct combinatorial crystal operators on the mentioned paths, and prove various properties of them.

The main application [21, 22] is that the new model uniformly describes tensor products of column shape Kirillov–Reshetikhin (KR) crystals [13], for all untwisted affine types. (KR crystals correspond to certain finite-dimensional representations of affine algebras.) More precisely, the model realizes the crystal operators on the mentioned tensor product, and also gives an efficient formula (based on the so-called height statistic) for the corresponding energy function [8]. (The energy can be viewed as an affine grading on a tensor product of KR crystals [28, 32].) This result, combined with the Ram–Yip formula for Macdonald polynomials [30], implies that the graded character of a tensor product of column shape KR modules (the grading being by the energy function) concides with the corresponding Macdonald polynomial specialized at $t=0$ [22].

In the second part of this paper, we specialize the quantum alcove model to types $A$ and $C$, and prove that the bijections constructed in [18], from the objects of the specialized quantum alcove model to the tensor products of the corresponding Kashiwara–Nakashima (KN) columns [12], are affine crystal isomorphisms. (A column shape KR crystal is realized by a KN column in these cases.) Note that this result has no overlap with the type-independent result in [21, 22], because the $\lambda$-chains on which the quantum alcove model is based in the two cases are different. Moreover, note that having such explicit bijections to models based on diagram fillings, which are known to be crystal isomorphisms, is important for the following reason. It is often easier to define extra structure on the quantum alcove model, which describes additional structure of the crystal, and only then translate it to the models based on fillings via the bijection. One such example was mentioned above, in connection with the energy function; the height statistic in the quantum alcove model has been translated to the so-called charge statistic for fillings in types $A$ and $C$ in [18], while type $B$ is currently under investigation in [1]. Another example is related to the combinatorial $R$-matrix mentioned below.

We also conjecture that, like the alcove model, its generalization given here is independent of the $\lambda$-chain of roots on which the whole construction is based, cf. [15]. This conjecture is currently under investigation in [19]. We intend to realize an affine crystal isomorphism between the models based on two $\lambda$-chains by extending to the quantum alcove model the alcove model version of Schützenberger’s jeu de taquin [5] on Young tableaux in [15]; the latter is based on so-called Yang–Baxter moves. Another application of this construction would be a uniform realization of the combinatorial $R$-matrix (i.e., the unique affine crystal isomorphism commuting factors in a tensor product of KR crystals).

2 Background

Let $\mathfrak {g}$ be a complex semisimple Lie algebra, and $\mathfrak {h}$ a Cartan subalgebra, whose rank is $r$. Let $\Phi \subset \mathfrak {h}^*$ be the corresponding irreducible root system, $\mathfrak {h}^*_{\mathbb {R}}\subset \mathfrak {h}$ the real span of the roots, and $\Phi ^{+} \subset \Phi$ the set of positive roots. Let $\Phi ^{-} := \Phi \backslash \Phi ^{+}$. For $\alpha \in \Phi$, we will say that $\alpha > 0$ if $\alpha \in \Phi ^{+}$, and $\alpha < 0$ if $\alpha \in \Phi ^{-}$. The sign of the root $\alpha$, denoted $\mathrm{sgn}(\alpha )$, is defined to be $1$ if $\alpha \in \Phi ^{+}$, and $-1$ otherwise. Let $| \alpha | = \mathrm{sgn}( \alpha ) \alpha$. Let $\rho := \frac{1}{2}(\sum _{\alpha \in \Phi ^{+}}\alpha )$. Let $\alpha _1, \ldots , \alpha _r \in \Phi ^{+}$ be the corresponding simple roots, and $s_i:=s_{\alpha _i}$ the corresponding simple reflections. We denote $\langle \cdot , \cdot \rangle$ the nondegenerate scalar product on $\mathfrak {h}^{*}_{\mathbb {R}}$ induced by the Killing form. Given a root $\alpha$, we consider the corresponding coroot $\alpha ^{\vee } := 2\alpha / \langle \alpha , \alpha \rangle$ and reflection $s_{\alpha }$. If $\alpha = \sum _i c_i \alpha _i$, then the height of $\alpha$, denoted by $\mathrm{ht}(\alpha )$, is given by $\mathrm{ht}(\alpha ):=\sum _i c_i$. We denote by $\widetilde{\alpha }$ the highest root in $\Phi ^{+}$; we let $\theta = \alpha _0 := -\widetilde{\alpha }$ and $s_0:=s_{\widetilde{\alpha }}$.

Let $W$ be the corresponding Weyl group. The length function on $W$ is denoted by $\ell (\cdot )$. The Bruhat order on $W$ is defined by its covers $w \lessdot ws_{\alpha }$, for $\alpha \in \Phi ^{+}$, if $\ell (ws_{\alpha }) = \ell (w) + 1$. Define $w \lhd ws_{\alpha }$, for $\alpha \in \Phi ^{+}$, if $\ell (ws_{\alpha }) = \ell (w) - 2\mathrm{ht}( \alpha ^\vee ) + 1$. The quantum Bruhat graph [6] is the directed graph on $W$ with edges labeled by positive roots

$$\begin{aligned} w \mathop {\longrightarrow }\limits ^{\alpha } ws_{\alpha } \quad \text { for } w \lessdot ws_{\alpha } \, \text{ or } \, w \lhd ws_{\alpha }\,; \end{aligned}$$

(1)

see Example 4.2.

The weight lattice $\Lambda$ is given by

$$\begin{aligned} \Lambda := \left\{ \lambda \in \mathfrak {h}_{\mathbb {R}}^{*} \,:\, \langle \lambda , \alpha ^{\vee } \rangle \in \mathbb {Z}\text { for any } \alpha \in \Phi \right\} . \end{aligned}$$

(2)

The weight lattice $\Lambda$ is generated by the fundamental weights $\omega _1, \ldots \omega _r$, which form the dual basis to the basis of simple coroots, i.e., $\langle \omega _i, \alpha _j^{\vee } \rangle = \delta _{ij}$. The set $\Lambda ^{+}$ of dominant weights is given by

$$\begin{aligned} \Lambda ^{+} := \left\{ \lambda \in \Lambda \, : \, \langle \lambda , \alpha ^{\vee } \rangle \ge 0 \text { for any } \alpha \in \Phi ^{+} \right\} . \end{aligned}$$

(3)

Given $\alpha \in \Phi$ and $k \in \mathbb {Z}$, we denote by $s_{\alpha ,k}$ the reflection in the affine hyperplane

$$\begin{aligned} H_{\alpha ,k}:= \left\{ \lambda \in \mathfrak {h}^{*}_{\mathbb {R}} \, : \, \langle \lambda , \alpha ^{\vee } \rangle = k \right\} . \end{aligned}$$

(4)

These reflections generate the affine Weyl group $W_\mathrm{aff }$ for the dual root system $\Phi ^{\vee }:= \left\{ \alpha ^{\vee } \, |\, \alpha \in \Phi \right\}$. The hyperplanes $H_{\alpha ,k}$ divide the real vector space $\mathfrak {h}^{*}_\mathbb {R}$ into open regions, called alcoves. The fundamental alcove $A_{\circ }$ is given by

$$\begin{aligned} A_{\circ } := \left\{ \lambda \in \mathfrak {h}_{\mathbb {R}}^{*} \, | \, 0 < \langle \lambda , \alpha ^{\vee } \rangle < 1 \text { for all } \alpha \in \Phi ^{+} \right\} . \end{aligned}$$

(5)

We will need the following properties of the quantum Bruhat graph which were proved in [21]; more precisely, Lemma 2.1 below is a simplified version of Proposition 5.4.2 in the mentioned paper, whereas Lemmas 2.2 and 2.3 below are simplified versions of different parts of the Diamond Lemma 5.5.2.

Lemma 2.1

Let $w \in W$. We have $w^{-1}(\theta )>0$ if and only if $w \lhd s_{\theta }w$. We also have $w^{-1}(\theta ) < 0$ if and only if $s_{\theta }w \lhd w$.

Lemma 2.2

Let $w \in W$, let $\alpha$ be a simple root, $\beta \in \Phi ^+$, and assume $s_{\alpha }w \ne ws_{\beta }$. Then $w \lessdot s_{\alpha }w$ and $w \longrightarrow ws_{\beta }$ if and only if $ws_{\beta } \lessdot s_{\alpha }ws_{\beta }$ and $s_{\alpha }w \longrightarrow s_{\alpha }ws_{\beta }$, cf. the diagram below. Furthermore, in this context we have $w \lessdot ws_{\beta }$ if and only if $s_{\alpha }w \lessdot s_{\alpha }ws_{\beta }$.

Lemma 2.3

Let $w \in W,\,\beta \in \Phi ^+$, and assume $s_{\theta }w \ne ws_{\beta }$. Then $w \lhd s_{\theta }w$ and $w \longrightarrow ws_{\beta }$ if and only if $ws_{\beta } \lhd s_{\theta }ws_{\beta }$ and $s_{\theta }w \longrightarrow s_{\theta }ws_{\beta }.$

2.1 Kirillov–Reshetikhin (KR) crystals

A $\mathfrak {g}$-crystal (for a symmetrizable Kac–Moody $\mathfrak {g}$) is a nonempty set $B$ together with maps $e_i,f_i:B\rightarrow B\cup \{ \mathbf {0}\}$ for $i\in I$ ($I$ indexes the simple roots, as usual, and $\mathbf {0}\not \in B$), and $\mathrm{wt}:B \rightarrow \Lambda$. We require $b^{\prime }=f_i(b)$ if and only if $b=e_i(b^{\prime })$, and $\mathrm{wt}(f_i(b))=\mathrm{wt}(b)-\alpha _i$. The maps $e_i$ and $f_i$ are called crystal operators and are represented as arrows $b \rightarrow b^{\prime }=f_i(b)$ colored $i$; thus they endow $B$ with the structure of a colored directed graph. For $b\in B$, we set $\varepsilon _i(b) := \max \{k \mid e_i^k(b) \ne \mathbf {0}\}$, and $\varphi _i(b) := \max \{k \mid f_i^k(b) \ne \mathbf {0}\}$. Given two $\mathfrak {g}$-crystals $B_1$ and $B_2$, we define their tensor product $B_1 \otimes B_2$ as follows. As a set, $B_1\otimes B_2$ is the Cartesian product of the two sets. For $b=b_1 \otimes b_2\in B_1 \otimes B_2$, the weight function is simply $\mathrm{wt}(b) := \mathrm{wt}(b_1) + \mathrm{wt}(b_2)$. The crystal operators are given by

$$\begin{aligned} f_i (b_1 \otimes b_2):={\left\{ \begin{array}{ll} f_i (b_1) \otimes b_2 &{} \text {if }\,\varepsilon _i(b_1) \ge \varphi _i(b_2)\\ b_1 \otimes f_i (b_2) &{} \text { otherwise, } \end{array}\right. } \end{aligned}$$

(6)

and similarly for $e_i$. The highest weight crystal $B(\lambda )$ of highest weight $\lambda \in \Lambda ^+$ is a certain crystal with a unique element $u_\lambda$ such that $e_i(u_\lambda )=\mathbf {0}$ for all $i\in I$ and $\mathrm{wt}(u_\lambda )=\lambda$. It encodes the structure of the crystal basis of the $U_q(\mathfrak {g})$-irreducible representation with highest weight $\lambda$ as $q$ goes to 0.

A Kirillov–Reshetikhin (KR) crystal [13] is a finite crystal $B^{r,s}$ for an affine algebra, associated to a rectangle of height $r$ and width $s$, where $r\in I{\setminus }\{0\}$ and $s$ is any positive integer. We refer, throughout, to the untwisted affine types $A_{n-1}^{(1)}-G_2^{(1)}$, and only consider column shape KR crystals $B^{r,1}$.

We now describe the models based on diagram fillings for KR crystals $B^{r,1}$ of type $A_{n-1}^{(1)}$ and $C_n^{(1)}$, where $r\in \{1,2,\ldots ,n-1\}$ and $r\in \{1,2,\ldots ,n\}$, respectively. As a classical type $A_{n-1}$ (resp. $C_n$) crystal, the KR crystal $B^{r,1}$ is isomorphic to the corresponding $B(\omega _r)$. Therefore, we can use the corresponding models in terms of fillings, as mentioned below.

In type $A_{n-1}^{(1)}$, an element $b\in B^{r,1}$ is represented by a strictly increasing filling of a height $r$ column, with entries in $[n]:=\{1, \ldots , n \}$. We will now describe the crystal operators on a tensor product of type $A_{n-1}^{(1)}$ KR crystals $B^{r,1}$ in terms of the so-called signature rule, which is just a translation of the tensor product rule (6). To apply $f_i$ (or $e_i$) on $b := b_1 \otimes \cdots \otimes b_k$ in $B^{i_1,1} \otimes \cdots \otimes B^{i_k,1}$, consider the word with letters $i$ and $i+1$, if $1\le i \le n-1$ (resp., the letters $n$ and $1$, if $i=0$) formed by recording these letters in $b_1,\ldots ,b_k$, which are scanned from left to right and bottom to top; we make the convention that if $i=0$ and a column contains both $1$ and $n$, then we discard this column. We replace the letter $i$ with the symbol $+$ and the letter $i+1$ with $-$ (resp., $n$ with $+$ and $1$ with $-$, if $i=0$). Then, we remove from our binary word adjacent pairs $-+$, as long as this is possible. At the end of this process, we are left with a word

$$\begin{aligned} \rho _i(b) = \underbrace{++ \ldots +}_x\underbrace{-- \ldots -}_y\,, \end{aligned}$$

(7)

called the $i$-signature of $b$.

Definition 2.4

• (1) If $y > 0$, then $e_i(b)$ is obtained by replacing in $b$ the letter $i+1$ which corresponds to the leftmost $-$ in $\rho _i(b)$ with the letter $i$ (resp., the letter $1$ with $n$, after which we sort the column, if $i=0$). If $y=0$, then $e_i(b) = \mathbf {0}$.

• (2) If $x > 0$, then $f_i(b)$ is obtained by replacing in $b$ the letter $i$ which corresponds to the rightmost $+$ in $\rho _i(b)$ with the letter $i+1$ (resp., the letter $n$ with $1$, after which we sort the column, if $i=0$). If $x=0$, then $f_i(b) =\mathbf {0}$.

Example 2.5

Let $n=3$,

, and has $+--$ as its $0$-signature. So we have

.

In type $C_{n}^{(1)}$, the elements of $B^{r,1}$ are represented by Kashiwara–Nakashima (KN) columns [12] of height $r$, with entries in the set $[\overline{n}]:= \left\{ 1< \cdots < n<\overline{n}< \cdots <\overline{1} \right\}$, which we will now describe.

