Combinatorial realizations of crystals via torus actions on quiver varieties
Abstract
Let V(λ) be a highest-weight representation of a symmetric Kac–Moody algebra, and let B(λ) be its crystal. There is a geometric realization of B(λ) using Nakajima’s quiver varieties. In many particular cases one can also realize B(λ) by elementary combinatorial methods. Here we study a general method of extracting combinatorial realizations from the geometric picture: we use Morse theory to index the irreducible components by connected components of the subvariety of fixed points for a certain torus action. We then discuss the case of \(\widehat{\mathfrak{sl}}_{n}\), where the fixed point components are just points, and are naturally indexed by multi-partitions. There is some choice in our construction, leading to a family of combinatorial realizations for each highest-weight crystal. In the case of B(Λ 0) we recover a family of realizations which was recently constructed by Fayers. This gives a more conceptual proof of Fayers’ result as well as a generalization to higher level crystals. We also discuss a relationship with Nakajima’s monomial crystal.
Keywords
Crystals Partitions Quiver varieties Torus actions1 Introduction
Kashiwara introduced a combinatorial object (a set along with certain operators) called a crystal associated with each irreducible highest-weight representation of a symmetrizable Kac–Moody algebras \(\mathfrak{g}\), which encodes various information about the representation. Kashiwara’s theory makes heavy use of the quantized universal enveloping algebra associated with \(\mathfrak{g}\), but the crystals themselves can often be realized by other means. For instance, in the case of the fundamental crystal B(Λ 0) for the affine Kac–Moody algebras \(\widehat {\mathfrak{sl}}_{n}\), Misra and Miwa [15] give a realization based on certain partitions. Recently Fayers [4], building on work of Berg [1], found an uncountable family of modifications to the Misra–Miwa realization, and hence many seemingly different realizations of the same crystal.
Fayers’ construction is purely combinatorial, and the motivation for the current work was to find a conceptual explanation of the existence of this family. We achieve this using Nakajima’s quiver varieties. Our construction also allows us to generalize Fayers’ results to give families of realizations of B(Λ) in terms of multi-partitions for any integrable highest-weight \(\widehat{\mathfrak {sl}}_{n}\)-module V(Λ). Most of our construction is actually carried out in the generality of symmetric Kac–Moody algebras, although the end result is less combinatorial in other cases.
In the current paper we mainly consider the case of \(\widehat {\mathfrak{sl}}_{n}\). Here the construction is particularly nice because each fixed point variety \(\mathfrak{F}(\mathbf{v}, W) \) is a finite collection of points. Even better, these points are naturally indexed by tuples of partitions. By taking various choices of ι, we get a large family of realizations for each highest-weight crystal B(Λ) where the vertices are certain multi-partitions. For many of these choices we give a simple characterization of the image of M ι and a combinatorial description of the corresponding crystal operators on multi-partitions (see Theorems 6.2 and 6.4). This gives a large family of combinatorial realizations of each highest-weight crystal B(Λ) for \(\widehat{\mathfrak{sl}}_{n}\). In the case Λ=Λ 0 we obtain exactly the realizations found by Fayers.
In Sect. 8, we give an application to the monomial crystal of Nakajima [22, Sect. 3], as generalized by Kashiwara [11, Sect. 4]. Specifically, we show that, for each of the realizations of B(Λ) discussed above, there is a map to a particular instance of the monomial crystal. One consequence of this is that certain instances of the monomial which have not previously been studied do in fact realize the crystals B(Λ).
Before beginning, we would like to mention related work of Savage [26] and Frenkel–Savage [5] discussing how to extract various combinatorial realizations of crystals from Nakajima’s quiver varieties. One special case is B(Λ 0) for \(\widehat{\mathfrak{sl}}_{n}\), where Savage demonstrates the appearance of the Misra–Miwa realization. Savage parameterizes the irreducible components of the varieties as conormal bundles of various orbits of representations of an undoubled cyclic quiver. By instead viewing the irreducible components as parameterized by torus fixed points, we gain the freedom to flow towards those fixed points in many different ways, thereby seeing a whole family of realizations where Savage only saw one. Even earlier work which uses ideas similar to the ones in this paper can be found in [18]. We also point to [6] for a survey of topics including affine type A quiver varieties, quot schemes, and torus actions, which discusses many of the tools used in this paper.
2 Notation
We give a table of important notation and where it is first used. We only include notation that is used in multiple sections.
| Notation | Description | First used |
|---|---|---|
| \(\mathfrak{M}(\mathbf{v},W), \mathfrak{L}(\mathbf{v},W)\) | Quiver varieties | Sect. 3.2 |
| λ; ω i | Highest weights; fundamental weights in general | Sect. 3.1 |
| B(λ),B(∞) | Crystals | Sect. 3.1 |
| e i , f i | Crystal operators | |
| Γ,I,A,Q,H | Quiver notation | Sect. 3.2 |
| \(\operatorname{Irr}X\) | The irreducible components of X | Sect. 3.3 |
| \(x_{\bar{\imath}}\), \(\overline{x}_{\bar{\imath}}\), s, t | Operators on quiver representations | Sect. 3.3 |
| \(\mathcal{T}\), T Ω , T W , T s | Tori in general framework | Sect. 3.4 |
| \(\mathfrak{F}(\mathbf{v},W)\) | \(\mathcal{T}\)-fixed points of \(\mathfrak{L}(\mathbf{v},W)\) | Sect. 4 |
| F ι (v,W) | \(\Bbb{C}^{*}\)-fixed points for a map \(\iota \colon \Bbb{C}^{*} \rightarrow\mathcal{T}\) | Sect. 4 |
| M ι , M ξ | Maps from \(\operatorname{Irr}\mathfrak{L}(\mathbf{v},W)\) to \(\operatorname{Irr}\mathfrak{F} (\mathbf{v},W)\) | |
| λ, p | Multi-partition and coloring function | Sect. 5.1 |
| \(\bar{c}(b)\), c(λ) | The color of a box b∈λ; the content of λ | Sect. 5.1 |
| \(\operatorname{arm}\), \(\operatorname{leg}\) | Arm and leg statistics | Sect. 5.1 |
| \(A_{\bar{\imath}}\), \(R_{\bar{\imath}}\) | Addable and removable \(\bar{\imath}\)-nodes | Definition 5.1 |
| ξ, ξ Ω , \(\xi_{\overline{\varOmega}}\), ξ i | Slope datum | Definition 5.2 |
| h ξ | The height function associated with ξ | Definition 5.3 |
| integral, general, aligned | Conditions on slope datum | Definition 5.3 |
| ξ-regular, ξ-illegal | Conditions on multi-partitions | Definition 5.4 |
| T, \((t_{\varOmega}, t_{\overline{\varOmega}}, t_{1}, \dots, t_{\ell})\) | \(\widehat{\mathfrak{sl}}_{n}\) specific torus and its coordinates | Sect. 5.2 |
| Λ; \(\varLambda_{\bar{\imath}}\) | Highest weights; fundamental weights for \(\widehat{\mathfrak{sl}}_{n}\) | Sect. 5.2 |
| p λ | T fixed point in \(\mathfrak{L}(\mathbf{v}, W)\) corresponding to λ | Fig. 3 |
| \(\mathcal{E}_{\boldsymbol{\lambda}}\) | Affine space locally containing \(\mathfrak{M}(\mathbf{v}, W)\) near p λ | Sect. 5.3 |
| T ξ | 1-parameter subgroup attached to slope datum | Sect. 6.1 |
| \(e_{\bar{\imath}}^{\boldsymbol{\xi}}\), \(f_{\bar{\imath}}^{\boldsymbol{\xi}}\) | Crystal operators attached to slope datum | Sect. 6.2 |
3 Background
3.1 Crystals
We refer the reader to [10] or [9] for more on this rich subject. Here we simply fix notation and state the properties of crystals we need. We only consider certain explicit realizations of crystals so do not need to discuss the general theory.