Definition 2.6

A column-strict filling $C = x_1 \ldots x_r$ with entries in $[\overline{n}]$ is a KN column if there is no pair $(z,\overline{z})$ of letters in $C$ such that

$$\begin{aligned} z=x_p,\; \overline{z} = x_q,\; q-p \le r-z. \end{aligned}$$

Crystal operators $f_i$ and $e_i$ are defined on tensor products of KN columns in a similar way to type $A_{n-1}^{(1)}$. To apply $f_i$ (or $e_i$) on $b := b_1 \otimes \cdots \otimes b_k$ in $B^{i_1,1} \otimes \cdots \otimes B^{i_k,1}$, consider the word with letters $i, i+1, \overline{\imath }, \overline{i+1}$, if $1\le i \le n-1$ (resp., the letters $n$ and $\overline{n}$, if $i=n$, or $\overline{1}$ and $1$, if $i=0$) formed by recording these letters in $b_1,\ldots ,b_k$, which are scanned from from left to right and bottom to top; note that the letters $1$ and $\overline{1}$ cannot simultaneously appear in a column, so we do not need an exception like in type $A_{n-1}^{(1)}$ if $i=0$. We replace the letters $i,\overline{i+1}$ with the symbol $+$ and the letters $i+1, \overline{\imath }$ with $-$, if $1\le i \le n-1$ (if $i=n$ we replace $n$ with $+$ and $\overline{n}$ with $-$, and if $i=0$ we replace $\overline{1}$ with $+$ and $1$ with $-$). We proceed like in type $A_{n-1}^{(1)}$ by canceling adjacent pairs $-+$ as long as possible, and we obtain the $i$-signature $\rho _i(b)$. The crystal operators $f_i$ and $e_i$ are again given in terms $\rho _i(b)$, by a similar procedure to the one in Definition 2.4. Namely, if $1\le i \le n-1$, changing $+$ to $-$ means changing $i$ to $i+1$, if $+$ corresponds to $i$, and changing $\overline{i+1}$ to $\overline{\imath }$, if $+$ corresponds to $\overline{i+1}$; similarly changing $-$ to $+$ means changing $i+1$ to $i$ or $\overline{\imath }$ to $\overline{i+1}$. On another hand, changing $+$ to $-$ means changing $n$ to $\overline{n}$ if $i=n$, and changing $\overline{1}$ to $1$ if $i=0$.

We will need a different definition of KN columns which was proved to be equivalent to the one above in [31].

Definition 2.7

Let $C$ be column and $I = \left\{ z_1 > \ldots > z_k \right\}$ the set of unbarred letters $z$ such that the pair $(z,\overline{z})$ occurs in $C$. The column $C$ can be split when there exists a set of $k$ unbarred letters $J = \left\{ t_1 > \ldots > t_k \right\} \subset [n]$ such that

• $t_1$ is the greatest letter in $[n]$ satisfying: $t_1 < z_1,\,t_1\not \in C$, and $\overline{t_1} \not \in C$,

• for $i = 2, \ldots , k$, the letter $t_i$ is the greatest one in $[n]$ satisfying $t_i < \min (t_{i-1}, z_i),\,t_i \not \in C$, and $\overline{t_i} \not \in C$.

In this case we write

• $rC$ for the column obtained by changing $\overline{z_i}$ into $\overline{t_i}$ in $C$ for each letter $z_i \in I$, and by reordering if necessary,

• $lC$ for the column obtained by changing $z_i$ into $t_i$ in $C$ for each letter $z_i \in I$, and by reordering if necessary.

The pair $(lC,rC)$ will be called a split column, which we well sometimes denote by $lCrC$.

Example 2.8

The following is a KN column of height 5 in type $C_n$ for $n \ge 5$, together with the corresponding split column:

We used the fact that $\left\{ z_1>z_2\right\} =\left\{ 5>4\right\}$, so $\left\{ t_1 > t_2 \right\} = \left\{ 2 > 1 \right\}$.

Proposition 2.9

• (1) [31] A column with entries in $[\overline{n}]$ is a KN column if and only if it can be split.

• (2) [10, Theorem 5.1] The splitting is compatible with the action of the crystal operators, i.e., if $f_i(C)=C^{\prime }$ then $lC^{\prime }rC^{\prime }=f_i^2(lCrC)$. This holds more generally, for tensor products of columns, i.e., if $f_i(C_1\ldots C_n)=C_1^{\prime }\ldots C_n^{\prime }$, then $lC_1^{\prime }rC_1^{\prime }\ldots lC_n^{\prime }rC_n^{\prime }=f_i^2(lC_1rC_1\ldots lC_nrC_n)$.

In what follows we will use Definition 2.7 as the definition of KN columns.

We refer again to (column shape) KR crystals of arbitrary (untwisted) type. Let $\lambda =(\lambda _1 \ge \lambda _2\ge \ldots )$ be a partition, which encodes a dominant weight in classical types; let $\lambda ^{\prime }$ be the conjugate partition. We define

$$\begin{aligned} B^{\otimes \lambda }:=\bigotimes _{i=1}^{\lambda _1} B^{\lambda _i^{\prime },1}\,, \end{aligned}$$

(8)

assuming that the corresponding column shape KR crystals exist. We denote such a tensor product generically by $B$.

Remarks 2.10

• (1) It is known that $B$ is connected as an affine crystal, but disconnected as a classical crystal (i.e., with the $0$-arrows removed).

• (2) Let ${\mathbf {p}^{\prime }}$ be a composition obtained from ${\mathbf {p}}$ by permuting its parts. There is an affine crystal isomorphism betwee $B^{\otimes {\mathbf {p}}}$ and $B^{\otimes {\mathbf {p^{\prime }}}}$, which is unique by the previous remark. This isomorphism is called the combinatorial $R$-matrix.

We need to distinguish certain arrows in $B$, which are related to affine Demazure crystals, as we shall explain.

Definition 2.11

An arrow $b\rightarrow f_i(b)$ in $B$ is called a Demazure arrow if $i \ne 0$, or $i=0$ and $\varepsilon _0(b)\ge 1$. An arrow $b\rightarrow f_i(b)$ in $B$ is called a dual Demazure arrow if $i \ne 0$, or $i=0$ and $\varphi _i(b)\ge 2$.

Remarks 2.12

• (1) By Fourier–Littelmann [3], in simply laced types, the tensor product of KR crystals $B$ is isomorphic, as a classical crystal (discard the affine $0$-arrows) with a certain Demazure crystal for the corresponding affine algebra. (Demazure modules are submodules of highest weight ones determined by a Borel subalgebra acting on an extremal weight vector.) Moreover, by [4], the $0$-arrows in the latter correspond precisely to the Demazure arrows in $B$.

• (2) In the case when all of the tensor factors in $B$ are perfect crystals [9], $B$ remains connected upon removal of the non-Demazure (resp. non-dual Demazure) $0$-arrows.

• (3) In classical types, $B^{k,1}$ is perfect as follows: in types $A^{(1)}_{n-1}$ and $D^{(1)}_{n}$ for all $k$, in type $B^{(1)}_{n}$ only for $k\ne n$, and in type $C^{(1)}_n$ only for $k=n$ (using the standard indexing of the Dynkin diagram); in other words, for all the Dynkin nodes in simply laced types, and only for the nodes corresponding to the long roots in non-simply-laced types. It was conjectured in [8] that the same is true in the exceptional types. In type $G_2^{(1)}$ this was confirmed in [33], while for types $E_{6,7}^{(1)}$ and $F_{4}^{(1)}$ it was checked by computer, based on a model closely related to the quantum alcove model, see Sect. 3 and [22].

The energy function $D=D_B$ is a function from $B$ to the integers, defined by summing the so-called local energies of all pairs of tensor factors [8]. We will only refer here to the so-called tail energy [22], so we will not make this specification. (There are two conventions in defining the local energy of a pair of tensor factors: commuting the right one toward the head of the tensor product, or the left one toward the tail; the tail energy corresponds to the second choice.) We will only need the following property of the energy function, which defines it as an affine grading on $B$.

Theorem 2.13

[28, 32] The energy is preserved by the classical crystal operators $f_i$, i.e., $i\ne 0$. If $b\rightarrow f_0(b)$ is a dual Demazure arrow, then $D(f_0(b))=D(b)-1$.

Remarks 2.14

Theorem 2.13 shows that the energy is determined up to a constant on the connected components of the subgraph of the affine crystal $B$ containing only the dual Demazure arrows. See also Remark 2.12 (2).

One can define a statistic called charge on the model based on KN columns for $B^{\otimes \lambda }$ in types $A$ and $C$. This was done in [18], by translating a certain statistic in the Ram–Yip formula for Macdonald polynomials (i.e., the height statistic in (11)) to the model based on KN columns, via certain bijections recalled in Sect. 4, cf. Remarks 4.8 (2) and 4.23 (2). In type $A$, this procedure leads to the same statistic that was originally defined by Lascoux and Schützenberger [14]. A similar procedure to the one in [18] is under investigation in type $B$ in [1]. The charge statistic is related to the energy function by the following theorem.

Theorem 2.15

[25, 29] Let $B^{\otimes \lambda }$ be a tensor product of KR crystals in type $A_{n-1}^{(1)}$ or type $C_{n}^{(1)}$. For all $b\in B^{\otimes \lambda }$, we have $D(b)=-\mathrm{charge}(b)$.

The charge gives a much easier method to compute the energy than the recursive one based on Theorem 2.13. See also Theorem 3.24 (2) for a much more general result.

3 The quantum alcove model

In this section we construct the quantum alcove model and study its main properties.

3.1 $\lambda$-Chains and admissible subsets

We say that two alcoves are adjacent if they are distinct and have a common wall. Given a pair of adjacent alcoves $A$ and $B$, we write $A \mathop {\longrightarrow }\limits ^{\beta } B$ if the common wall is of the form $H_{\beta ,k}$ and the root $\beta \in \Phi$ points in the direction from $A$ to $B$.

Definition 3.1

[23] An alcove path is a sequence of alcoves $(A_0, A_1, \ldots , A_m)$ such that $A_{j-1}$ and $A_j$ are adjacent, for $j=1,\ldots m.$ We say that an alcove path is reduced if it has minimal length among all alcove paths from $A_0$ to $A_m$.

Let $A_{\lambda }=A_{\circ }+\lambda$ be the translation of the fundamental alcove $A_{\circ }$ by the weight $\lambda$.

Definition 3.2

[23] The sequence of roots $(\beta _1, \beta _2, \ldots , \beta _m)$ is called a $\lambda$-chain if

$$\begin{aligned} A_0=A_{\circ } \mathop {\longrightarrow }\limits ^{-\beta _1} A_1 \mathop {\longrightarrow }\limits ^{-\beta _2}\cdots \mathop {\longrightarrow }\limits ^{-\beta _m} A_m=A_{-\lambda } \end{aligned}$$

is a reduced alcove path.

We now fix a dominant weight $\lambda$ and an alcove path $\varPi =(A_0, \ldots , A_m)$ from $A_0 = A_{\circ }$ to $A_m = A_{-\lambda }$. Note that $\varPi$ is determined by the corresponding $\lambda$-chain $\Gamma :=(\beta _1, \ldots , \beta _m)$, which consists of positive roots. A specific choice of a $\lambda$-chain, called a lex $\lambda$-chain and denoted $\Gamma _\mathrm{lex}$, is given in [24] [Proposition 4.2]; this choice depends on a total order on the simple roots. We let $r_i:=s_{\beta _i}$, and let $\widehat{r_i}$ be the affine reflection in the hyperplane containing the common face of $A_{i-1}$ and $A_i$, for $i=1, \ldots , m$; in other words, $\widehat{r}_i:= s_{\beta _i,-l_i}$, where $l_i:=|\left\{ j<i \, ; \, \beta _j = \beta _i \right\} |$. We define $\widetilde{l}_i:= \langle \lambda , \beta _i^{\vee } \rangle -l_i = |\left\{ j \ge i \, ; \, \beta _j = \beta _i \right\} |$.

Example 3.3

Consider the dominant weight $\lambda =3\varepsilon _1+2\varepsilon _2$ in the root system $A_2$ (cf. Sect. 4.1 and the notation therein). A $\lambda$-chain is $\Gamma =(\alpha _{23},\alpha _{13},\alpha _{23},\alpha _{13},\alpha _{12},\alpha _{13})$. The corresponding $l_i$ are $(0,0,1,1,0,2)$ and $\widetilde{l}_i$ are $\left\{ 2,3,1,2,1,1 \right\}$. The alcove path is shown in Fig. 1a; here $A_\circ$ is shaded, and $A_{-\lambda }$ is the alcove at the end of the path.

Fig. 1

Unfolded and folded $\lambda$-chain. a $\Gamma$ for $\lambda =3\varepsilon _1+2\varepsilon _2$. b $\Gamma (J)$ for $J=\left\{ 1,2,3,5 \right\}$

Let $J=\left\{ j_1 < j_2 < \cdots < j_s \right\}$ be a subset of $[m]$. The elements of $J$ are called folding positions. We fold $\varPi$ in the hyperplanes corresponding to these positions and obtain a folded path, see Example 3.6 and Fig. 1b. Like $\varPi$, the folded path can be recorded by a sequence of roots, namely $\Gamma (J)=\left( \gamma _1,\gamma _2, \ldots , \gamma _m \right)$, where

$$\begin{aligned} \gamma _k:=r_{j_1}r_{j_2}\ldots r_{j_p}(\beta _k)\,, \end{aligned}$$

(9)

with $j_p$ the largest folding position less than $k$. We define $\gamma _{\infty } := r_{j_1}r_{j_2}\ldots r_{j_s}(\rho )$. Upon folding, the hyperplane separating the alcoves $A_{k-1}$ and $A_k$ in $\varPi$ is mapped to

$$\begin{aligned} H_{|\gamma _k|,-l^{J}_{k}}=\widehat{r}_{j_1}\widehat{r}_{j_2}\ldots \widehat{r}_{j_p}(H_{\beta _k,-l_k})\,, \end{aligned}$$

(10)

for some $l^{J}_{k}$, which is defined by this relation.

Given $i \in J$, we say that $i$ is a positive folding position if $\gamma _i>0$, and a negative folding position if $\gamma _i<0$. We denote the positive folding positions by $J^{+}$, and the negative ones by $J^{-}$. We call $\mu =\mu (J):=-\widehat{r}_{j_1}\widehat{r}_{j_2}\ldots \widehat{r}_{j_s}(-\lambda )$ the weight of $J$. We define

$$\begin{aligned} \mathrm{height}(J):= \sum _{j \in J^{-}} \widetilde{l}_j. \end{aligned}$$

(11)

Definition 3.4

A subset $J=\left\{ j_1 < j_2 < \cdots < j_s \right\} \subseteq [m]$ (possibly empty) is an admissible subset if we have the following path in the quantum Bruhat graph on $W$:

$$\begin{aligned} 1 \mathop {\longrightarrow }\limits ^{\beta _{j_1}} r_{j_1} \mathop {\longrightarrow }\limits ^{\beta _{j_2}} r_{j_1}r_{j_2} \mathop {\longrightarrow }\limits ^{\beta _{j_3}} \cdots \mathop {\longrightarrow }\limits ^{\beta _{j_s}} r_{j_1}r_{j_2}\cdots r_{j_s}\,. \end{aligned}$$

(12)

We call $\Gamma (J)$ an admissible folding. We let $\mathcal {A}(\Gamma )$ be the collection of admissible subsets.

Remarks 3.5

If we restrict to admissible subsets for which the path (12) has no down steps, we recover the classical alcove model in [23, 24].

Example 3.6

We continue Example 3.3. Let $J=\left\{ 1,2,3,5 \right\}$, then $\Gamma (J)=(\alpha _{23},\alpha _{12}, \alpha _{31}, \alpha _{23},\alpha _{21},\alpha _{13})$. The folded path is shown in Fig. 1b. We have $J^{+}=\left\{ 1,2\right\} ,\,J^{-}=\left\{ 3,5 \right\} ,\,\mu (J)=-\varepsilon _3$, and $\mathrm{height}(J)=2$. In Sect. 4.1 we will describe an easy way to verify that $J$ is admissible.

3.2 Crystal operators

In this section we define the crystal operators on $\mathcal {A}( \Gamma )$. Given $J\subseteq [m]$ and $\alpha \in \Phi$, we will use the following notation:

$$\begin{aligned} I_\alpha = I_{\alpha }(J):= \left\{ i \in [m] \, | \, \gamma _i = \pm \alpha \right\} \,, \qquad \widehat{I}_\alpha = \widehat{I}_{\alpha }(J):= I_{\alpha } \cup \{\infty \}\,, \end{aligned}$$

and $l_{\alpha }^{\infty }:= \langle \mu (J), \mathrm{sgn}(\alpha )\alpha ^{\vee } \rangle$. The following graphical representation of the heights $l_i^J$ for $i\in {I}_\alpha$ and $l_{\alpha }^{\infty }$ is useful for defining the crystal operators. Let

$$\begin{aligned} \widehat{I}_{\alpha }= \left\{ i_1 < i_2 < \cdots < i_n <i_{n+1}=\infty \right\} \, \text { and } \varepsilon _i := {\left\{ \begin{array}{ll} 1 &{}\quad \text {if } i \not \in J\\ -1 &{} \quad \text{ if } i \in J \end{array}\right. }.\, \end{aligned}$$

If $\alpha > 0$, we define the continuous piecewise linear function $g_{\alpha }:[0,n+\frac{1}{2}] \rightarrow \mathbb {R}$ by

$$\begin{aligned} g_\alpha (0)= -\frac{1}{2}, \quad g^{\prime }_{\alpha }(x)= {\left\{ \begin{array}{ll} \mathrm{sgn}(\gamma _{i_k}) &{}\quad \text {if } x \in (k-1,k-\frac{1}{2}),\, k = 1, \ldots , n\\ \varepsilon _{i_k}\mathrm{sgn}(\gamma _{i_k}) &{}\quad \text {if } x \in (k-\frac{1}{2},k),\, k=1,\ldots ,n \\ \mathrm{sgn}( \langle \gamma _{\infty }, \alpha ^{\vee } \rangle ) &{} \quad \text { if } x \in (n,n+\frac{1}{2}). \end{array}\right. } \end{aligned}$$

(13)

If $\alpha <0$, we define $g_{\alpha }$ to be the graph obtained by reflecting $g_{-\alpha }$ in the $x$-axis. By [24][Propositions 5.3 and 5.5], for any $\alpha$ we have

$$\begin{aligned} \mathrm{sgn}(\alpha )l^{J}_{i_k}\!=\!g_\alpha \left( k-\frac{1}{2}\right) , k\!=\!1, \ldots , n, \, \text { and }\, \mathrm{sgn}(\alpha )l_{\alpha }^{\infty }\!:=\! \langle \mu (J), \alpha ^{\vee } \rangle \!=\! g_{\alpha }\left( n\!+\!\frac{1}{2}\right) \!. \end{aligned}$$

(14)

Example 3.7

We continue Example 3.6. The graphs of $g_{\alpha _2}$ and $g_{\theta }$ are given in Fig. 2.

Fig. 2

Examples of $g_{\alpha }$

Let $J$ be an admissible subset. Let $\delta _{i,j}$ be the Kronecker delta function. Fix $p$ in $\{0,\ldots ,r\}$, so $\alpha _p$ is a simple root if $p>0$, or $\theta$ if $p=0$. Let $M$ be the maximum of $g_{{\alpha }_p}$. Let $m$ be the minimum index $i$ in $\widehat{I}_{{\alpha }_p}$ for which we have $\mathrm{sgn}({\alpha }_p)l^{J}_{i}=M$. By Proposition 3.22, if $M\ge \delta _{p,0}$, then we have either $m\in J$ or $m=\infty$; furthermore, if $M>\delta _{p,0}$, then $m$ has a predecessor $k$ in $\widehat{I}_{{\alpha }_p}$, and we have $k\not \in J$. We define

$$\begin{aligned} f_p(J):= {\left\{ \begin{array}{ll} (J \backslash \left\{ m \right\} ) \cup \{ k \} &{}\quad \text { if }\,M>\delta _{p,0} \\ \mathbf {0}&{}\quad \text { otherwise }. \end{array}\right. } \end{aligned}$$

(15)

Now we define $e_p$. Again let $M:= \max g_{{\alpha }_p}$. Assuming that $M> \langle \mu (J), {\alpha }_p^{\vee } \rangle$, let $k$ be the maximum index $i$ in $I_{{\alpha }_p}$ for which we have $\mathrm{sgn}({\alpha }_p)l^{J}_{i}=M$, and let $m$ be the successor of $k$ in $\widehat{I}_{{\alpha }_p}$. Assuming also that $M\ge \delta _{p,0}$, by Proposition 3.23 we have $k\in J$, and either $m\not \in J$ or $m=\infty$. Define

$$\begin{aligned} e_p(J):={\left\{ \begin{array}{ll} (J \backslash \left\{ k \right\} ) \cup \{ m \} &{}\quad \text { if } M> \langle \mu (J), {\alpha }_p^{\vee } \rangle \text { and } M \ge \delta _{p,0} \\ \mathbf {0}&{} \quad \text { otherwise. } \end{array}\right. } \end{aligned}$$

(16)

In the above definitions, we use the convention that $J\backslash \left\{ \infty \right\} = J \cup \left\{ \infty \right\} = J$.

Example 3.8

We continue Example 3.7. We find $f_2(J)$ by noting that $\widehat{I}_{\alpha _2}=\left\{ 1,4,\infty \right\}$. From $g_{\alpha _2}$ in Fig. 2 we can see that the heights $l^{J}_{i}$ and $l_{\alpha _2}^\infty$ corresponding to these positions are $0,0,1$, so $k=4,\,m=\infty$, and $f_2(J)=J \cup \{ 4 \} = \left\{ 1,2,3,4,5 \right\}$. We can see from Fig. 2 that the maximum of $g_{\theta }=1$, hence $f_0(J)=\mathbf {0}$. To compute $e_0(J)$ observe that $\widehat{I}_{\theta }=\left\{ 3,6\right\}$ with $k = 3$ and $m = 6$. So $e_0(J) = ( J \backslash \{k\} ) \cup \{ m \} = \left\{ 1,2,5,6 \right\}$.

The following theorem is one of our main results, and will be proved in Sect. 3.3.

Theorem 3.9

• (1) If $J$ is an admissible subset and if $f_p(J) \ne \mathbf {0}$, then $f_p(J)$ is also an admissible subset. Similarly for $e_p(J)$. Moreover, $f_p(J)=J^{\prime }$ if and only if $e_p(J^{\prime })=J$.

• (2) We have $\mu (f_p(J)) = \mu (J) - \alpha _p$. Moreover, if $M\ge \delta _{p,0}$, then

  $$\begin{aligned} \varphi _p(J)=M-\delta _{p,0}\,,\quad \varepsilon _p(J)=M-\langle \mu (J),\alpha _p^\vee \rangle \,, \end{aligned}$$

  while otherwise $\varphi _p(J)=\varepsilon _p(J)=0$.

3.3 Proofs

In this section we collect necessary results for the proof of Theorem 3.9. The techniques are similar to those in [24]; we go in detail over the parts of the proofs where there are notable differences, and we refer to the mentioned paper for the remaining parts.

Lemma 3.10

Let $w\in W,\,\alpha$ a simple root or $\theta$, and $\beta$ a positive root. If we have $w \longrightarrow ws_{\beta }$, as well as $w^{-1}(\alpha )>0$ and $s_{\beta }w^{-1}(\alpha )<0$, then $w^{-1}(\alpha )=\beta$.

Proof

If $s_{\alpha }w=ws_{\beta }$, then $w^{-1}(\alpha )=\pm \beta$, and $w^{-1}(\alpha )>0$ implies $w^{-1}(\alpha )=\beta$. Suppose by way of contradiction that $s_{\alpha }w\ne ws_{\beta }$. First suppose $\alpha$ is a simple root. Since $w^{-1}(\alpha )>0$, then $w \lessdot s_{\alpha }w$. By assumption we have $w \longrightarrow ws_{\beta }$, hence by Lemma 2.2 we have $ws_{\beta } \lessdot s_{\alpha }ws_{\beta }$. But $s_{\beta }w^{-1}(\alpha )<0$ implies $s_{\alpha }ws_{\beta }\lessdot ws_{\beta }$, which is a contradiction.

Suppose $\alpha =\theta$. Since $w^{-1}(\theta )>0$, we deduce $w \lhd s_{\theta }w$, by Lemma 2.1, and then $ws_{\beta } \lhd s_{\theta }ws_{\beta }$, by Lemma 2.3. On another hand, since $s_{\beta }w^{-1}(\theta )<0$, Lemma 2.1 implies that $s_{\theta }ws_{\beta } \lhd ws_{\beta }$, which is a contradiction. $\square$

Lemma 3.11

Let $J = \left\{ j_1 < j_2 < \ldots < j_s \right\}$ be an admissible subset. Assume that $r_{j_a}\ldots r_{j_1}(\alpha )>0$ and $r_{j_{b}}\ldots r_{j_1}(\alpha )<0$, where $\alpha$ is a simple root or $\theta$, and $0\le a < b$ (if $a=0$, then the first condition is void). Then there exists $i$ with $a \le i < b$ such that $\gamma _{j_{i+1}}=\alpha$.

Proof

Find $i$ with $a \le i < b$ such that $r_{j_i}\ldots r_{j_1}(\alpha )>0$ and $r_{j_{i+1}}\ldots r_{j_1}(\alpha )<0$. By Lemma 3.10, we have $\beta _{j_{i+1}}= r_{j_i}\ldots r_{j_1}(\alpha )$. This means that $\gamma _{j_{i+1}}=r_{j_1}\ldots r_{j_i}(\beta _{j_{i+1}})=\alpha$. $\square$

Proposition 3.12

Let $J = \left\{ j_1 < j_2 < \cdots < j_s \right\}$ be an admissible subset. Assume that $\alpha$ is a simple root or $\theta$, with $I_{\alpha }\ne \emptyset$. Let $m \in I_{\alpha }$ be an element for which its predecessor $k$ (in $I_{\alpha }$) satisfies $(\gamma _k,\varepsilon _k)\in \{(\alpha ,1),(-\alpha ,-1)\}.$ Then we have $\gamma _m=\alpha$.

Proof

First suppose that $(\gamma _k,\varepsilon _k)=(\alpha ,1)$. Assume that $\gamma _m=-\alpha$. Let us define the index $b$ by the condition $j_b < m \le j_{b+1}$ (possibly $b=s$, in which case the second inequality is dropped). We define the index $a$ by the condition $j_{a} < k < j_{a+1}$ (possibly $a=0$, in which case the first inequality is dropped). We clearly have $r_{j_1}\ldots r_{j_b}(\beta _m)=-\alpha$, which implies $r_{j_b}\ldots r_{j_1}(\alpha )<0$. We also have $r_{j_1}\ldots r_{j_a}(\beta _k)=\alpha$, so $r_{j_a}\ldots r_{j_1}(\alpha )>0$ (hence $a<b$). Note that if $\alpha =\theta$, then $a>0$. We can now apply Lemma 3.11 to conclude that $\gamma _{j_i} =\alpha$ for some $i \in [a+1,b]$. Since $k<j_{a+1} \le j_{b} < m$, we contradicted the assumption that $\gamma _k$ is the predecessor of $\gamma _m$ in $I_{\alpha }$.

Now suppose that $(\gamma _k,\varepsilon _k)=(-\alpha ,-1)$. Assume that $\gamma _m = -\alpha$ and define $b$ as in the previous case. Again we have $r_{j_b}\ldots r_{j_1}(\alpha )<0$. Define $a$ by the condition $j_a = k < j_{a+1}$. Hence $r_{j_1}\ldots r_{j_{a-1}}(\beta _{j_a})=-\alpha$, so $r_{j_a}\ldots r_{j_1}(\alpha )>0$. This leads to a contradiction by a similar reasoning to the one above.

Proposition 3.13

Let $J$ be an admissible subset. Assume that $\alpha$ is a simple root for which $I_{\alpha }\ne \emptyset$. Let $m \in I_{\alpha }$ be the minimum of $I_{\alpha }$. Then we have $\gamma _m=\alpha$.

Proof

The proof of Proposition 3.12 carries through with $a=0$. $\square$

Proposition 3.14

Let $J = \left\{ j_1 < j_2 < \cdots < j_s \right\}$ be an admissible subset. Assume that $\alpha$ is a simple root or $\theta$. Suppose that $I_{\alpha }\ne \emptyset$, and $(\gamma _m,\varepsilon _m) \in \{ (\alpha ,1), (-\alpha ,-1) \}$ for $m=\max I_{\alpha }.$ Then we have $\langle \gamma _{\infty },\alpha ^{\vee } \rangle >0$.

Proof

Assume that the conclusion fails, which means that $r_{j_s}\ldots r_{j_1}(\alpha )<0$. First suppose that $(\gamma _m,\varepsilon _m)=(\alpha ,1)$. Define the index $a$ by the condition $j_a < m < j_{a+1}$. (If $a=0$ or $a=s$ one of the two inequalities is dropped). We have $r_{j_1}\ldots r_{j_{a}}(\beta _m)=\alpha$, so $r_{j_a}\ldots r_{j_1}(\alpha )>0$ (hence $a \ne s$). Note that if $\alpha =\theta$, then $a>0$. We now apply Lemma 3.11 to conclude that $\gamma _{j_i}=\alpha$ for $i \in [a+1,s]$. Since $m<j_{a+1}\le j_s$, this contradicts that $m=\max I_{\alpha }$.