A crystal for a symmetric Kac–Moody algebra \(\mathfrak{g}\) consists of a set B along with operators e i ,f i :B→B∪{0} for each i∈I, which satisfy various axioms. There is a crystal B(λ) associated with each integral highest weight λ, which records certain combinatorial information about the irreducible highest-weight representation V(λ). Often the definition of a crystal includes three functions \(\operatorname{wt}, \varphi, \varepsilon\colon B \rightarrow P\), where P is the weight lattice. In the case of crystals of B(λ), these functions can be recovered (up to shifts in null directions) from knowledge of the e i and f i , so we will not count them as part of the data.
When λ−μ is a dominant integral weight, there is a unique embedding B(μ)↪B(λ) that commutes with all the operators e i (although not with the operators f i ). In this way {B(λ)} becomes a directed system. This system has a limit called the infinity crystal, which we denote by B(∞).
3.2 Quiver varieties
Here we review the quiver varieties originally constructed by Lusztig [13] and Nakajima [17, 19]. We work only with quiver varieties over the field \(\Bbb{C}\) of complex numbers.
A point x∈E(V) is called nilpotent if, for some N and all paths a N ⋯a 2 a 1 in Q of length N, \(x_{a_{N}} \circ\cdots \circ x_{a_{2}} \circ x_{a_{1}} =0\) as a map from \(V_{t(a_{1})}\) to \(V_{h(a_{N})}\). The Lusztig quiver variety Λ(V) is the subvariety of E(V) consisting of nilpotent points which also satisfy the preprojective relations μ(x)=(0). As discussed in [13], Λ(V) is a Lagrangian subvariety of E(V).
Definition 3.1
We call (x,s,t)∈E(V,W) stable if \(\operatorname{im}(t)\) generates V under the action of x. Denote the subset of E(V,W) consisting of stable representations by E(V,W)st.
Theorem 3.3
[17, Theorem 5.8]
Assume that Q does not have any loops (i.e., edges starting and ending at the same vertex). Then \(\mathfrak{L}(\mathbf{v},W)\) is a Lagrangian subvariety of \(\mathfrak{M}(\mathbf{v},W)\). In particular, it is equidimensional of dimension \(\frac{1}{2}\dim\mathfrak{M}(\mathbf{v},W)\).
Remark 3.4
The varieties \(\mathfrak{M}(\mathbf{v}, W)\) and \(\mathfrak{L}(\mathbf{v}, W)\) constructed using different spaces V with the same graded dimension v are canonically isomorphic, which is why we only record the dimension v. Up to isomorphism, the space also only depends on the graded dimension of W, but that isomorphism is not canonical, and it is useful to keep track of a fixed vector space W. On occasion we will need to refer to a choice of vector space V associated with a point in \(\mathfrak{M}(\mathbf{v}, W)\), in which case we will refer to the point by [V,x,s,t] instead of just [x,s,t].
3.3 Crystal structure on quiver varieties
The following construction is due to Kashiwara and Saito [12, 25], although we have rephrased things slightly.
3.4 Torus actions
One of the main tools in the current paper is a large torus acting on each variety \(\mathfrak{L}(\mathbf{v}, W)\). This torus is the product of three smaller tori, which we now define.
It is clear that all these actions preserve \(\mathfrak{L}(\mathbf{v}, W) \subset \mathfrak{M}(\mathbf{v}, W)\). Let \(\mathcal{T}= T_{\varOmega}\times T_{W} \times T_{s}\).
Remark 3.5
The torus \(T_{I} := (\Bbb{C}^{*})^{I}\) naturally embeds in \(\mathcal{T}\), and the induced action of T I on \(\mathfrak{M}(\mathbf{v},W)\) is trivial. Thus we actually have an action of the quotient \(\mathcal{T}/T_{I}\). If Γ is a tree, one can see that \(\mathcal{T}/T_{I} \cong(T_{W} \times T_{s})/ (T_{I} \cap(T_{W} \times T_{s}))\), so the orbit of any point under \(\mathcal{T}\) is the same as the orbit under T W ×T s . Thus T Ω only contributes non-trivially when Γ has at least one cycle.
4 A framework for extracting combinatorics
Let \(\mathcal{T}= T_{\varOmega}\times T_{W} \times T_{s}\), which acts on \(\mathfrak {M}(\mathbf{v},W)\) and \(\mathfrak{L}(\mathbf{v},W)\) as in Sect. 3.4. Consider a 1-parameter subgroup \(\iota\colon\Bbb{C}^{*} \hookrightarrow \mathcal{T}\), and denote the induced \(\Bbb{C}^{*}\) action on \(\mathfrak {M}(\mathbf{v} ,W)\) by T ι . Consider \(\pi_{s} \circ\iota\colon\Bbb{C}^{*} \rightarrow\Bbb{C}^{*}\), where π s is projection onto T s . Define \(\operatorname{wt}(T_{\iota})\) to be the weight of π s ∘ι. The following is clear from the definitions:
Lemma 4.1
T ι acts with weight \(\operatorname{wt}(T_{\iota})\) on the symplectic form from Sect. 3.2.
Proposition 4.2
If \(\operatorname{wt}T_{\iota}>0\) then M ι is injective.
Proof
Pick \(C \in\operatorname{Irr}F_{\iota}(\mathbf{v},W)\) and x∈C. The symplectic form 〈⋅,⋅〉 on the tangent space \({\rm T}_{x} \mathfrak{M}(\mathbf{v},W)\) is non-degenerate and T ι acts with positive weight, so the tangent vectors in A C at x form an isotropic subspace of \({\rm T}_{x} \mathfrak{M}(\mathbf{v},W)\). Hence \(\dim A_{C} \leq\frac{1}{2} \dim\mathfrak{M}(\mathbf{v},W) = \dim \mathfrak {L}(\mathbf{v},W)\). But A C is irreducible and \(\mathfrak {L}(\mathbf{v}, W)\) is equidimensional, so A C cannot contain a dense subset of two distinct irreducible components. □
Remark 4.3
Proposition 4.2 shows that, for any ι of positive weight, we can transport the crystal structure on \(\coprod_{\mathbf{v}}\operatorname{Irr} \mathfrak{L}(\mathbf{v},W)\) to a crystal structure on some subset of \(\coprod_{\mathbf{v}} \operatorname{Irr}F_{\iota}(\mathbf{v}, W)\), and hence this gives a realization of B(Λ).
- (F1)
For any fixed v and all sufficiently large N, the fixed points of \(T_{\iota^{(N)}}\) acting on \(\mathfrak{M}(\mathbf{v},W)\) are exactly the fixed points of \(\mathcal{T}\) acting on \(\mathfrak {M}(\mathbf{v},W)\), i.e., we have \(F_{\iota^{(N)}}(\mathbf{v},W)= \mathfrak {F}(\mathbf{v}, W)\).