Now suppose that $(\gamma _m,\varepsilon _m)=(-\alpha ,-1)$. In this case we define the index $a$ by $j_a = m <j_{a+1}$. We have $r_{j_1}\ldots r_{j_{a-1}}(\beta _{j_a})=-\alpha$, so $r_{j_a}\ldots r_{j_1}(\alpha )>0$. This leads to a contradiction by a similar reasoning to the one above. $\square$

Proposition 3.15

Let $J$ be an admissible subset. Assume that, for some simple root $\alpha$, we have $I_{\alpha }= \emptyset$. Then $\langle \gamma _{\infty },\alpha ^{\vee } \rangle >0$.

Proof

The proof of Proposition 3.14 carries through with $a=0$.

Let us now fix a simple root $\alpha$. We will rephrase some of the above results in a simple way in terms of $g_{\alpha }$, and we will deduce some consequences. Assume that $I_{\alpha }=\left\{ i_1 < i_2 < \cdots < i_n \right\}$, so that $g_{\alpha }$ is defined on $[0,n+\frac{1}{2}]$, and let $M$ be the maximum of $g_{\alpha }$. Note first that the function $g_{\alpha }$ is determined by the sequence $(\sigma _1, \ldots , \sigma _{n+1})$, where $\sigma _j = (\sigma _{j,1},\sigma _{j,2}):= (\mathrm{sgn}(\gamma _{i_j}), \varepsilon _{i_j}\mathrm{sgn}(\gamma _{i_j}))$ for $1\le j\le n$, and $\sigma _{n+1}= \sigma _{n+1,1}:=\mathrm{sgn}(\langle \gamma _{\infty }, \alpha ^{\vee } \rangle )$. From Propositions 3.12, 3.13, 3.14 and 3.15 we have the following restrictions.

• (C1) $\sigma _{1,1}=1$.

• (C2) $\sigma _{j,2}=1 \Rightarrow \sigma _{j+1,1}=1$.

Proposition 3.16

If $g_{\alpha }(x)=M$, then $M \in \mathbb {Z}_{\ge 0},\,x=m+\frac{1}{2}$ for $0 \le m \le n$, and $\sigma _{m+1} \in \left\{ (1,-1),1 \right\}$.

Proof

By (C1), we have $M\ge 0$, therefore $g_{\alpha }(0) = - \frac{1}{2} \ne M$. For $m \in \left\{ 1, \ldots , n \right\}$, if $g_{\alpha }(m)=M$ then $\sigma _{m,2}=1$, and (C2) leads to a contradiction. The last statement is obvious. $\square$

We use Proposition 3.16 implicitly in the proof of Proposition 3.17 and Proposition 3.18.

Proposition 3.17

Assume that $M>0$, and let $m$ be such that $m+ \frac{1}{2} = \min g_{\alpha }^{-1}(M)$. We have $m>0,\,\sigma _m = (1,1)$, and $g_{\alpha }(m-\frac{1}{2})= M-1$. Moreover, we have $g_{\alpha }(x) \le M-1$ for $0 \le x \le m- \frac{1}{2}$.

Proof

By (C1) we have $g_{\alpha }(\frac{1}{2})=0$, so $m>0$. If $\sigma _m \in \{(-1,-1), (1,-1)\}$, then we have $g_{\alpha }(m-\frac{1}{2})=M$, which contradicts the definition of $m$. If $\sigma _m=(-1,1)$, then $g_{\alpha }(m-1)=M-\frac{1}{2}>-\frac{1}{2}$. By (C1) we have $m \ge 2$, and by (C2) we have $\sigma _{m-1,2}=-1$. This implies that $g_{\alpha }(m-\frac{3}{2})=M$, contradicting the definition of $m$. Hence $\sigma _m=(1,1)$.

Suppose by way of contradiction that the last statement in the corollary fails. Then there exists a $k$ with $1 \le k \le m-1$ such that $g_{\alpha }(k-1)=M-\frac{1}{2}>-\frac{1}{2}$ and $\sigma _{k,1}=-1$. Condition (C1) implies that $k \ge 2$ and Condition (C2) implies $\sigma _{k-1,2}=-1$. This implies $g_{\alpha }(k-\frac{3}{2})=M$, contradicting the definition of $m$. $\square$

Proposition 3.18

Assume that $M> g_{\alpha }(n+\frac{1}{2})$, and let $k$ be such that $k-\frac{1}{2}= \max g_{\alpha }^{-1}(M)$. We have $k \le n, \sigma _{k+1} \in \{ (-1, -1), -1\}$, and $g_{\alpha }(k+\frac{1}{2}) =M-1$. Moreover, we have $g_{\alpha }(x) \le M-1$ for $k + \frac{1}{2} \le x \le n+ \frac{1}{2}$.

Proof

Since $M> g_{\alpha }(n+\frac{1}{2})$, it follows that $k\le n$. If $\sigma _{k+1} \in \{ (1,1),(1,-1),1 \}$ then $g_{\alpha }(k+\frac{1}{2})=M$, contradicting the choice of $k$. If $\sigma _{k+1}=(-1,1)$, then by (C2) we have $\sigma _{k+2,1}=1$, and $g_{\alpha }(k+\frac{3}{2})=M$, contradicting the choice of $k$. Hence $\sigma _{k+1} \in \{(-1,-1),-1 \}$.

Suppose by way of contradiction the last statement in the corollary fails. Then there exists an $m$ with $k+2 \le m \le n$ such that $g_{\alpha }(m)=M-\frac{1}{2}$ and $\sigma _{m,2}=1$. Condition (C2) implies that $\sigma _{m+1,1}=1$, so $g_{\alpha }(m+\frac{1}{2})=M$, contradicting the choice of $k$.$\square$

We now consider $g_{\theta }$. Since $\theta <0$, the definition of the piecewise linear function $g_\theta$ requires us to define its linear steps by $\sigma _j = (\sigma _{j,1},\sigma _{j,2}):= (-\mathrm{sgn}(\gamma _{i_j}),-\varepsilon _{i_j}\mathrm{sgn}(\gamma _{i_j}))$ for $1 \le j \le n$, and $\sigma _{n+1}=\sigma _{n+1,1}:=\mathrm{sgn}(\langle \gamma _{\infty }, \theta ^{\vee } \rangle )$. From Propositions 3.12 and 3.14 we conclude that condition (C2) holds for $g_{\theta }$. We can replace condition (C1) by restricting to admissible subsets $J$ where $M$ is large enough, as we will now explain. In the proof of Proposition 3.16, condition (C1) is needed to conclude that $g_{\alpha }(0) \ne M$. It is possible that $g_{\theta }(0)=M$, but if we restrict to $g_{\theta }$ where $M\ge 1$ we can conclude that $g_{\theta }(0)=\frac{1}{2} \ne M$, and the rest of the proof follows through. In the proof of Proposition 3.17, condition (C1) first implies $m\ge 1$; we can conclude the same thing if we assume $M\ge 2$, since $g_\theta (\frac{1}{2})\le 1$. Then we need to derive $m\ge 2$ from $g_\theta (m-1)=M-\frac{1}{2}$; again, if $M\ge 2$, then $M-\frac{1}{2}>g_\theta (0)=\frac{1}{2}$, so $m-1>0$. Note that Proposition 3.18 depends on Proposition 3.16 so we need to assume $M \ge 1$ here too. We have therefore proved the following propositions.

Proposition 3.19

Suppose $M\ge 1$. If $g_{\theta }(x)=M$, then $M \in \mathbb {Z}_{\ge 1},\,x=m+\frac{1}{2}$ for $0 \le m \le n$, and $\sigma _{m+1} \in \left\{ (1,-1),1 \right\}$.

Proposition 3.20

Assume that $M\ge 2$, and let $m$ be such that $m+ \frac{1}{2} = \min g_{\theta }^{-1}(M)$. We have $m>0,\,\sigma _m = (1,1)$, and $g_{\theta }(m-\frac{1}{2})= M-1$. Moreover, we have $g_{\theta }(x) \le M-1$ for $0 \le x \le m- \frac{1}{2}$.

Proposition 3.21

Assume $M\ge 1$, and also that $M> g_{\theta }(n+\frac{1}{2})$. Let $k$ be such that $k-\frac{1}{2}= \max g_{\theta }^{-1}(M)$. We have $k \le n, \sigma _{k+1} \in \{ (-1, -1), -1\}$, and $g_{\theta }(k+\frac{1}{2}) =M-1$. Moreover, we have $g_{\theta }(x) \le M-1$ for $k + \frac{1}{2} \le x \le n+ \frac{1}{2}$.

Recall from from Sect. 3.2 the definitions of the finite sequences $I_{\alpha }(J)$ and $\widehat{I}_{\alpha }(J)$, where $\alpha$ is a root, of $g_{\alpha }$, as well as the related notation.

Fix $p$, so $\alpha _p$ is a simple root if $p>0$, or $\theta$ if $p=0$. Let $M$ be the maximum of $g_{\alpha _p}$, and suppose that $M \ge \delta _{p,0}$. Note this is always true for $p \ne 0$ by Proposition 3.16. Let $m$ be the minimum index $i$ in $\widehat{I}_{\alpha _p}(J)$ for which we have $\mathrm{sgn}(\alpha _p)l^{J}_{i}=M$. The following proposition is an immediate consequence of Propositions 3.16, 3.17, 3.19, 3.20.

Proposition 3.22

Given the above setup, the following hold.

• (1) If $m \ne \infty$, then $\gamma _m=\alpha _p$ and $m \in J$.

• (2) If $M>\delta _{p,0}$, then $m$ has a predecessor $k$ in $\widehat{I}_{\alpha _p}(J)$ such that

  $$\begin{aligned} \gamma _k=\alpha _p,\, k \not \in J, \, \text{ and } \, \mathrm{sgn}(\alpha _p)l^{J}_{k} = M-1. \end{aligned}$$

Now assume that $M> \langle \mu (J), {\alpha }_p^{\vee } \rangle$. Let $k$ be the maximum index $i$ in $I_{\alpha _p}(J)$ for which we have $\mathrm{sgn}(\alpha _p)l^{J}_{i}=M$, and let $m$ be the successor of $k$ in $\widehat{I}_{\alpha _p}(J)$. The following analog of Proposition 3.22 is proved in a similar way, based on Propositions 3.16, 3.18, 3.19, 3.21.

Proposition 3.23

Given the above setup, and assuming also that $M\ge \delta _{p,0}$, the following hold.

• (1) We have $\gamma _k=\alpha _p$ and $k \in J$.

• (2) If $m\ne \infty$, then

  $$\begin{aligned} \gamma _m=-\alpha _p,\, m \not \in J,\, \text{ and } \mathrm{sgn}(\alpha _p)l^{J}_{m} = M-1. \end{aligned}$$

Proof of Theorem 3.9

Suppose $p \ne 0$. We consider $f_p$ first. The cases corresponding to $m\ne \infty$ and $m=\infty$ can be proved in similar ways, so we only consider the first case. Let $J=\left\{ j_1 < j_2< \ldots < j_s \right\}$, and let $w_i := r_{j_1}r_{j_2} \ldots r_{j_i}$. Based on Proposition 3.22, let $a<b$ be such that

$$\begin{aligned} j_a < k < j_{a+1} < \cdots < j_b = m < j_{b+1}\,; \end{aligned}$$

if $a=0$ or $b+1>s$, then the corresponding indices $j_a$, respectively $j_{b+1}$, are missing. To show that $(J \backslash \left\{ m \right\} ) \cup \left\{ k \right\}$ is an admissible subset, it is enough to prove that we have the path in the quantum Bruhat graph

$$\begin{aligned} w_a \longrightarrow w_ar_{k} \longrightarrow w_ar_k r_{j_{a+1}} \longrightarrow \cdots \longrightarrow w_ar_k r_{j_{a+1}}\ldots r_{j_{b-1}} = w_b\,. \end{aligned}$$

(17)

By our choice of $k$, we have

$$\begin{aligned} w_a(\beta _k)=\alpha _p \,\iff \, w_a^{-1}(\alpha _p)=\beta _k >0 \,\iff \, w_a \lessdot s_pw_a = w_ar_k\,. \end{aligned}$$

(18)

So we can rewrite (17) as

$$\begin{aligned} w_a \longrightarrow s_pw_a \longrightarrow s_pw_{a+1} \longrightarrow \cdots \longrightarrow s_pw_{b-1} = w_b\,. \end{aligned}$$

(19)

We will now prove that (19) is a path in the quantum Bruhat graph. Observe that, for $a<i\le b$, we have

$$\begin{aligned} s_p w_{i-1}=w_i=w_{i-1}r_{j_i} \,\iff \,w_{i-1}(\beta _{j_i})=\pm \alpha _p \,\iff \; j_i \in I_{\alpha }\,. \end{aligned}$$

Our choice of $k$ and $b$ implies that we have

$$\begin{aligned} s_p w_{i-1} \ne w_i \,\text { for } a<i<b\,,\quad \text { and } s_p w_{b-1}=w_b\,. \end{aligned}$$

(20)

Since $J$ is admissible, we have

$$\begin{aligned} w_{i-1} \longrightarrow w_i\,. \end{aligned}$$

(21)

Starting from (18), and then using (20)–(21), we can apply Lemma 2.2 repeatedly to conclude that

$$\begin{aligned} s_pw_{i-1} \longrightarrow s_pw_{i} \;\text { and }\; w_{i} \lessdot s_pw_{i}\,,\;\; \text { for } a<i<b\,. \end{aligned}$$

(22)

The proof for $e_p(J)$ is similar. The main difference is that we need the “if” part of Lemma 2.2, whereas above we used the “only if” part.

The above proof follows through for $p=0$, based on Lemma 2.1, which is used to derive the analog of (18), and Lemma 2.3, which replaces Lemma 2.2.

We can prove that $f_p(J)=J^{\prime }$ if and only if $e_p(J^{\prime })=J$ based on [24] [Proposition 7.4 (1)]; this still holds in the above setup (for any $p$, including $p=0$), based on Propositions 3.16–3.21. The same result can be invoked to derive the formulas for $\varphi _p(J)$ and $\varepsilon _p(J)$.

In order to show that $f_p$ changes weights by $-\alpha _p$, the proof of [24][Proposition 7.1 (3)] can be applied in our context. In essence, we note that $\mu (f_p(J))$ is $-\widehat{ t}_k \widehat{ t}_m (-\mu )$ if $m \ne \infty$, and $-\widehat{ t}_k (-\mu )$ otherwise, where $\widehat{ t }_j:=s_{|\gamma _j|, -l^{J}_{j}}=s_{\gamma _j,-\mathrm{sgn}(\gamma _j)l_j^J}$. By Proposition 3.22 and (14), we have $\widehat{t}_k=s_{\alpha _p,-M}$ and $\widehat{t}_m=s_{\alpha _p,-(M-1)}$. The rest of the calculation is the same as in the proof mentioned above. $\square$

3.4 Main application

We summarize the main results in [22], cf. also [20, 21]. The setup is that of untwisted affine root systems.

Theorem 3.24

[22] Consider a composition ${\mathbf {p}}=(p_1,\ldots ,p_k)$ and the corresponding crystal $B:=\bigotimes _{i=1}^{k} B^{p_i,1}$. Let $\lambda :=\omega _{p_1}+\ldots +\omega _{p_k}$, and let $\Gamma _\mathrm{lex}$ be a corresponding lex $\lambda$-chain (see above).

• (1) The (combinatorial) crystal $\mathcal {A}(\Gamma _\mathrm{lex})$ is isomorphic to the subgraph of $B$ consisting of the dual Demazure arrows, via a specific bijection which preserves the weights of the vertices.

• (2) If the vertex $b$ of $B$ corresponds to $J$ under the isomorphism in part (1), then the energy is given by $D_B(b)-C=-\mathrm{height}(J)$, where $C$ is a global constant.

Remarks 3.25

• (1) The entire crystal $B$ is realized in [22] in terms of the so-called quantum LS path model. If we identify the two, the bijection in Theorem 3.24 (1) is the “forgetful map” from the quantum alcove model to the quantum LS path model, so it is a very natural map. Therefore, we think of the former model as a mirror image of the latter, via this bijection. However, if we use this identification to construct the non-dual Demazure arrows in the quantum alcove model, we quickly realize that, in general, the constructions are considerably more involved than (15)–(16), cf. Remark 4.8 (1) and Example 4.9.

• (2) Although the quantum alcove model so far misses the non-dual Demazure arrows, it has the advantage of being a discrete model. Therefore, combinatorial methods are applicable, for instance in proving the independence of the model from the choice of an initial alcove path (or $\lambda$-chain of roots), see below, including the application in Remark 3.27 (2). This should be compared with the continuous arguments used for the similar purpose in the Littelmann path model [27].

• (3) Theorem 3.24, combined with the Ram–Yip formula for Macdonald polynomials [30], implies that the graded character of a tensor product of column shape KR modules (the grading being by the energy function) concides with the corresponding Macdonald polynomial specialized at $t=0$ [22].

Based on Theorem 3.24 (1), as well as on the realization of the same subgraph of $B$ in types $A$ and $C$ in terms of a different $\lambda$-chain (see Theorems 4.7 and 4.22), we make the following conjecture.

Conjecture 3.26

Theorem 3.24 holds for any choice of a $\lambda$-chain (instead of a lex $\lambda$-chain).

We plan to prove this conjecture in [19] by using Theorem 3.24 as the starting point. Then, given two $\lambda$-chains $\Gamma$ and $\Gamma ^{\prime }$, we would construct a bijection between $\mathcal {A}(\Gamma )$ and $\mathcal {A}(\Gamma ^{\prime })$ preserving the dual Demazure arrows, the weights of the vertices, and the height statistic; this would mean that the quantum alcove model does not depend on the choice of a $\lambda$-chain. This construction will be based on generalizing to the quantum alcove model the so-called Yang–Baxter moves in [15]. As a result, we would obtain a collection of a priori different affine crystal isomorphisms between $B$ and $\mathcal {A}(\Gamma )$.

Remarks 3.27

• (1) We believe that the bijections mentioned above would be identical. In fact, this would clearly be the case if all the tensor factors of $B$ are perfect crystals, see Sect. 2.1. Indeed, then the subgraph of $B$ consisting of the dual Demazure arrows is connected, so there is no more than one isomorphism between it and $\mathcal {A}(\Gamma )$.

• (2) In the case when all the tensor factors of $B$ are perfect crystals, a corollary of the work in [19] would be the following application of the quantum alcove model, cf. Remark 3.27 (1). By making specific choices for the $\lambda$-chains $\Gamma$ and $\Gamma ^{\prime }$, the bijection between $\mathcal {A}(\Gamma )$ and $\mathcal {A}(\Gamma ^{\prime })$ mentioned above would give a uniform realization of the combinatorial $R$-matrix (see Remark 2.10 (2)); the details will appear in the forthcoming paper [19]. In fact, we believe that the previous statement would hold in full generality, rather than just the perfect case.

4 The quantum alcove model in types $A$ and $C$

In this section we specialize the quantum alcove model to types $A$ and $C$, and prove that the bijections constructed in [18], from the objects of the specialized quantum alcove model to the tensor products of the corresponding KN columns (see Sect. 2.1), are affine crystal isomorphisms.

4.1 Type $A$

We start with the basic facts about the root system of type $A_{n-1}$. We can identify the space $\mathfrak {h}_\mathbb {R}^*$ with the quotient $V:=\mathbb {R}^n/\mathbb {R}(1,\ldots ,1)$, where $\mathbb {R}(1,\ldots ,1)$ denotes the subspace in $\mathbb {R}^n$ spanned by the vector $(1,\ldots ,1)$. Let $\varepsilon _1,\ldots ,\varepsilon _n\in V$ be the images of the coordinate vectors in $\mathbb {R}^n$. The root system is $\Phi =\{\alpha _{ij}:=\varepsilon _i-\varepsilon _j \,:\, i\ne j,\ 1\le i,j\le n\}$. The simple roots are $\alpha _i=\alpha _{i,i+1}$, for $i=1,\ldots ,n-1$. The highest root $\widetilde{\alpha }=\alpha _{1n}$. We let $\alpha _0=\theta =\alpha _{n1}$. The weight lattice is $\Lambda =\mathbb {Z}^n/\mathbb {Z}(1,\ldots ,1)$. The fundamental weights are $\omega _i = \varepsilon _1+\ldots +\varepsilon _i$, for $i=1,\ldots ,n-1$. A dominant weight $\lambda =\lambda _1\varepsilon _1+\ldots +\lambda _{n-1}\varepsilon _{n-1}$ is identified with the partition $(\lambda _{1}\ge \lambda _{2}\ge \ldots \ge \lambda _{n-1}\ge \lambda _n=0)$ having at most $n-1$ parts. Note that $\rho =(n-1,n-2,\ldots ,0)$. Considering the Young diagram of the dominant weight $\lambda$ as a concatenation of columns, whose heights are $\lambda _1^{\prime },\lambda _2^{\prime },\ldots$, corresponds to expressing $\lambda$ as $\omega _{\lambda _1^{\prime }}+\omega _{\lambda _2^{\prime }}+\ldots$ (as usual, $\lambda ^{\prime }$ is the conjugate partition to $\lambda$).

The Weyl group $W$ is the symmetric group $S_n$, which acts on $V$ by permuting the coordinate vectors $\varepsilon _1,\ldots ,\varepsilon _n$. Permutations $w\in S_n$ are written in one-line notation $w=w(1)\ldots w(n)$. For simplicity, we use the same notation $(i,j)$ with $1\le i<j\le n$ for the root $\alpha _{ij}$ and the reflection $s_{\alpha _{ij}}$, which is the transposition $t_{ij}$ of $i$ and $j$. We recall a criterion for the edges of the type $A$ quantum Bruhat graph. We need the circular order $\prec _i$ on $[n]$ starting at $i$, namely $i\prec _i i+1\prec _i\ldots \prec _i n\prec _i 1\prec _i\ldots \prec _i i-1$. It is convenient to think of this order in terms of the numbers $1,\ldots ,n$ arranged on a circle clockwise. We make the convention that, whenever we write $a\prec b\prec c\prec \ldots$, we refer to the circular order $\prec =\prec _a$.

Proposition 4.1

[18] For $1\le i<j\le n$, we have an edge $w \mathop {\longrightarrow }\limits ^{(i,j)} w(i,j)$ if and only if there is no $k$ such that $i<k<j$ and $w(i) \prec w(k) \prec w(j)$.

Example 4.2

The quantum Bruhat graph of type $A_2$, i.e., on the symmetric group $S_3$, is indicated in Fig. 3.

Fig. 3

The quantum Bruhat graph on $S_3$

We now consider the specialization of the quantum alcove model to type $A$. For any $k=1, \ldots , n-1$, we have the following $\omega _k$-chain, from $A_{\circ }$ to $A_{-\omega _k}$, denoted by $\Gamma (k)$ [23]:

$$\begin{aligned} \begin{array}{llll} ( (k,k+1),&{} (k,k+2)&{}, \ldots ,&{} (k,n), \\ (k-1,k+1),&{} (k-1,k+2)&{}, \ldots ,&{} (k-1,n), \\ \vdots &{} \vdots &{} &{} \vdots \\ (1,k+1),&{} (1, k+2)&{}, \ldots ,&{} (1,n)) \,. \end{array} \end{aligned}$$

(23)

Example 4.3

We specialize (23) to $n=4$ and $k=1,2,3$. It is best to visualize $\Gamma (k)$ based on a column of height $n$ broken into two pieces, with the top part of height $k$ and the bottom one of height $n-k$; then $\Gamma (k)$ is obtained by pairing row numbers in the top and bottom parts, in the prescribed order.

Fix a dominant weight/partition $\lambda$ for the remainder of this section. We construct a $\lambda$-chain $\Gamma =\left( \beta _1, \beta _2, \ldots , \beta _m \right)$ as the concatenation $\Gamma := \Gamma ^{1}\ldots \Gamma ^{\lambda _1}$, where $\Gamma ^{j}=\Gamma (\lambda ^{\prime }_j)$. Let $J=\left\{ j_1 < \cdots < j_s \right\}$ be a set of folding positions in $\Gamma$, not necessarily admissible, and let $T$ be the corresponding list of roots of $\Gamma$, also viewed as transpositions. The factorization of $\Gamma$ induces a factorization of $T$ as $T=T^{1}T^2 \ldots T^{\lambda _1}$, and of $\Delta =\Gamma (J)$ as $\Delta =\Delta ^1 \ldots \Delta ^{\lambda _1}$. Recalling that the roots in $\Delta$ were denoted $\gamma _k$, we use the notation $\gamma _k\in \Delta ^q$ to indicate that the $k$th root in $\Delta$ falls in the segment $\Delta ^q$ (rather than the fact that $\Delta ^q$ contains a root equal to $\gamma _k$). We denote by $T^{1} \ldots T^{j}$ the permutation obtained by composing the transpositions in $T^{1}, \ldots , T^{j}$ left to right. For $w\in W$, let $w_i = w(i)$. For $w$ written in one-line notation as $w=w_1w_2\ldots w_n$, let $w[i,j]=w_i\ldots w_j$.

We now recall from [18] the construction of the correspondence between the type $A$ quantum alcove model and model based on diagram fillings.

Definition 4.4

Let $\pi _{j}=\pi _{j}(T):=T^1 \ldots T^j$. We define the filling map, which associates with each $J\subseteq [m]$ a filling of the Young diagram $\lambda$, by

$$\begin{aligned} \mathrm{fill}(J)=\mathrm{fill}(T):=C_{1}\ldots C_{\lambda _1}\,, \; \text{ where } C_{i}:=\pi _i[1,\lambda ^{\prime }_i]\,, \end{aligned}$$

(24)

see the notation above. We define the sorted filling map $\mathrm{sfill}(J)$ by sorting ascendingly the columns of $\mathrm{fill}(J)$.

In other words, the $i$th column $C_i$ of $\mathrm{fill}(J)$ consists of the first $\lambda _i^{\prime }$ entries of the permutation $\pi _i$, written in one-line notation; see Example 4.5.

Example 4.5

Let $n=3$ and $\lambda =(4,3,0)$, which is identified with $4\varepsilon _1 + 3\varepsilon _2 = 3\omega _2 +\omega _1$, and corresponds to the Young diagram

. We have

$$\begin{aligned} \Gamma&= \Gamma ^1 \Gamma ^2 \Gamma ^3\Gamma ^4 = \Gamma (2)\Gamma (2)\Gamma (2)\Gamma (1)\\&=(\underline{(2,3)},\underline{(1,3)} \;|\; \underline{(2,3)}, (1,3) \;|\; \underline{(2,3)},(1,3)\;|\;\underline{(1,2)},(1,3) ), \end{aligned}$$

where we underlined the roots in positions $J=\{1,2,3,5,7 \}$. Then

$$\begin{aligned} T&= ((2,3),(1,3)\;|\;(2,3)\;|\;(2,3)\;|\;(1,2) )\,,\, \text{ and } \nonumber \\ \Gamma (J)&= \Delta \nonumber \\ {}&= \Delta ^1\Delta ^2\Delta ^3\Delta ^4 = (\underline{(2,3)}, \underline{(1,2)} \;|\; \underline{(3,1)}, (2,3) \;|\; \underline{(1,3)}, (2,1)\;|\; \underline{(2,3)},(3,1) ),\nonumber \\ \end{aligned}$$

(25)

where we again underlined the folding positions, and indicated the factorizations of $T$ and $\Delta$ by vertical lines. It is easy to check that $J$ is admissible; indeed, the sequence of permutations (12) corresponding to $J$ is a path in the quantum Bruaht graph, cf. Proposition 4.1 and Example 4.2:

(26)

Here each permutation in (12) is written vertically in one-line notation, with the entries in bold to be transposed; moreover, if the transposition to be applied lies in $\Gamma ^i$, then the corresponding permutation is represented as a broken column with the top part of height $\lambda _i^{\prime }$; see the structure of $\Gamma ^i$ in (23) and Example 4.3. By considering the top part of the last column in each segment and by concatenating these columns left to right, we obtain $\mathrm{fill}(J)$, i.e.,

.

Theorem 4.6

[18] [Theorem 4.1] The map $\mathrm{sfill}$ is a bijection between $\mathcal {A}(\Gamma )$ and $B^{\otimes \lambda }$, see (8).

We now state the main result of this section.

Theorem 4.7

The map $\mathrm{sfill}$ is an affine crystal isomorphism between $\mathcal {A}(\Gamma )$ and the subgraph of $B^{\otimes \lambda }$ consisting of the dual Demazure arrows. In other words, given $\mathrm{sfill}(J)=b$, there is a dual Demazure arrow $b\rightarrow f_p(b)$ if and only if $f_p(J)\ne \mathbf {0}$, and we have $f_i(b)=\mathrm{sfill}(f_i(J))$.

Remarks 4.8

• (1) The affine crystal isomorphism in Theorem 4.7 is unique, cf. Remark 3.27 (1). Therefore, this isomorphism gives the unique way to realize the non-dual Demazure arrows in $\mathcal {A}(\Gamma )$, cf. Remark 3.25 (1) and Example 4.9 below.

• (2) In [17] it was proved that the map $\mathrm{sfill}$ preserves weights, cf. Lemma 4.10. Furthermore, in [18] it was shown this map translates the height statistic to the charge statistic, which is known to express the energy function in the model based on diagram fillings, cf. Theorem 2.15. This should be compared with Theorem 3.24 (2), where the constant $C$ is $0$ in this case.