- (F2)
For any fixed v, the map \(M_{\iota^{(N)}} \colon \operatorname{Irr}\mathfrak{M}(\mathbf{v},W) \rightarrow \operatorname{Irr}F_{\iota^{(N)}}(\mathbf{v}, W)\) stabilizes for N large enough (recalling that for large N we have \(\operatorname{Irr}F_{\iota^{(N)}}(\mathbf{v}, W)= \operatorname{Irr}\mathfrak{F}(\mathbf{v}, W)\)).
- (F3)
\(\operatorname{wt}\iota^{(N)}\) is positive for all N.
Remark 4.4
Remark 4.5
We will see that, if Γ=A n or \(\mathrm{A}^{(1)}_{n}\), then all \(\mathfrak{F}(\mathbf{v}, W)\) are finite sets of points. In fact, these are the only cases with connected Γ where this happens. To see this, consider type D4, where the branch node is labeled 2. For W of dimension (0,1,0,0), one can explicitly calculate that the irreducible component of \(\mathfrak{L}(\mathbf{v}, W)\) corresponding to f 2 f 1 f 3 f 4 f 2 v +∈B(∞) is isomorphic to \({\Bbb{P}}^{1}\), and that this whole component is fixed under the torus action. Any connected simply-laced Dynkin diagram other than A n or \(\mathrm{A}^{(1)}_{n}\) contains D4 as a subdiagram, so there is a fixed point component isomorphic to \({\Bbb{P}}^{1}\).
5 \({\widehat{\mathfrak{sl}}}_{n}\) specific background and definitions
The \(\widehat{\mathfrak{sl}}_{n}\) Dynkin diagram and the associated doubled quiver Q. In Q, the negatively oriented arrows (the ones of the form i→i+1) are colored red. These are the arrows for which ϵ(a)=−1 (Color figure online)
5.1 Partitions and multi-partitions
The Ferrers diagram of the partition (7,6,5,5,5,3,3,1). The parts are the lengths of the “rows” of boxes sloping up and to the left. The (x,y) coordinates are normalized so that the vertex of the partition has coordinates (0.5,0.5), and all boxes have unit side lengths. The center of each box has coordinate (i,j) for some \(i,j \in\Bbb{Z}\). For the box labeled b, i=3 and j=2. Here \(\operatorname{hook}(b)= 8\), \(\operatorname{arm}(b)=3\), and \(\operatorname{leg}(b)=4\)
The coordinates of a box b in a Ferrers diagram are the coordinates (i,j) of the center of b, using the axes shown in Fig. 2. Let b=(i,j) be a box and λ a partition. The arm length of b relative to λ is \(\operatorname{arm}_{\lambda}(b):= \lambda_{i}-j\). The leg length of b relative to λ is \(\operatorname{leg}_{\lambda}(b) = \lambda'_{j} - i\). These quantities are non-negative if and only if b∈λ.
Let \(\mathcal{P}^{\ell}\) denote the set of ℓ-tuples of partitions λ=(λ(1),…,λ(ℓ)), which we call multi-partitions. To distinguish boxes for different λ(k) in a multi-partition λ, we will sometimes use the notation (k;i,j) to denote the box (i,j) associated with λ(k), so a box b is associated with a triple of coordinates (k b ;i b ,j b ). A colored multi-partition is a multi-partition λ=(λ(1),…,λ(ℓ)) along with a chosen function \(p \colon\{1, \dots, \ell\} \to\Bbb{Z}/n\). For \(w= (w_{\overline{0}}, \ldots, w_{\overline{n-1}})\), we say a colored multi-partition is of type w if for all \(\overline{k} \in\Bbb{Z}/n\), \(\#\{ 1 \leq j \leq\ell\mid p(j) = \overline {k} \}=w_{\overline{k}}\).
Definition 5.1
Definition 5.2
For a fixed ℓ, a slope datum is an (ℓ+2)-tuple of positive real numbers \(\boldsymbol{\xi}= (\xi_{\varOmega}, \xi_{\overline{\varOmega}}, \xi_{1}, \dots, \xi_{\ell})\).
Definition 5.3
Fix a multi-partition λ=(λ(1),…,λ(ℓ)) and a slope datum ξ. We define the height of a box b=(k;i,j) by \(h^{\boldsymbol{\xi}}(b) = \xi_{k} + \xi_{\varOmega} i + \xi_{\overline {\varOmega}} j\). We call such a datum general if b≠b′ implies that h ξ (b)≠h ξ (b′). We call such a datum integral if \(\xi_{\varOmega}, \xi_{\overline{\varOmega}}, \xi_{1}, \ldots, \xi_{\ell}\in\Bbb{Z}_{>0}\). We call such a datum aligned if, for all i,j, \(|\xi_{i}-\xi_{j}| < \xi_{\varOmega}+ \xi_{\overline {\varOmega}}\).
Definition 5.4
- (i)
b∈λ(i)
- (ii)
n divides \(p(i) - p(j) + \operatorname{arm}_{\lambda(i)}(b) + \operatorname{leg} _{\lambda(j)}(b) +1\), and
- (iii)
\(-\xi_{\varOmega}< \xi_{j}-\xi_{i}+ \xi _{\varOmega} \operatorname{leg}_{\lambda(j)}(b) - \xi_{\overline {\varOmega}} \operatorname{arm} _{\lambda(i)}(b) < \xi_{\overline{\varOmega}}\).
5.2 Torus actions
Fix ℓ>0, and \(\varLambda= \varLambda_{\bar{\imath}_{1}} + \cdots+ \varLambda_{\bar{\imath}_{\ell}}\), where \(\varLambda_{\bar{\imath}}\) is the \(\bar{\imath}\)th fundamental weight. We now apply the framework from Sect. 4 to realize B(Λ), so fix a graded vector space with dimension \((w_{\bar{\imath}})_{\bar{\imath}\in\Bbb {Z}/n}\), where \(w_{\bar{\imath}}= |\{ 1 \leq j \leq\ell\mid\bar{\imath}_{j} = \bar{\imath}\}|\).
Remark 5.5
In the above construction, T is subtorus of the torus \(\mathcal{T}\) from Sect. 3.4. Thus the construction below is really the application of the methods from Sect. 4 to the \(\widehat{\mathfrak{sl}}_{n}\) case.
Remark 5.6
This same torus action has been considered in e.g. [24]. Here we have made a slight change of conventions; to match that paper we should instead define \(s'=t_{\varOmega}t_{\overline{\varOmega}}Ds\), t′=tD −1. However, our conventions are more convenient in Sect. 6 below.
Proposition 5.7
[24, Proposition 2.9]
The fixed point p λ associated with a multi-partition λ=((3,2),(2,1),(2,2),(2)). In this example, n=3, and we take \((w_{\,\overline{0}}, w_{\,\overline{1}}, w_{\,\overline{2}})=(1,2,1)\) and \((v_{\,\overline{0}}, v_{\,\overline{1}}, v_{\,\overline{2}})= (5,4,5)\). Each \(\bar{\imath}\)-colored box of the multi-partition corresponds to a basis element of \(V_{\bar{\imath}}\). The symbols w k represent a basis for W respecting the I-grading. When representing the \(x_{\bar{\imath}}\) and \({}_{\bar{\imath}}x\) as matrices in this basis, the arrows give non-zero entries equal to 1 (the color of the boxes at the head and tail of a uniquely specify which arrow). The arrows pointing up represent matrix elements of 1 for t. All other matrix elements are 0
The multi-partition for Proposition 5.7 naturally comes with a coloring map p from the component partitions to \(\Bbb{Z}/n\), where the partition corresponding to w k is colored \(\bar{\imath}\) where \(w_{k} \in W_{\bar{\imath}}\).