Example 4.9

In type $A_2$, consider $\lambda =(3,2,0)$, the $\lambda$-chain in Example 3.3, and the admissible subset $J=\{1,2,3,5\}$, cf. Examples 3.6, 3.7, and 3.8. We have

, and

, cf. Example 2.5. However, $b \rightarrow f_0(b)$ is not a dual Demazure arrow, and indeed $f_0(J)=\mathbf {0}$, cf. Example 3.8. In order to realize this arrow in the quantum alcove model, we would have to define $f_0(J)=\emptyset$. This shows that, in general, the changes in an admissible subset corresponding to non-dual Demazure arrows are hard to control. Nevertheless, such arrows are sometimes still realized by our construction (15), assuming that we drop the corresponding condition $M>1$. For an example, still in type $A_2$, consider $\lambda =(3,0,0)$, the $\lambda$-chain $(\alpha _{12},\alpha _{13},\alpha _{12},\alpha _{13},\alpha _{12},\alpha _{13})$, and $J=\{3,4\}$. We have

, and

is not a dual Demazure arrow. Now note that the corresponding arrow in the quantum alcove model is $J\mapsto J\cup \{5\}$, which is given by the mentioned relaxed version of (15).

The main idea of the proof of Theorem 4.7 is the following. The signature of a filling, used to define the crystal operator $f_p$, can be interpreted as a graph similar to the graph of $g_{\alpha _p}$, which is used to define the crystal operator on the corresponding admissible subsequence. The link between the two graphs is given by Lemma 4.12 below, called the height counting lemma, which we now explain.

Let $N_c(\sigma )$ denote the number of entries $c$ in a filling $\sigma$. Let $\mathrm{ct}(\sigma )=(N_1(\sigma ), \ldots , N_n(\sigma ))$ be the content of $\sigma$, which is identified with a type $A_{n-1}$ weight. Let $\sigma [q]$ be the filling consisting of the columns $1,2, \ldots , q$ of $\sigma$. Given a $\lambda$-chain and a corresponding sequence $J$ (not necessarily admissible), recall the related notation, including the heights $l_k^J$ in (10), the sequence of roots $\Delta$, and its factorization illustrated in (25).

Lemma 4.10

[17] [Proposition 3.6] Let $J \subseteq [m]$, and $\sigma =\mathrm{sfill}(J)$. Then we have $\mu (J)=\mathrm{ct}(\sigma )$.

Corollary 4.11

Let $J \subseteq [m],\,\sigma =\mathrm{fill}(J)$, and $\alpha \in \Phi$. Then $\mathrm{sgn}(\alpha )l_{\alpha }^{\infty } = \langle \mathrm{ct}(\sigma ), \alpha ^{\vee } \rangle$.

The height counting lemma can be viewed as an extension of Corollary 4.11.

Lemma 4.12

[17] [Proposition 4.1] Let $J \subseteq [m]$, and $\sigma =\mathrm{fill}(J)$. For a fixed $k$, let $\gamma _k=(c,d)$ be a root in $\Delta ^{q+1}$. We have

$$\begin{aligned} \mathrm{sgn}(\gamma _k)\,l^{J}_{k} = \langle \mathrm{ct}(\sigma [q]),\gamma _k^{\vee }\rangle = N_c(\sigma [q]) - N_d(\sigma [q]). \end{aligned}$$

We now introduce notation to be used for the remainder of this section. Let $p \in \{1, \ldots , n-1 \}$. Let $J$ be an admissible subset and let $\sigma =\mathrm{sfill}(J)=C_1\ldots C_{\lambda _1}$. For $i=1,\ldots ,\lambda _1$, let $a_i:= \langle \mathrm{ct}(C_i), \alpha _p^{\vee } \rangle$, and note that $a_i\in \left\{ 1,-1,0\right\}$; here $a_i =1$ (resp. $a_i=-1$) corresponds to $C_i$ containing $p$ but not $p+1$ (resp. $p+1$ but not $p$), while $a_i = 0$ corresponds to $C_i$ containing both $p$ and $p+1$, or neither of them.

The sequence $a_i$ corresponds to the $p$-signature from Sect. 2.1, as we now explain. For $j=0,\ldots ,\lambda _1$, let $h_j:= \langle \mathrm{ct}(\sigma [j]), \alpha _p^{\vee } \rangle =\sum _{i=0}^j a_i$, where we set $a_0=h_0:=0$. It is useful to think of this sequence as a piecewise linear function, by analogy with the function $g_{\alpha _p}$ used to define the crystal operator $f_p$ in the quantum alcove model. Let $M^{\prime }$ be the maximum of $h_j$, and let $m^{\prime }$ be minimal with the property $h_{m^{\prime }}=M^{\prime }$. We clearly have $M^{\prime } \ge 0$. If $M^{\prime }>0$ then $a_{m^{\prime }}=1$, and $m^{\prime }$ is the position of the rightmost $+$ in the $p$-signature of $\sigma$. It follows that $f_p$ changes the $p$ in column $m^{\prime }$ of $\sigma$ to $p+1$. The previous observations hold if we replace $\alpha _p,\,f_p$, and the $p$-signature with $\alpha _0,\,f_0$, and the $0$-signature, while at the same time we replace the entries $p$ and $p+1$ in a filling with $n$ and $1$, respectively. Therefore, from now on we assume that the index $p$ is in $\{ 0,1, \ldots , n-1\}$.

Example 4.13

We continue Example 4.5. Let

, then

. Let $p=2$ and refer to Fig. 4. From the graph $g_{\alpha _2}$ for $J$, we can see that $M=1$. We note that $m=7$, with $\gamma _7 \in \Delta ^4$, and $k=4$ with $\gamma _k \in \Delta ^{2}$. So $f_2(J)=(J \backslash \{ 7 \}) \cup \{ 4 \} = \{1,2,3,4,5 \}$, and

$$\begin{aligned} \Gamma (f_2(J))= ( \underline{(2,3)},\underline{(1,2)} \;|\; \underline{(3,1)}, \underline{(2,3)} \;|\; \underline{(1,2)},(3,1)\;|\;(3,2),(3,1 )), \end{aligned}$$

where we underlined roots in positions $f_2(J)$. From the graph corresponding to $h_j$ for $J$, we can see that $M^{\prime }=1$ and $m^{\prime }=2$.

Fig. 4

Effects of $f_2$ on $g_{\alpha }$ and $h_j$

Lemma 4.14

If $\alpha _p=\gamma _k \in \Delta ^q$ with $k \not \in J$ then $a_q=1$.

Proof

Recall that $\alpha _p=\gamma _k=w(\beta _k)$ for the corresponding $w$ defined in (9), and let $\beta _k = (b,c)$. The result follows from the claim that $w(b)=\pi _q(b)$ and $w(c)=\pi _q(c)$ (cf. Definition 4.4), which is a consequence of the structure of $\Gamma ^q$ (cf. (23)), as we now explain. The only reflections in $\Gamma ^q$ to the right of $\beta _k$ that affect values in positions $b$ or $c$ are $(b,c^{\prime })$ for $c^{\prime }>c$, and $(b^{\prime },c)$ for $b^{\prime }<b$. Applying these reflections (on the right) to any $w^{\prime }$ satisfying $w^{\prime }(b)=w(b)$ and $w^{\prime }(c)=w(c)$ does not give an edge in the quantum Bruhat graph, by the corresponding criterion in Proposition 4.1.

Fix an admissible subset $J$, and recall Proposition 3.22, including the notation therein. In particular, $M$ is the maximum of $g_{\alpha _p}$. Moreover, if $M> \delta _{p,0}$ we defined $k\not \in J$ and $m\in J\cup \{\infty \}$ with $\gamma _k=\alpha _p,\,\mathrm{sgn}(\alpha _p)l^{J}_{k}=M-1$, and $\gamma _m=\alpha _p$ when $m \ne \infty$. We will implicitly use the following observation when applying Lemma 3.11 in the next two proofs: if $a_i \ne 0$ then $\mathrm{sgn}(a_i) = \mathrm{sgn}(\pi _i^{-1}(\alpha _p))$, where $\pi _i$ is given in Definition 4.4.

Proposition 4.15

We have $M\ge M^{\prime }$. If $M \ge \delta _{p,0}$ then $M=M^{\prime }$.

Proof

We first prove that $M\ge M^{\prime }$. By Corollary 4.11 we have $h_{\lambda _1}=\mathrm{sgn}(\alpha )l_{\alpha _p}^{\infty }$, so the case $M^{\prime }=h_{\lambda _1}$ is clear. The case $M^{\prime }=0$ is trivial, since $M\ge 0$. Therefore, we can assume that the maximum of the sequence $h_i$ does not occur at its endpoints $i=0$ and $i=\lambda _1$. Then we can find $i<j$ such that $h_i=M^{\prime },\,a_i>0,\,a_j<0$, and $a_t=0$ for $t \in (i,j)$. By Lemma 3.11 there exists $\gamma _{k^{\prime }} = \alpha _p \in \Delta ^{q}$ with $q \in (i,j]$, and by Lemma 4.12 we have $\mathrm{sgn}(\alpha _p)l^{J}_{k^{\prime }}=h_{q-1}=h_i=M^{\prime }$. Hence $M\ge M^{\prime }$.

By (14), Lemma 4.12, and Corollary 4.11, all the values of $g_{\alpha _p}$ at points in ${\mathbb Z}+\frac{1}{2}$ are among the values $h_i$. If $M\ge \delta _{p,0}$ then, by Propositions 3.16 and 3.19, the maximum $M$ is attained at a point in ${\mathbb Z}+\frac{1}{2}$. It follows that $M \le M^{\prime }$, hence $M=M^{\prime }$.

The previous proposition states that $M=M^{\prime }$ except in a few corner cases that occur when $p=0$. We will sometimes use one symbol in favor of the other to allude to the corresponding graph.

Proposition 4.16

Assume that $M>\delta _{p,0}$, so $M=M^{\prime }$ (by Proposition 4.15) and $f_p(J)\ne \mathbf {0}$. Then $\gamma _k\in \Delta ^{m^{\prime }}$. If $m \ne \infty$, so $\gamma _m \in \Delta ^{m^{\prime \prime }}$ for some $m^{\prime \prime }\ge m^{\prime }$, then $a_i=0$ for $i \in (m^{\prime },m^{\prime \prime })$; if $m= \infty$ then $a_i=0$ for $i>m^{\prime }$.

Proof

Assuming $\gamma _k \in \Delta ^j$, by Proposition 3.22 (2), Lemma 4.14, and Lemma 4.12, we have $a_j=1$ and $\mathrm{sgn}(\alpha _p)l^{J}_{k}=M-1=h_{j-1}$. It follows that $h_j=M=M^{\prime }$. By the definition of $m^{\prime }$ and the fact that $M^{\prime }>0$, we have $1\le m^{\prime }\le j$ and $a_{m^{\prime }}=1$. By way of contradiction suppose that $m^{\prime }<j$. It follows that the set $\left\{ i \in (m^{\prime },j] \,|\, a_i \ne 0 \right\}$ is not empty, so let $t$ be its minimum. We have $a_t=-1$ and $t<j$, because $a_t=1$ would imply $h_t>h_{m^{\prime }}=M^{\prime }$. We can now apply Lemma 3.11 to show that there exists $\gamma _{k^{\prime }}=\alpha _p\in \Delta ^q$ with $q\in (m^{\prime },t]$ and $k^{\prime }\in J$. By Lemma 4.12, we have $\mathrm{sgn}(\alpha _p)l^{J}_{k^{\prime }}=h_{q-1}=h_{m^{\prime }}=M^{\prime }=M$. Thus, since $k^{\prime }<k<m$, the minimality of $m$ is contradicted. We conclude that $j=m^{\prime }$. If $m\ne \infty$, we can use a similar proof to conclude that the set $\left\{ i \in (m^{\prime },m^{\prime \prime }) \,|\, a_i \ne 0 \right\}$ is empty. The case $m=\infty$ is done similarly.$\square$

Proof of Theorem 4.7

We continue to use the notation from the above setup. Recall that $b=\mathrm{sfill}(J)$. The statement that there is a dual Demazure arrow $b \rightarrow f_p(b)$ if and only if $f_p(J)\ne \mathbf {0}$ follows from Proposition 4.15; indeed, it is clear that $\varphi _p(b)=M^{\prime }$, whereas $\varphi _p(J)=M-\delta _{p,0}$ in the quantum alcove model if $M\ge \delta _{p,0}$, by Theorem 3.9 (2).

We next show that $f_p(b) = \mathrm{sfill}(f_p(J))$, when $f_p(J)\ne \mathbf {0}$. Since $f_p(b) \ne \mathbf {0}$, we have $M^{\prime }>0$, and $f_p$ changes the $p$ in column $m^{\prime }$ to $p+1$ ($f_0$ changes $n$ to $1$ and sorts the column). Now let us turn to $f_p(J)$, where we write the admissible subset $J$ as $\{j_1 < \cdots < j_s \}$. Let $w_i:=r_{j_1} \ldots r_{j_i}$ be the corresponding sequence of permutations. (Recall that the filling $\mathrm{fill}(J)$ is constructed from a subsequence of $w_i$, see Definition 4.4.) We assume $m\ne \infty$, as the case $m=\infty$ is proved similarly. There exist $a<b$ such that

$$\begin{aligned} j_a < k < j_{a+1} < \cdots < j_b = m < j_{b+1}\,; \end{aligned}$$

if $a=0$ or $b=s$, then the corresponding indices $j_a$, respectively $j_{b+1}$ are missing. The sequence of permutations associated to $f_p(J)$ is

$$\begin{aligned} 1,\,w_1,\, \ldots ,\, w_a,\, s_pw_a,\, s_pw_{a+1},\, \ldots ,\, s_pw_{b-1}=w_b,\, w_{b+1},\, \ldots ,\, w_s \end{aligned}$$

(see (19)). By the first part of Proposition 4.16 and by using the notation therein, we conclude that $\mathrm{sfill}(f_p(J))$ is obtained from $\mathrm{sfill}(J)$ by interchanging $p$ and $p+1$ in columns $i$ for $i \in [m^{\prime },m^{\prime \prime })$ (interchange $n$ with $1$ if $p=0$). By the second part of Proposition 4.16, this amounts to changing the $p$ in column $m^{\prime }$ to $p+1$ ($n$ to $1$ if $p=0$).$\square$

4.2 Type $C$

We start with the basic facts about the root system of type $C_n$. We can identify the space $\mathfrak {h}^*_{\mathbb {R}}$ with $V:= \mathbb {R}^n$, the coordinate vectors being $\varepsilon _1, \ldots , \varepsilon _n$. The root system is $\Phi = \left\{ \pm \varepsilon _i \pm \varepsilon _j \, :\, 1 \le i < j \le n \right\} \cup \left\{ \pm 2 \varepsilon _i \, : \, 1 \le i \le n \right\}$. The simple roots are $\alpha _i = \varepsilon _i - \varepsilon _{i+1}$, for $i= 1, \ldots , n-1,$ and $\alpha _n=2\varepsilon _n$. The highest root $\widetilde{\alpha }=2\varepsilon _1$. We let $\alpha _0=\theta =-2\varepsilon _1$. The weight lattice is $\Lambda = \mathbb {Z}^n$. The fundamental weights are $\omega _i = \varepsilon _1 + \cdots + \varepsilon _i$, for $i=1, \ldots , n$. A dominant weight $\lambda = \lambda _1\varepsilon _1 + \cdots + \lambda _n\varepsilon _n$ is identified with the partition $(\lambda _1 \ge \lambda _2 \ge \cdots \ge \lambda _{n-1}\ge \lambda _n \ge 0)$ of length at most $n$. Note that $\rho = (n, n-1, \ldots , 1)$. Like in type $A$, writing the dominant weight $\lambda$ as a sum of fundamental weights corresponds to considering the Young diagram of $\lambda$ as a concatenation of columns. We fix a dominant weight $\lambda$ throughout this section.

The Weyl group $W$ is the group of signed permutations $B_n$, which acts on $V$ by permuting the coordinates and changing their signs. A signed permutation is a bijection $w$ from $[\overline{n} ]:= \{ 1 < 2 < \cdots < n < \overline{n} < \overline{n-1} < \cdots < \overline{1} \}$ to $[\overline{n}]$ satisfying $w(\overline{\imath }) = \overline{w(i)}$. Here $\overline{\imath }$ is viewed as $-i$, so $\overline{\overline{\imath }}= i,\,|\overline{\imath }| = i$, and $\mathrm{sign}(\overline{\imath })=-1$. We use both the window notation $w=w_1 \ldots w_n$ and the full one-line notation $w=w(1)\ldots w(n)w(\overline{n})\ldots w(\overline{1})$ for signed permutations. For simplicity, given $1 \le i < j \le n$, we denote by $(i,j)$ the root $\varepsilon _i - \varepsilon _j$ and the corresponding reflection, which is identified with the composition of transpositions $t_{ij}t_{\overline{\jmath \imath }}$. Similarly, we denote by $(i,\overline{\jmath })=(j,\overline{\imath })$, for $1\le i<j \le n$, the root $\varepsilon _i + \varepsilon _j$ and the corresponding reflection, which is identified with the composition of transpositions $t_{i\overline{\jmath }} t_{j\overline{\imath }}$. Finally, we denote by $(i,\overline{\imath })$ the root $2\varepsilon _i$ and the corresponding reflection, which is identified with the transposition $t_{i\overline{\imath }}$.

We recall a criterion for the edges of the type $C$ quantum Bruhat graph. We need the circular order $\prec _i$ on $[\overline{n}]$ starting at $i$, which is defined in the obvious way, cf. Sect. 4.1. It is convenient to think of this order in terms of the numbers $1, \ldots , n, \overline{n}, \ldots , \overline{1}$ arranged on a circle clockwise. We make the same convention as in Sect. 4.1 that, whenever we write $a \prec b \prec c \prec \ldots ,$ we refer to the circular order $\prec = \prec _a$.

Proposition 4.17

[18]

• (1) Given $1 \le i < j \le n$, we have an edge $w \mathop {\longrightarrow }\limits ^{(i,j)} w(i,j)$ if and only if there is no $k$ such that $i < k < j$ and $w(i) \prec w(k) \prec w(j)$.

• (2) Given $1 \le i < j \le n$, we have an edge $w\mathop {\longrightarrow }\limits ^{(i,\overline{\jmath })}$ if and only if $w(i) < w(\overline{\jmath }),\,\mathrm{sign}(w(i)) = \mathrm{sign}(w(\overline{\jmath }))$, and there is no $k$ such that $i<k<\overline{\jmath }$ and $w(i) < w(k) < w(\overline{\jmath })$.

• (3) Given $1 \le i \le n$, we have an edge $w \mathop {\longrightarrow }\limits ^{(i,\overline{\imath })} w(i,\overline{\imath })$ if and only if there is no $k$ such that $i < k < \overline{\imath }$ (or, equivalently, $i<k\le n$) and $w(i) \prec w(k) \prec w(\overline{\imath })$.

We now consider the specialization of the quantum alcove model to type $C$. For any $k=1, \ldots , n$, we have the following $\omega _k$-chain, from $A_{\circ }$ to $A_{-\omega _k}$, denoted by $\Gamma (k)$ [16]:

$$\begin{aligned} \Gamma (k):=\,&\Gamma _l(k)\Gamma _r(k) \text { where }\nonumber \\ \Gamma _l(k):=\,&\Gamma _{kk} \ldots \Gamma _{k1},\;\;\; \Gamma _r(k):=\, \Gamma _k \ldots \Gamma _2,\, \\ \Gamma _i:=\,&( (i,\overline{i-1}), (i, \overline{i-2}), \ldots , (i, \overline{1})),\,\nonumber \\ \begin{matrix} \Gamma _{ki} :=\, \\ \\ \\ \end{matrix}&\begin{array}{llll} ( (i,k+1),&{} (i,k+2),&{} \ldots ,&{} (i,n), \nonumber \\ (i,\overline{\imath }), &{} &{} &{} \nonumber \\ (i,\overline{n}),&{} (i, \overline{n-1}),&{} \ldots ,&{} (i, \overline{k+1}),\nonumber \\ (i,\overline{i-1}),&{} (i,\overline{i-2}),&{} \ldots ,&{} (i,\overline{1}))\,.\nonumber \end{array} \end{aligned}$$

(27)

Example 4.18

We specialize (27) to $n=3$ and $k=2$. In this case

$$\begin{aligned} \Gamma (2) = \Gamma _l(2)\Gamma _r(2) = ( (2,3), (2, \overline{ 2 }), (2, \overline{ 3 }), (2, \overline{ 1 }), (1,3), (1, \overline{ 1 }), (1, \overline{ 3 }) \;|\; (2, \overline{ 1 }) ). \end{aligned}$$

On the right hand side we use a vertical line to denote the end of $\Gamma _l(2)$. For a graphical interpretation similar to the one in Example 4.3, based on broken columns, we refer to [18][eq. (7.1)].

Fix a dominant weight/partition $\lambda$ for the remainder of this section. We construct a $\lambda$-chain $\Gamma =(\beta _1, \beta _2, \ldots , \beta _m)$ as a concatenation $\Gamma := \Gamma ^1 \ldots \Gamma ^{\lambda _1}$, where $\Gamma ^j= \Gamma (\lambda ^{\prime }_j)$; we also let $\Gamma _l^j:=\Gamma _l(\lambda ^{\prime }_j)$ and $\Gamma _r^j:=\Gamma _r(\lambda ^{\prime }_j)$. Like in type $A$, given a set $J=\left\{ j_1 < \cdots < j_s \right\}$ of folding positions in $\Gamma$, not necessarily admissible, we let $T$ be the corresponding list of roots of $\Gamma$, also viewed as type $C$ transpositions. We factor $\Gamma$ as $\Gamma =\widetilde{\Gamma }^1\ldots \widetilde{\Gamma }^{2\lambda _1}$, where $\widetilde{\Gamma }^{2i-1}=\Gamma ^i_l$ and $\widetilde{\Gamma }^{2i}=\Gamma ^i_r$, for $1\le i \le \lambda _1$. This factorization of $\Gamma$ induces a factorization of $T$ as $T^1T^2 \ldots T^{2\lambda _1}$, and of $\Delta =\Gamma (J)$ as $\Delta =\Delta ^1 \ldots \Delta ^{2\lambda _1}$. Like in type $A$, we use the notation $\gamma _k\in \Delta ^q$ to indicate that the $k$th root in $\Delta$ falls in the segment $\Delta ^q$. We denote by $T^1T^2 \ldots T^j$ the signed permutation obtained by composing the type $C$ transpositions in $T^1, \ldots , T^j$ left to right. For $w \in W$, let $w_i = w(i)$. For $w$ written in the window notation as $w=w_1w_2 \ldots w_n$, let $w[i,j]=w_i \ldots w_j$.

We now recall from [18] the construction of the correspondence between the type $C$ quantum alcove model and model based on diagram fillings; it is useful to compare with Definition 4.4 in type $A$.

Definition 4.19

Let $\pi _{j}=\pi _{j}(T):=T^1 \ldots T^j$. We define the filling map, which associates with each $J\subseteq [m]$ a filling of the Young diagram $2\lambda$, by

$$\begin{aligned} \mathrm{fill}(J)=\mathrm{fill}(T):=C_{1}\ldots C_{2\lambda _1}\,,\; \text{ where } C_{i}:=\pi _i[1,\lambda ^{\prime }_{\lceil \frac{i}{2}\rceil }]\,, \end{aligned}$$

(28)

see the notation above. We define the sorted filling map $\mathrm{sfill}(J)$ by sorting ascendingly the columns of the filling $\mathrm{fill}(J)$.

Example 4.20

Let $n=3$ and $\lambda = (2,1,0)$, which is identified with $2\varepsilon _1 + \varepsilon _2 = \omega _2 + \omega _1$ and corresponds to the Young diagram

. We have the following $\lambda$-chain, see Example 4.18:

$$\begin{aligned} \Gamma&\!=\! \Gamma (2) \Gamma (1) = \Gamma _l(2)\Gamma _r(2)\Gamma _l(1)\Gamma _r(1)\\&\!=\!( \underline{(2,3)}, \underline{(2, \overline{ 2 })}, \underline{(2, \overline{ 3 })}, (2, \overline{ 1 }), (1,3), (1, \overline{ 1 }), (1, \overline{ 3 }) \;|\; \underline{(2, \overline{ 1 })} \;||\; \underline{(1,2)}, (1,3), \underline{(1, \overline{ 1 })}, (1, \overline{ 3 }), (1, \overline{ 2 }) \;|\;), \end{aligned}$$

where we underlined the roots in positions $J = \left\{ 1,2,3, 8,9,11 \right\}$. On the right hand side we used double vertical lines to indicate the end of $\Gamma (2)$, and single vertical lines indicate the end of $\Gamma _l(k)$. Then

$$\begin{aligned} T = ( (2,3), (2, \overline{ 2 }), (2, \overline{ 3 }) \;|\; (2, \overline{ 1 }) \;||\; (1,2), (1, \overline{ 1 }) \;|\; ), \end{aligned}$$

and

$$\begin{aligned} \Gamma (J)&\!=\! \Delta \!=\! \Delta ^1\Delta ^2\Delta ^3\Delta ^4 \\&\!=\!(\underline{(2,3)}, \underline{(3, \overline{ 3 })}, \underline{( \overline{ 3 }, \overline{ 2 })}, (\overline{ 2 }, \overline{ 1 }), (1,3), (1, \overline{ 1 }), (1, \overline{ 3 }) \;|\; \underline{( \overline{ 2 }, \overline{ 1 })} \;||\; \underline{(2, \overline{ 1 })}, (\overline{ 1 }, 3), \underline{( \overline{ 1 },1)}, (1, \overline{ 3 }), (1, \overline{ 2 }) \;|\; )\,; \end{aligned}$$

here we again underlined the folding positions, and indicated the factorizations of $T$ and $\Delta$ by vertical lines.

It is easy to check that $J$ is admissible using Proposition 4.17; indeed, the sequence of signed permutations (12) corresponding to $J$ is a path in the quantum Bruhat graph:

Here, like in Example 4.5 in type $A$, each signed permutation in (12) is written vertically in the window notation, with the entries in bold to be changed by the corresponding reflection; moreover, if this reflection lies in $\Gamma ^i_l$ or $\Gamma ^i_r$, then the corresponding signed permutation is represented as a broken column with the top part of height $\lambda _i^{\prime }$; see the structure of $\Gamma ^i_l$ and $\Gamma ^i_r$ in (27) and Example 4.18. By considering the top part of the last column in each segment and by concatenating these columns left to right, we obtain $\mathrm{fill}(J)$, i.e.,

.

Recall from (8) the definition of $B^{\otimes \lambda }$, which is now realized with split KN columns, see Sect. 2.1. As such, its arrows are given by $f_p^2$, according to Proposition 2.9 (2).

Theorem 4.21

[18] [Theorem 6.1] The map $\mathrm{sfill}$ is a bijection between $\mathcal {A}(\Gamma )$ and $B^{\otimes \lambda }$.

We now state the main result of this section, cf. Theorem 4.7 in type $A$.

Theorem 4.22

The map $\mathrm{sfill}$ is an affine crystal isomorphism between ${\mathcal A}(\Gamma )$ and the subgraph of $B^{\otimes \lambda }$ consisting of the dual Demazure arrows.

Remarks 4.23

• (1) The affine crystal isomorphism in Theorem 4.22 is not guaranteed to be unique in general. However, we believe that it coincides with the bijection that we plan to construct (in a type-independent setup) in order to prove Conjecture 3.26. See Remark 3.27 (1).

• (2) In [16] it was proved that the map $\mathrm{sfill}$ preserves weights, cf. Lemma 4.24. Furthermore, in [18] it was shown that this map translates the height statistic to the charge statistic, which is known to express the energy function in the model based on KN columns, cf. Theorem 2.15. This should be compared with Theorem 3.24 (2), where the constant $C$ is $0$ in this case.

The proof of Theorem 4.21 is parallel to the proof of Theorem 4.7. In this case, we use the height counting lemma in type $C_n$, namely Lemma 4.26. As before let $N_i(\sigma )$ denote the number of entries $i$ in a filling $\sigma$. Let $c_i=c_i(\sigma ) := \frac{1}{2} ( N_i(\sigma ) - N_{\overline{\imath }}(\sigma )$) and define the content of a filling $\sigma$ as $\mathrm{ct}(\sigma ):=(c_1,c_2,\ldots ,c_n)$, which is identified with a type $C_n$ weight. Let $\sigma [q]$ be the filling consisting of the columns $1,2, \ldots , q$ of $\sigma$. Given a $\lambda$-chain and a corresponding sequence $J$ (not necessarily admissible), recall the related notation, including the heights $l_k^J$ in (10), the sequence of roots $\Delta$, and its factorization.

Lemma 4.24

[16][Proposition 4.6 (2)] Let $J \subseteq [m]$, and $\sigma =\mathrm{sfill}(J)$. Then we have $\mu (J)=\mathrm{ct}(\sigma )$.

Corollary 4.25

Let $J \subseteq [m],\,\sigma =\mathrm{fill}(J)$, and $\alpha \in \Phi$. Then $\mathrm{sgn}(\alpha )l_{\alpha }^{\infty } = \langle \mathrm{ct}(\sigma ), \alpha ^{\vee } \rangle$.