5.3 Local coordinates
Fix a \(\Bbb{Z}/n\)-graded vector space W with dimW=w, and a colored multi-partition λ of type w with c(λ)=v. Introduce the type-specific notation \({}_{\bar{\imath}\pm\bar{1}} x_{\bar{\imath}}\) to mean x a where a is the arrow from node \(\bar{\imath}\) to node \(\bar{\imath}\pm \bar{1}\).
- (i)
The matrix element of \({}_{\bar{\imath}-\bar{1}} \bar{x}_{\bar {\imath}}\) from b=b k;i,j to b′=b k′;i′,j′ where \(\overline {c} (b') = \overline{c}(b)-1\), and either j≠1 or b k;i+1,j ∉λ.
- (ii)
The matrix element of \({}_{\bar{\imath}+ \bar{1}} x_{\bar{\imath }}\) from b=b k;i,j to b′=b k′;i′,j′ where \(\overline {c} (b') = \overline {c}(b)+1\) and b k;i,j+1∉λ.
- (iii)
The matrix coefficients of the maps \(s_{\bar{\imath}}\).
-
\(\overline{c} (b') = \overline{c}(b)+1\) and (k;i,j+1)∉λ, or
-
\(\overline{c} (b') = \overline{c}(b)-1\), and either j≠1 or (k;i+1,j)∉λ.
Remark 5.8
A similar description of local charts in the case |w|=1 is given in [7, Proposition 2.1] using the language of Hilbert schemes, and the general case is discussed in [20].
5.4 The \(\widehat{\mathfrak{sl}}_{n}\) quiver variety and the loop quiver
Choose coordinates \((t_{\varOmega}, t_{\overline{\varOmega}}, t_{1}, \ldots, t_{\ell})\) for T, where (t 1,t 2,…,t ℓ )=T W and each t k is homogeneous with respect to the \(\Bbb{Z}/n\)-grading on W. Then T acts on the tangent space to p λ in \(\mathfrak{M}_{\widetilde{Q}}(v,W)\), and this action preserves the tangent space to p λ in \(\mathfrak{M}(\mathbf{v},W)\). The next result follows from [24, Theorem 2.11] (where we have modified the statement to match our conventions, as mentioned in Remark 5.6).
Proposition 5.10
Lemma 5.11
Proof
Notice that the automorphism ϕ n of \(\mathfrak{M}_{\tilde{Q}}(v, W)\) is in fact the action of \((\zeta_{n}, \zeta_{n}^{-1}, D_{n}) \in T\), where D n acts on W i as \(\zeta_{n}^{i}\). Thus the ϕ n fixed subspace of the tangent space to p λ consists exactly of those tangent vectors from Proposition 5.10 such that (5.12) holds. Thus we need only show that \({\mathrm {T}}_{p_{\boldsymbol{\lambda}}} (X^{\Bbb{Z}/n}) = (\mathrm{T}_{p_{\boldsymbol{\lambda}}} X)^{\Bbb {Z}/n}\), where \(X = \mathfrak{M}_{\widetilde{Q}}(v,W)\). The inclusion ⊆ is clear. For the other direction, we may find an analytic neighborhood around p λ so that the action of \(\Bbb{Z}/n\) is linear. Given \(\gamma\in(\mathrm{T}_{p_{\boldsymbol{\lambda}}} X)^{\Bbb{Z}/n}\), choose a 1-parameter family γ(t) whose derivative is γ. Then \(n^{-1} \sum_{g \in\Bbb{Z}/n} g \cdot\gamma(t)\) lies in \(X^{\Bbb{Z}/n}\) and its derivative is γ. □
6 Resulting combinatorial realizations of \(\widehat{\mathfrak{sl}}_{n}\) crystal
In this section we precisely describe some combinatorial models that arise from our construction in type \(\widehat{\mathfrak{sl}}_{n}\). Proofs are delayed until the next section.
6.1 The image of M ι
Theorem 6.2
Fix a general slope datum ξ. The image of M ξ is contained in the set of ξ-regular multi-partitions. If ξ is aligned, then \(\operatorname{im}M_{\boldsymbol{\xi}}\) consists of exactly the ξ-regular multi-partitions of type prescribed by p.
6.2 Combinatorial description of the crystal structure on multi-partitions
For each integral highest weight Λ and each general aligned slope datum ξ, Theorem 6.2 gives a bijection between \(\coprod_{\mathbf{v}}\operatorname{Irr}\mathfrak{L}(\mathbf {v},W)\) and the set of ξ-regular multi-partitions. Transporting the crystal structure from Sect. 3.3 gives a realization of the crystal B(Λ) where the underlying set is the ξ-regular multi-partitions. We now give a purely combinatorial description of the crystal operators on ξ-regular multi-partitions.
Definition 6.3
Let \(B^{\boldsymbol{\xi}}\subset\mathcal{P}^{\ell}\) be the set of multi-partitions which can be obtained from the empty multi-partition (∅,…,∅) by applying a sequence of operators \(f_{\bar{\imath}}^{\boldsymbol{\xi}}\) for various \(\bar{\imath}\).
Theorem 6.4
Fix a general aligned slope datum ξ. With the notation above, \(B^{\boldsymbol{\xi}}= \operatorname{im}M_{\boldsymbol{\xi}}\) (which by Theorem 6.2 is the set of ξ-regular multi-partitions). The operators \(e_{\bar{\imath}}^{\boldsymbol{\xi}}\) and \(f_{\bar{\imath}}^{\boldsymbol{\xi}}\) are exactly the crystal operators inherited from the crystal structure on \(\coprod_{\mathbf{v}}\operatorname{Irr}\mathfrak {L}(\mathbf{v}, W)\).
Remark 6.5
One can easily see that the set of ξ-regular multi-partitions depends on ξ, so we do see many different combinatorial realizations of B(Λ). In fact, one always obtains an uncountable family of realizations, since for any irrational number 1/(n−1)<z<n−1 one can find a general aligned slope datum ξ with \(\xi_{\varOmega}/\xi_{\overline{\varOmega}}=z\), and one can easily argue that these all lead to different realizations.
Remark 6.6
When λ=λ is a single partition (i.e., in level 1), the combinatorial operators defined above appeared in Fayers’ recent work [4]. To describe the exact relationship, set \(y = \frac {n \xi_{\varOmega}}{\xi_{\varOmega}+ \xi_{\overline{\varOmega}}}\) and consider the arm sequence A y from [4, Lemma 7.4]. Our notion of ξ-regular then agrees with Fayers’ notion of A y -regular, and our crystal operators agree with Fayers’ crystal operators.
Remark 6.7
The operators \(e_{\bar{\imath}}^{\boldsymbol{\xi}}\) and \(f_{\bar{\imath }}^{\boldsymbol{\xi}}\) are well-defined on any multi-partition, but they do not define the structure of an \(\widehat{\mathfrak{sl}}_{n}\)-crystal on the set of all multi-partitions (this was noted in [4, Sect. 7] in the level 1 case).
6.3 Rational slopes
- (i)b≻ ξ b′ if and only if
-
h ξ (b)>h ξ (b′) or
-
h ξ (b)=h ξ (b′) and i 1>i 2 or
-
h ξ (b)=h ξ (b′) and i 1=i 2 and k 1>k 2,
-
- (ii)\(b \succ'_{\boldsymbol{\xi}}b'\) if and only if
-
h ξ (b)>h ξ (b′) or
-
h ξ (b)=h ξ (b′) and i 1<i 2 or
-
h ξ (b)=h ξ (b′) and i 1=i 2 and k 1<k 2.
-
- (i)
\(h^{\boldsymbol{\xi}^{(N)}}(b) > h^{\boldsymbol{\xi }^{(N)}}(b')\) for all sufficiently large N if and only if b≻ ξ b′ or
- (ii)
\(h^{\boldsymbol{\xi}^{(N)}}(b) > h^{\boldsymbol{\xi }^{(N)}}(b')\) for all sufficiently large N if and only if \(b \succ'_{\boldsymbol{\xi}}b'\).
Corollary 6.8
Define operators \(\tilde{e}_{i}, \tilde{f}_{i}\) on \(\mathcal{P}^{\ell}\) as in Sect. 6.2, but ordering boxes according to ≻ ξ . Then the subset B ξ of \(\mathcal{P}^{\ell}\) which can be obtained from the empty multi-partition by applying operators \(\tilde {e}_{i}\) and \(\tilde{f}_{i}\) is a copy of B(Λ).
Proof
Follows immediately from Theorem 6.4 by taking a limit. □
7 Proofs of results from Sect. 6
7.1 Proof of Theorem 6.2
Fix a general slope datum ξ and a sequence ξ (k) of integral slope data converging to ξ. Fix a colored multi-partition λ with c(λ)=v, and choose N large enough so that for all pairs of boxes b, b′∈λ, \(h^{\boldsymbol {\xi}^{(N)}}(b) > h^{\boldsymbol{\xi} ^{(N)}}(b')\) if and only if h ξ (b)>h ξ (b′). Fix a \(\Bbb{Z} /n\)-graded vector space W, where \(\dim W_{\bar{\imath}}= |\{ k : p(k) = \bar{\imath}\}|\).
Lemma 7.1
The dimension of the attracting set of p λ under the action of \(T_{\boldsymbol{\xi}^{(N)}}\) on \(\mathfrak{M}(\mathbf{v}, W)\) is \(\dim\mathfrak {M}(\mathbf{v}, W)/2\) if and only if λ is ξ-regular.
Proof
Using the coordinates from Sect. 5.3, the fixed points of \(T_{\boldsymbol{\xi}^{(N)}}\) acting on \(\mathfrak{M}(\mathbf{v}, W)\) are exactly the fixed points of T, and in particular they are isolated.
Lemma 7.2
If ξ is aligned, then the attracting set of p λ under the action of \(T_{\boldsymbol{\xi}^{(N)}}\) is contained in \(\mathfrak{L}(\mathbf{v}, W)\).
Proof
Let [x,s,t] belong to the attracting set of p λ . Since \(\xi _{\varOmega}, \xi_{\overline{\varOmega}}\) are both positive x must be nilpotent, since otherwise some path acts with a non-zero eigenvalue and this eigenvalue goes to infinity as z→∞ (z as in (6.1)). Furthermore, we must have s=0, since otherwise it follows from the stability condition that there is some path π in Q such that s∘π∘t≠0, and using the alignment condition this map goes to infinity as z→∞. Thus \([x,s,t] \in\mathfrak{L}(\mathbf{v}, W)\) by definition. □
Since \(\mathfrak{L}(\mathbf{v}, W)\) is equidimensional of dimension \(\dim\mathfrak {M}(\mathbf{v}, W)/2\), Lemma 7.1 implies that \(\operatorname{im}M_{\boldsymbol{\xi}}\) is contained in the set of ξ-regular multi-partitions. Lemma 7.1 also shows that the attracting set of p λ for any ξ-regular λ has dimension exactly \(\dim \mathfrak{M}(\mathbf{v}, W)/2\), so, in the case when ξ is aligned, Lemma 7.2 shows that \(\operatorname{im}M_{\boldsymbol{\xi}}\) consists exactly of all ξ-regular multi-partitions. This completes the proof of Theorem 6.2.
7.2 Proof of Theorem 6.4
First, we need several technical lemmas:
Lemma 7.4
Remark 7.5
Lemma 7.4 can be interpreted geometrically: If λ is ξ-regular then, for any \(\bar{\imath}\in\Bbb {Z}/n\), \(a \in A_{\bar{\imath}}(\boldsymbol{\lambda})\) and \(r \in R_{\bar{\imath}}(\boldsymbol{\lambda})\), comparing the height function h ξ on the centers of a and r orders them in the same way as comparing h ξ on the top corner of r with h ξ on the bottom corner of a.
Proof of Lemma 7.4
If j≥j′, then b 1∈λ(k) and a similar argument shows that (b 1,k,k′) is ξ-illegal. □
Recall from (5.9) that \(V_{\bar{\imath}}^{\geq H}\) is the span of the set of basis vectors of V at height ≥H, using the coordinates from Sect. 5.3, so \(\dim V_{\bar{\imath}}^{\geq H}\) is the number of \(\bar{\imath}\)-boxes in λ of height at least H. Define \(R_{\bar{\imath}}^{\ge H}\) be the number of removable \(\bar{\imath}\)-nodes at height at least H, and define \(A_{\bar{\imath}}^{\ge H}\) to be the number of addable \(\bar{\imath}\)-nodes at height at least H.
Lemma 7.7
Proof
-
If an \((\bar{\imath}+\bar{1})\)-box is added at height less than H+ξ Ω or an \((\bar{\imath}-\bar{1})\)-box is added at height less than \(H+\xi_{\overline{\varOmega}}\), there is no change.
-
If an \((\bar{\imath}+\bar{1})\)-box is added at height at least H+ξ Ω or an \((\bar{\imath}-\bar{1})\)-box is added at height at least \(H+\xi_{\overline{\varOmega}}\), then both sides decrease by 1.
-
If an \(\bar{\imath}\)-box is added at height less than \(H + \xi _{\varOmega} + \xi_{\overline{\varOmega}}\), then both sides increase by 1.
-
If an \(\bar{\imath}\)-box is added at height at least \(H + \xi_{\varOmega } + \xi_{\overline{\varOmega}}\), then both sides increase by 2.
-
If \(c(b')=c(b)+ \overline{1}\), then \(z \cdot e_{b\rightarrow b'}= z^{ \xi^{(N)}_{k} +\xi^{(N)}_{\varOmega}i + \xi^{(N)}_{\overline {\varOmega}} j -\xi^{(N)}_{k'} - \xi^{(N)}_{\varOmega}i' - \xi^{(N)}_{\overline {\varOmega}} j'+ \xi^{(N)}_{\overline{\varOmega}}} e_{b\rightarrow b'}\),
-
If \(c(b')=c(b)- \overline{1}\), then \(z \cdot e_{b\rightarrow b'}= z^{ \xi^{(N)}_{k} +\xi^{(N)}_{\varOmega}i + \xi^{(N)}_{\overline {\varOmega}} j -\xi^{(N)}_{k'} - \xi^{(N)}_{\varOmega}i' - \xi^{(N)}_{\overline {\varOmega}} j'+ \xi^{(N)}_{\varOmega}} e_{b\rightarrow b'}\),
-
\(z \cdot e_{ b \rightarrow r} =z^{ -\xi_{k} -\xi^{(N)}_{\varOmega}i - \xi^{(N) }_{\overline{\varOmega}}j +\xi_{p(r)}} e_{b \rightarrow r}\).