Lemma 4.26

[16][Proposition 6.1] Let $J \subseteq [m]$, and $\sigma =\mathrm{fill}(J)$. For a fixed $k$, let $\gamma _k$ be a root in $\Delta ^{q+1}$. We have

$$\begin{aligned} \mathrm{sgn}(\gamma _k)\,l^{J}_{k} = \langle \mathrm{ct}(\sigma [q]),\gamma _k^{\vee }\rangle . \end{aligned}$$

We now introduce notation to be used for the remainder of this section. Let $p \in \{0,1, \ldots , n \}$. Let $J$ be an admissible subset, and $\sigma =\mathrm{sfill}(J)=C_1\ldots C_{2\lambda _1}$, which is guaranteed to be in $B^{\otimes \lambda }$ by Theorem 4.21. Let $a_i:= \langle \mathrm{ct}(C_i), \alpha _p^{\vee } \rangle$. We have $a_i\in \left\{ -1,-\frac{1}{2},0,\frac{1}{2},1\right\}$, for $1\le p \le n-1$, and $a_i \in \left\{ -\frac{1}{2},0,\frac{1}{2} \right\}$ for $p\in \left\{ 0,n \right\}$.

Remarks 4.27

• (1) The value of $a_i$ indicates which entries related to the action of $f_p$ are contained in column $C_i$, as we now explain. Assuming first that $1 \le p \le n-1$, the relevant entries are ${\mathcal P}:=\{p,\,p+1,\,\overline{p+1},\,\overline{p}\}$. If $a_i = 1$ (resp. $a_i=-1$), then $C_i$ contains both $p$ and $\overline{p+1}$ (resp. $p+1$ and $\overline{p}$). If $a_i = \frac{1}{2}$ (resp. $a_i=-\frac{1}{2}$), then $C_i$ contains only one of $p,\,\overline{p+1}$ (resp. only one of $p+1,\,\overline{p}$), while if $a_i=0$ then $C_i$ contains both $p$ and $p+1$, or both $\overline{p+1}$ and $\overline{p}$, or none of these elements. For $p = n$, the relevant entries are $n$ and $\overline{n}$. If $a_i = \frac{1}{2}$ (resp. $a_i=-\frac{1}{2}$), then $C_i$ contains $n$ (resp. $\overline{n}$), while if $a_i=0$ then $C_i$ contains both $n$ and $\overline{n}$, or none of these elements. The case $p=0$ is similar to $p=n$: just replace $n$ and $\overline{n}$ with $\overline{1}$ and $1$, respectively.

• (2) The sequence $a_i$ corresponds to the $p$-signature of the filling $\sigma$. To be more precise, associate with this sequence a $(+,-)$-word by replacing $\pm \frac{1}{2}$ with $\pm$ and $\pm 1$ with $\pm \pm$ (the $0$’s are ignored). If $1\le p\le n-1$, this is the same as the $(+,-)$-word associated with $\sigma$ (see Sect. 2.1) after canceling $-+$ pairs corresponding to the entries $p$ and $p+1$ (or $\overline{p+1}$ and $\overline{p}$) in a column; similarly for $p=n$ and $p=0$.

Let $h_j:= \langle \mathrm{ct}(\sigma [j]), \alpha _p^{\vee } \rangle =\sum _{i=0}^j a_i$, with $a_0=h_0:=0$. Like in type $A$, let $M^{\prime }\ge 0$ be the maximum of $h_j$, and let $m^{\prime }$ be minimal with the property $h_{m^{\prime }}=M^{\prime }$. If $M^{\prime }>0$ then $m^{\prime }$ is the number of the column containing the entry changed by $f_p$. Recall that we need to apply $f_p$ twice; the way in which this can happen is described below.

Proposition 4.28

If $1\le p\le n-1$, then we always have one of the following cases related to the action of $f_p^2$ on the filling $\sigma$.

• (i) $m^{\prime }=2i-1$ and $a_{m^{\prime }}=1$: $p$ and $\overline{p+1}$ in column $m^{\prime }$ are changed to $p+1$ and $\overline{p}$.

• (ii) same as (i) with $m^{\prime }=2i$.

• (iii) $m^{\prime }=2i$ and $a_{m^{\prime }}=a_{m^{\prime }-1}=\frac{1}{2}$: columns $m^{\prime }$ and $m^{\prime }-1$ both contain an entry $p$ (or both contain $\overline{p+1}$), and these entries are changed to $p+1$ (resp. $\overline{p}$).

If $p=n$ or $p=0$, then the analog of case (iii) always holds, with $n$ changed to $\overline{n}$, resp. $\overline{1}$ changed to $1$.

Proof

We implicitly use the following observation, which is immediate from the construction of the splitting $(lC,rC)$ of a column $C$ in Definition 2.7: given $x\in [n]$, the column $lC$ contains $x$ or $\overline{x}$ if and only if $rC$ does.

We consider only $1\le p\le n-1$, as the proof is simpler for $p=n$ and $p=0$. We first prove the following claim: if $a_{2i-1}=\pm \frac{1}{2}$ (or $a_{2i}=\pm \frac{1}{2}$), then $C_{2i-1}$ and $C_{2i}$ contain a single element in ${\mathcal P}$, the two elements have the same absolute value, and the pair $(a_{2i-1},\,a_{2i})$ can take only the following values: $\left( \frac{1}{2},\,\frac{1}{2}\right) ,\,\left( -\frac{1}{2},-\,\frac{1}{2}\right)$, or $\left( -\frac{1}{2},\,\frac{1}{2}\right)$.

We consider only the case $a_{2i-1}=\frac{1}{2}$, as the others are completely similar. The assumption implies that column $C_{2i-1}$ contains a single element in $\mathcal P$, namely $p$ or $\overline{p+1}$. In the first case, it is clear that $C_{2i}$ does not contain $p+1$ or $\overline{p+1}$, but it contains either $p$ or $\overline{p}$. It suffices to rule out the occurence of $\overline{p}$. Assuming it, we deduce that the column $C$ whose splitting is $(C_{2i-1},C_{2i})$ contains $z_j>p$ and $\overline{z_j}$, and the corresponding $t_j<z_j$ is $p$ (cf. Definition 2.7). But $C$ cannot contain $p+1$ or $\overline{p+1}$, so $z_j>p+1$, and the maximality of $t_j$ is contradicted. In the second case, we need to rule out the occurence of $p+1$ in $C_{2i}$. Assuming it, we deduce that $C$ contains $z_j=p+1$ and $\overline{z_j}$, and we have $t_j<p$, so again the maximality of $t_j$ is contradicted.

Now consider the $(+,-)$-word associated with the sequence $a_j$, see Remark 4.27 (2). Cancel pairs $-+$ corresponding to the case $(a_{2i-1},\,a_{2i})=\left( -\frac{1}{2},\,\frac{1}{2}\right)$ mentioned above. The above claim implies that the resulting word is a concatenation of pairs $++$ and $--$ which come from $a_j=\pm 1$ and $(a_{2i-1},\,a_{2i})=\left( \pm \frac{1}{2},\,\pm \frac{1}{2}\right)$; recall that for the latter pairs, the claim also gives the corresponding entries in $\mathcal P$. The statement of the proposition now follows. $\square$

The following is the analog of Lemma 4.14.

Lemma 4.29

If $\alpha _p=\gamma _k \in \Delta ^{2i-1}$ with $k \not \in J$ then we have either $a_{2i-1}=\frac{1}{2}$ and $a_{2i}=\frac{1}{2}$, or $a_{2i-1}=1$. If $\alpha _p=\gamma _k \in \Delta ^{2i}$ with $k \not \in J$ then $a_{2i}=1$.

Proof

We only consider the case corresponding to $\alpha _p\in \Delta ^{2i-1}$ and $1\le p\le n-1$, as the others are simpler. We use freely the structure of the chain of roots $\widetilde{\Gamma }^{2i-1}$, see (27). Recall that $\alpha _p=\gamma _k=w(\beta _k)$ for the corresponding $w$ defined in (9). We have the following cases:

• (1) $\beta _k=(b,c)$ with $b\le \lambda _i^{\prime },\,c>\lambda _i^{\prime }$, and $w(b)=p,\,w(c)=p+1$ (or $w(b)=\overline{p+1},\,w(c)=\overline{p}$);

• (2) $\beta _k=(b,\overline{c})$ with $b\le \lambda _i^{\prime },\,c>\lambda _i^{\prime }$, and $w(b)=p,\,w(c)=\overline{p+1}$ (or $w(b)=\overline{p+1},\,w(c)={p}$);

• (3) $\beta _k=(b,\overline{c})$ with $c<b\le \lambda _i^{\prime }$, and $w(b)=p,\,w(c)=\overline{p+1}$ (or $w(b)=\overline{p+1},\,w(c)={p}$).

We will only consider the first case with $w(b)=p,\,w(c)=p+1$, as the others are completely similar. We first claim that $w(b)=\pi _{2i-1}(b)$ (cf. Definition 4.4), i.e., the entry $p$ is not moved by the reflections $s_{\beta _{k^{\prime }}}$ for $\beta _{k^{\prime }}$ in $\widetilde{\Gamma }^{2i-1}$ with $k^{\prime }>k$. These reflections are $(b,c^{\prime })$ with $c^{\prime }>c,\,(b,\overline{b}),\,(b,\overline{c^{\prime }})$ with $c^{\prime }>\lambda _i^{\prime }$, and $(b,\overline{c^{\prime }})$ with $c^{\prime }<b$. So if the claim failed, the quantum Bruhat graph criterion in Proposition 4.17 would be violated, because the entry $p+1$ is still in position $c$ when these reflections are applied.

Let us now track the entry $p+1$ as we apply the subsequent reflections not involving position $b$, by freely using the quantum Bruhat graph criterion. If this entry is not moved by any of these reflections, then $\pi _{2i-1}(c)=p+1$, so $a_{2i-1}=\frac{1}{2}$. The first reflection which can move $p+1$ is of the form $(b^{\prime },\overline{c})$ with $b^{\prime }<b$, which means that in position $b^{\prime }$ we will now have the entry $\overline{p+1}$. If this entry is not moved by any of the subsequent reflections $(b^{\prime },\overline{c^{\prime }})$ with $\lambda _i^{\prime }<c^{\prime }<c$, then it is not moved by any of the remaining reflections either, so $\pi _{2i-1}(b^{\prime })=\overline{p+1}$, and $a_{2i-1}=1$. Otherwise, we will have the entry $p+1$ in a position $c^{\prime }$ with $\lambda _i^{\prime }<c^{\prime }<c$, and the above reasoning can be applied again (a finite number of times).

Finally, the fact that if $a_{2i-1}=\frac{1}{2}$ then $a_{2i}=\frac{1}{2}$ was deduced in the proof of Proposition 4.28. $\square$

Recall Proposition 3.22 and the notation therein. $M$ is the maximum of $g_{\alpha _p}$, and suppose $M> \delta _{p,0}$; then $\gamma _k=\alpha _p$ with $k \not \in J,\,\mathrm{sgn}(\alpha _p)l^{J}_{k}=M-1$, and if $m \ne \infty$ then $\gamma _m=\alpha _p$ with $m \in J$. The following result is the analog of Proposition 4.15, and its proof is identical. Indeed, the following key fact is still true: if $a_i \ne 0$ then $\mathrm{sgn}(a_i) = \mathrm{sgn}(\pi _i^{-1}(\alpha _p))$, where $\pi _i$ is given in Definition 4.19 (simply note that $a_i=\langle \pi _i(\omega _k),\alpha _p^\vee \rangle =\langle \omega _k,\pi _i^{-1}(\alpha _p^\vee )\rangle$, where $k=\lambda _{\left\lceil \frac{i}{2}\right\rceil }^{\prime }$). This will be needed in the proof of Proposition 4.31 as well.

Proposition 4.30

We have $M\ge M^{\prime }$. If $M \ge \delta _{p,0}$, then $M=M^{\prime }$.

The following result is the analog of Proposition 4.16.

Proposition 4.31

Assume that $M>\delta _{p,0}$, so $M=M^{\prime }$ (by Proposition 4.30) and $f_p(J)\ne \mathbf {0}$. If $a_{m^{\prime }}=1$ then $\gamma _k\in \Delta ^{m^{\prime }}$, otherwise $\gamma _k \in \Delta ^{m^{\prime }-1}$. If $m \ne \infty$, so $\gamma _m \in \Delta ^{m^{\prime \prime }}$, then $a_i=0$ for $i \in (m^{\prime },m^{\prime \prime })$; if $m= \infty$, then $a_i=0$ for $i>m^{\prime }$.

Proof

Assuming $\gamma _k \in \Delta ^j$, by Proposition 3.22 (2), Lemma 4.29, and Lemma 4.26, we have $\mathrm{sgn}(\alpha _p)l^{J}_{k}=M-1=h_{j-1}$, and either $a_j=1$ or $a_j=a_{j+1}=\frac{1}{2}$. In the first case, the rest of the proof is essentially identical to that of Proposition 4.16; in particular, we show that $j=m^{\prime }$. In the second case, we have $j=2i-1$, so it follows that $h_{2i}=M=M^{\prime }$. Once again, essentially the same proof as that of Proposition 4.16 applies; in particular, we show that $2i=m^{\prime }$. Note that in both situations we implicitly used the cases in Proposition 4.28.

Proof of Theorem 4.22

The proof is similar to that of Theorem 4.7, so we only point out the extra complexity in type $C$. This has to do with showing that, if $b=\mathrm{sfill}(J)$ and $f_p(J)\ne \mathbf {0}$, then $f_p^2(b) = \mathrm{sfill}(f_p(J))$. We continue to use the notation from the above setup, and we consider only the case $1\le p\le n-1$, as the others are simpler.

Since $f_p(b) \ne \mathbf {0}$, we have $M^{\prime }>0$, and $f_p^2$ acts on $b$ in one of the ways indicated in Proposition 4.28, cases (i)–(iii). Now let us turn to $f_p(J)$, and use the same setup as in the proof of Theorem 4.7, to which we refer. By the first part of Proposition 4.31, we conclude that $\mathrm{sfill}(f_p(J))$ is obtained from $\mathrm{sfill}(J)$ by applying $s_p$ to columns $i$ for $i \in [m^{\prime },m^{\prime \prime })$ in cases (i)–(ii), resp. $i \in [m^{\prime }-1,m^{\prime \prime })$ in case (iii). By the second part of Proposition 4.31, this amounts to applying $s_p$ to column $m^{\prime }$, resp. to columns $m^{\prime }-1$ and $m^{\prime }$. By Proposition 4.28, this is the same as the action of $f_p^2$ on $\mathrm{sfill}(J)$, which concludes the proof. Note that in the above reasoning we implicitly used Remark 4.27 (1). $\square$