Lemma 7.13
For any [V,x,0,t]∈Z and H>max k {ξ k }, \(\dim\ker x_{\bar{\imath}}|_{V_{\bar{\imath}}^{\geq H}} \geq R_{\bar{\imath }}^{{\geq} H} -A_{\bar{\imath}}^{\geq H +\xi _{\varOmega} + \xi_{\overline{\varOmega}}}\).
Proof
We are now ready to construct the tangent vectors we need:
Proposition 7.14
- (i)
x r,a′−x r,a″=ε;
- (ii)
For all other pairs \(\tilde {a} \in A_{\bar{\imath }}(\boldsymbol{\lambda})\) and \(\tilde{r} \in R_{\bar{\imath}}(\boldsymbol{\lambda})\), \(x^{\tilde{r}, \tilde{a}'} -x^{\tilde{r},\tilde{a}''} = 0\), where \(\tilde{a}'\) is the box with coordinates \((k_{\tilde{a}};i_{\tilde{a}}-1,j_{\tilde{a}})\) and \(\tilde{a}''\) is the box with coordinates \((k_{\tilde{a}};i_{\tilde{a}},j_{\tilde{a}}-1)\);
- (iii)
The resulting tangent vector t a,r in \(\mathrm{T}_{p_{\boldsymbol{\lambda}}} \mathcal{E}_{\boldsymbol{\lambda}}\) lies in \(\mathrm{T}_{p_{\boldsymbol{\lambda}}} Z\).
Proof
The one-parameter family in \(\mathcal{E}_{\boldsymbol{\lambda}}\) from Proposition 7.14. The blue arrows are matrix elements for \({}_{\overline{k}-1} x_{\overline{k}}\), and the red arrows are matrix elements for \({}_{\overline{k}+1} x_{\overline{k}}\), for various \(\overline{k}\). All other coordinates are 0. In general, to define these new matrix elements, one follows the rim of λ(k r ), beginning at r, putting in matrix elements of −ε as shown. As soon as one would need to draw an arrow to a box which is not present in λ(k a ), one may stop; or if one reaches the end of the rim of λ(k r ), one may stop. One of these must happen at some point because j a −1≥j r (Color figure online)
The first case is handled by a symmetric construction. □
Following the definition of the crystal operators in Sect. 3.3, to obtain a generic point in \(e_{\bar{\imath}}(Z)\), one should proceed as follows: start with a generic point z∈Z, and choose a representative [V,x,t]. Take the quotient of this representation by a generic 1-dimensional module in the \(\bar{\imath}\)-socle of V (that is, the socle of z intersected with \(V_{\bar{\imath}}\)) to obtain [V′,x′,t′]. This will be a representative of a generic point z′ in \(e_{\bar{\imath}}(Z)\).
The \(\bar{\imath}\)-socle of \(V_{\bar{\imath}}\) is exactly \(\ker x_{\bar{\imath}}\). Quotienting out by a element of \(\ker x_{\bar{\imath}}\) will decrease \(\dim V_{\bar {\imath}}^{\geq H}\) by 1 for low heights H, and will not change \(\dim V_{\bar{\imath }}^{\geq H}\) for high values of H. If this is done generically, the cutoff value of H between these two behaviors will be as low as possible, which is to say that \(\dim V_{\bar{\imath}}^{\geq H}\) will be unchanged for all \(H>\bar {H}\), where \(\bar{H}\) is the lowest value of H such that \(\dim\ker x_{\bar{\imath}}^{>H} < \dim\ker x_{\bar{\imath}}^{\geq H}\).
- (i)
For generic [x,0,t]∈Z, \(\dim\ker x_{\bar{\imath}}\) is the number of uncanceled “)” brackets in \(S_{\bar{\imath}}^{\xi}(\boldsymbol {\lambda})\) (see Sect. 6.2);
- (ii)
Let r be the removable box corresponding to the first uncanceled “)” from the right in \(S_{\bar{\imath}}^{\xi}(\boldsymbol {\lambda})\), and let H=h ξ (r). Then \(\dim\ker x_{\bar{\imath}}|_{V_{\bar{\imath}}^{\geq H}} = \dim\ker x_{\bar{\imath}} \) and \(\dim\ker x_{\bar{\imath}}|_{V_{\bar{\imath}}^{> H}}=\dim\ker x_{\bar{\imath}}-1\).
8 Application to the monomial crystal
8.1 Background on the monomial crystal for \(\widehat {\mathfrak{sl}}_{n}\)
Here we describe the monomial crystal of Nakajima [22, Sect. 3] in the case of \(\widehat{\mathfrak{sl}}_{n}\), as generalized and modified by Kashiwara in [11, Sect. 4]. In fact the definition below is even more general than Kashiwara’s, since Kashiwara assumes that our K is 1.
Example: The string of brackets \(S^{M}_{\overline{1}}\)
Theorem 8.3
[11, Theorem 4.3]
Assume K=1. Fix a monomial D which is highest weight for the crystal structure and let B D be the subset generated by D. Then B D along with these operators is isomorphic to the crystal B(Λ), where \(\varLambda= \operatorname{wt}D\).
It is not true that all of \(\mathcal{M}\) forms an \(\widehat{\mathfrak {sl}}_{n}\) crystal under these operations. This was noted by Nakajima in [22, Sect. 3] for some specific cases with K=2. However, here we will see that some elements of \(\mathcal{M}\) do generate \(\widehat {\mathfrak{sl}}_{n}\) crystals, even for K>1.
8.2 Results on monomial crystals
In this section we construct a map of crystals from certain instances of our crystals B ξ to certain monomial crystals. This is a generalization of [27, Theorem 5.1], and the proof proceeds as in that paper. We then use this result to prove that some instances of the monomial crystal which have not previously been understood do in fact realize B(Λ).
-
h ξ (b)>h ξ (b′) or
-
h ξ (b)=h ξ (b′) and i 1>i 2 or
-
h ξ (b)=h ξ (b′) and i 1=i 2 and k 1>k 2.
Theorem 8.5
For all λ∈B ξ , we have \(\varPsi(\tilde{e}_{i}^{\boldsymbol{\xi}}(\boldsymbol{\lambda})) = \tilde{e}_{i}^{\textbf{c}}(\varPsi(\boldsymbol{\lambda}))\), \(\varPsi (\tilde{f}_{i}^{\boldsymbol{\xi}}(\boldsymbol{\lambda} )) = \tilde{f}_{i}^{\textbf{c}}(\varPsi(\boldsymbol{\lambda}))\), and \(\operatorname{wt}(\varPsi(\boldsymbol{\lambda}))= \operatorname{wt}(\boldsymbol{\lambda})\).
The proof of Theorem 8.5 will take most of the rest of this section.
Lemma 8.6
Let λ and μ be multi-partitions such that μ=λ⊔b for some box b. Then \(\varPsi(\boldsymbol{\mu}) = A_{\textbf{c}; \bar{\imath}, h^{\boldsymbol{\xi}} (b)}^{-1} \varPsi(\boldsymbol{\lambda})\), where \(\bar{\imath}= \bar {c}(b)\).
Proof
-
\(b \in A_{\bar{\imath}}(\boldsymbol{\lambda}) \setminus A_{\bar{\imath}}(\boldsymbol{\mu})\).
-
\(b \in R_{\bar{\imath}}(\boldsymbol{\mu}) \setminus R_{\bar {\imath}}(\boldsymbol{\lambda})\).
-
Either (i): \(A_{\bar{\imath}+ \bar{1}}(\boldsymbol{\mu}) \setminus A_{\bar{\imath}+ \bar{1}}(\boldsymbol{\lambda})= b'\) and \(R_{\bar{\imath}+ \bar{1}}(\boldsymbol{\lambda}) =R_{\bar{\imath}+ \bar{1}}(\boldsymbol{\mu })\) for some box b′ with \(h^{\boldsymbol{\xi}}(b') = h^{\boldsymbol{\xi }}(b)+\xi_{\overline{\varOmega}}\), or (ii): \(R_{\bar{\imath}+ \bar{1}}(\boldsymbol{\lambda}) \setminus R_{\bar {\imath}+ \bar{1}}(\boldsymbol{\mu}) = b'\) and \(A_{\bar{\imath}+ \bar{1}}(\boldsymbol{\lambda}) =A_{\bar{\imath}+ \bar{1}}(\boldsymbol{\mu})\) for some box b′ with h ξ (b′)=h ξ (b)−ξ Ω .
-
Either (i): \(A_{\bar{\imath}- \bar{1}}(\boldsymbol{\mu}) \setminus A_{\bar{\imath}- \bar{1}}(\boldsymbol{\lambda})= b''\) and \(R_{\bar{\imath}- \bar{1}}(\boldsymbol{\lambda}) =R_{\bar{\imath}- \bar{1}}(\boldsymbol{\mu})\) for some box b″ with h ξ (b″)=h ξ (b)+ξ Ω , or (ii): \(R_{\bar{\imath}- \bar{1}}(\boldsymbol{\lambda}) \setminus R_{\bar{\imath}- \bar{1}}(\boldsymbol{\mu}) = b''\) and \(A_{\bar{\imath}- \bar{1}}(\boldsymbol{\lambda}) =A_{\bar {\imath}- \bar{1}}(\boldsymbol{\mu})\) for some box b″ with \(h^{\boldsymbol{\xi}}(b'')= h^{\boldsymbol{\xi}}(b)-\xi _{\overline{\varOmega}}\).
Lemma 8.7
Fix λ∈B ξ . For any \(a \in A_{\bar{\imath}}(\boldsymbol{\lambda})\) and \(r \in R_{\bar{\imath}} (\boldsymbol{\lambda})\), we have a≺ ξ r if and only if a≺ ξ b, where if r=(k;i,j) we set b=(k;i+1,j+1).
Proof
Any pair violating this will lead to an illegal triple (Lemma 7.4), and hence is not in B ξ . □
Proof of Theorem 8.5
Fix λ∈B ξ and \(\bar{\imath }\in\Bbb{Z}/n\). Let \(S^{M}_{\bar{\imath}}(\varPsi(\boldsymbol{\lambda} ))\) denote the string of brackets used in Sect. 8.1. Let \(S_{\bar{\imath}}^{\boldsymbol{\xi}}(\boldsymbol{\lambda})\) denote the string of brackets used in Sect. 6.2, and define the height of a bracket in \(S^{\boldsymbol{\xi}}_{\bar{\imath}}(\boldsymbol{\lambda} )\) to be h ξ (b) for the corresponding box.
It follows immediately from Lemma 8.7 that, for each k, all “(” in \(S^{\boldsymbol{\xi}}_{\bar{\imath}}(\boldsymbol {\lambda})\) of height \(k+\xi_{\varOmega}+\xi_{\overline{\varOmega}}\) are immediately to the left of all “)” of height k. Let T be the string of brackets obtained from \(S^{\boldsymbol {\xi}}_{\bar{\imath}}(\boldsymbol{\lambda} )\) by, for each k, canceling as many “(” of height \(k+\xi _{\varOmega}+ \xi_{\overline{\varOmega}}\) with “)” of height k as possible. One can use T instead of \(S^{\boldsymbol{\xi}}_{\bar{\imath}}(\boldsymbol {\lambda})\) to calculate \(\tilde {e}_{\bar{\imath}} ^{\boldsymbol{\xi}}(\boldsymbol{\lambda})\) and \(\tilde {f}_{\bar{\imath }^{\boldsymbol{\xi}}}(\boldsymbol{\lambda})\) without changing the result.
- (i)
The “(” in T of height \(k+\xi_{\varOmega}+\xi_{\overline {\varOmega}}\) correspond exactly to the factors of \(Y_{\bar{\imath}, k+K}\) in Ψ(λ).
- (ii)
The “)” in T of height k correspond exactly to the factors of \(Y_{\bar{\imath}, k+K}^{-1}\) in Ψ(λ).
Definition 8.8
Theorem 8.5 has the following consequence, which shows that certain new instances of the monomial crystal do in fact still realize the crystal B(Λ). This is related to [11, Problem 2].
Corollary 8.9
Fix \(\xi_{\varOmega},\xi_{\overline{\varOmega}} \in\Bbb{Z}_{>0}\). For all \(\bar{\imath}\in \Bbb{Z}/n\), let \(c_{\bar{\imath},\bar{\imath}+1}= \xi_{\overline{\varOmega}}\) and \(c_{\bar{\imath}+1,\bar{\imath}}=\xi_{\varOmega}\). Let M be a \((\xi_{\varOmega}+\xi_{\overline{\varOmega}})\)-aligned dominant monomial. Then the component of the monomial crystal generated by M under the operators \(\tilde{e}_{\bar{\imath}}\) and \(\tilde {f}_{\bar{\imath}}\) is a copy of B(Λ) for \(\varLambda = \operatorname{wt}(M)\).
Proof
One can easily find an aligned slope datum ξ with ξ Ω , \(\xi_{\overline{\varOmega}}\) as specified, and such that Ψ sends the empty multi-partition to M, and by Corollary 6.8, B ξ is a copy of the crystal B(Λ). Thus the corollary follows immediately from Theorem 8.5. □
9 Construction in terms of punctual quot schemes
Another reason the case of \(\widehat{\mathfrak{sl}}_{n}\) is special is that in this case \(\mathfrak{L}(\mathbf{v}, W)\) can be realized using punctual quot schemes. We now briefly explain how our results translate into that language. Our references for this section are [20] and [23].
9.1 The punctual quot scheme
Fix a finite dimensional \(\Bbb{C}\)-vector space W. Choose coordinates on \(\Bbb{A}^{2}\) so that we can identify its coordinate ring \(\mathcal {O}_{\Bbb{A}^{2}} =\Bbb{C}[x,y]\). The punctual quot scheme Quot 0(W,m) is the moduli space of \(\Bbb{C}[x,y]\)-submodules \(K \subset\Bbb{C}[x,y] \otimes W\) where the quotient \((\Bbb{C}[x,y] \otimes W )/ K\) is m-dimensional and supported at the origin, i.e., annihilated by some power of the maximal ideal (x,y).
Remark 9.3
9.2 Torus actions
One can choose the isomorphism in (9.2) so that it intertwines this action with the action of T on \(\mathfrak{L}(\mathbf{v}, W)\) from Sect. 5.2 and identifies fixed points corresponding to the same multi-partition.
Fix an integral slope datum ξ and consider T ξ ⊂T. The action of T ξ induces a Białynicki-Birula stratification of Quot 0(W,m) as follows: For any K∈Quot 0(W,m), lim t→∞ t⋅ ξ K is the submodule of \(\Bbb{C}[x,y] \otimes W\) generated by the lowest degree (ordered by ξ) terms of elements of K. One can think of this as a “reverse initial submodule,” and the strata are “reverse Gröbner strata,” which we will denote by RG λ . Note that this limit only makes sense since all K∈Quot 0(W,m) are supported at the origin.
10 Questions
We feel that the construction in Sect. 4 should have more applications. We finish by formulating some questions related to this construction.
Question 10.1
Do interesting combinatorics arise when the construction from Sect. 4 is carried out in other cases?
In types other than \(\mathfrak{sl}_{n}\) and \(\widehat{\mathfrak{sl}}_{n}\) the fixed-point components defined in Sect. 4 are more complicated than just points, but perhaps it is still possible to index them combinatorially. Even the case of \(\mathfrak{sl}_{n}\), where the fixed point components are just points, may lead to potentially new combinatorial realizations of B(λ); one has freedom to choose the weights for the torus T W , and this choice should lead to different realizations.
Question 10.2
In type \(\widehat{\mathfrak{sl}}_{n}\), can one describe the situation combinatorially for more general ξ?
For instance, it would be natural to consider the case when ξ is general but not aligned. By a combinatorial description we would mean a combinatorial characterization of \(\operatorname{im}M_{\boldsymbol{\xi}}\) and of the induced operators e i ,f i on \(\operatorname{im}M_{\boldsymbol{\xi}}\). By Theorem 6.2, \(\operatorname{im}M_{\boldsymbol{\xi}}\) consists only of ξ-regular multi-partitions, but one can easily find examples where it is a proper subset of these. Also, the construction in Sect. 4 only requires \(\xi_{\varOmega}+ \xi_{\overline{\varOmega}}>0\), not both individually to be positive. It may be interesting to understand our construction for this more general notion of slope datum.
Question 10.3
Is there a natural crystal structure on a set of fixed point components larger than \(\operatorname{im}M_{\iota}\)? Or more geometrically, is there is natural crystal structure on the attracting sets of these components in the various \(\mathfrak{M}(\mathbf{v}, W)\)?
- (i)
that its attracting set has dimension \(\dim(\mathfrak {M}(\mathbf{v},W))/2\), and
- (ii)
that its attracting set is contained in \(\mathfrak{L}(\mathbf{v},W)\).
Choose a decomposition W=W 1⊕W 2. Consider ι defined by ι(z)=D where D∈T W is the diagonal matrix which acts on W 1 as the identity, and acts on W 2 as multiplication by z N for some large N (this is roughly the action used by Nakajima in [21]). Deform ι by allowing non-trivial weights of z embedding into \(T_{\overline{\varOmega}}\) and T s , and changing the weights in D, but such that all changes to weights are much smaller than N. Then (at least in finite type; in other types one should be more careful with limits) this action preserves the tensor-product variety \(\mathcal{E}\) from [21], so the map M ι can be extended to a map \(\widetilde {M}_{\iota}\) from \(\operatorname{Irr}\mathcal{E}\) to fixed-point components, and R ι is exactly \(\operatorname{im}\widetilde{M}_{\iota}\). Nakajima defines a crystal structure on \(\operatorname{Irr}\mathcal{E}\), so this can be pushed to a crystal structure on R ι . The result is the crystal for a tensor product of two highest-weight representations.
One interpretation of Question 10.3 is to ask if Nakajima’s tensor product variety can be generalized in the following sense: Choose a generic family ι (N) as in Sect. 4. For each v, and large enough N, consider the subset \(\mathcal{E}(\mathbf{v}, W)\) of \(\mathfrak{M}(\mathbf{v}, W)\) such that lim z→∞ ι (N)(z) exists. We conjecture that this is a subvariety of \(\mathfrak{M}(\mathbf{v}, W)\) of pure dimension \(\dim(\mathfrak{M}(\mathbf{v}, W))/2\). Is there a natural crystal structure on \(\operatorname{Irr}\mathcal{E}(\mathbf{v}, W)\)? If yes, is this the crystal of some \(\mathfrak{g}\) representation? We have not seriously considered these questions beyond the cases covered by Nakajima’s previous work, but they seem like a natural course for further study.
Notes
Acknowledgements
We thank Pavel Etingof, Matthew Fayers, Monica Vazirani and Ben Webster for interesting discussions. We also thank Hiraku Nakajima, Alistair Savage, Ben Webster, and two anonymous referees for helpful comments on earlier versions of this paper. The first author was supported by an NDSEG fellowship and a Miller research fellowship. The second author was supported by NSF grants DMS-0902649, DMS-1162385 and DMS-1265555.
References
- 1.Berg, C.: The ladder crystal. Electron. J. Comb. 17(1) (2010). Research Paper 97, 20 pp. arXiv:0901.3565v3
- 2.Białynicki-Birula, A.: Some theorems on actions of algebraic groups. Ann. Math. (2) 98, 480–497 (1973) MATHCrossRefGoogle Scholar
- 3.Chriss, N., Ginzburg, V.: Representation Theory and Complex Geometry. Birkhäuser, Boston (1997) MATHGoogle Scholar
- 4.Fayers, M.: Partition models for the crystal of the basic \({U}_{q}(\widehat {\frak{sl}}_{n})\)-module. J. Algebr. Comb. 32(3), 339–370 (2010). arXiv:0906.4129v3 MATHMathSciNetCrossRefGoogle Scholar
- 5.Frenkel, I.B., Savage, A.: Bases of representations of type A affine Lie algebras via quiver varieties and statistical mechanics. Int. Math. Res. Not. 28, 1521–1547 (2003). arXiv:math/0211452v2 MathSciNetCrossRefGoogle Scholar
- 6.Fujii, S.: Minabe, S.: A combinatorial study on quiver varieties. arXiv:math/0510455v2
- 7.Haiman, M.: t,q-Catalan numbers and the Hilbert scheme. Discrete Math. 193(1–3), 201–224 (1998). Selected papers in honor of Adriano Garsia (Taormina, 1994) MATHMathSciNetCrossRefGoogle Scholar
- 8.Harris, J.: Algebraic Geometry, a First Course. Graduate Texts in Math., vol. 133. Springer, Berlin (1992) MATHCrossRefGoogle Scholar
- 9.Hong, J., Kang, S.-J.: Introduction to Quantum Groups and Crystal Bases. Graduate Studies in Mathematics, vol. 42. Am. Math. Soc., Providence (2002) MATHGoogle Scholar
- 10.Kashiwara, M.: On crystal bases. In: Representations of Groups, Banff, AB, 1994. CMS Conf. Proc., vol. 16, pp. 155–197. Am. Math. Soc., Providence (1995) Google Scholar
- 11.Kashiwara, M.: Realizations of crystals. In: Combinatorial and Geometric Representation Theory, Seoul, 2001. Contemp. Math., vol. 325, pp. 133–139. Am. Math. Soc., Providence (2003). arXiv:math/0202268v1 CrossRefGoogle Scholar
- 12.Kashiwara, M., Saito, Y.: Geometric construction of crystal bases. Duke Math. J. 89(1), 9–36 (1997) MATHMathSciNetCrossRefGoogle Scholar
- 13.Lusztig, G.: Quivers, perverse sheaves, and quantized enveloping algebras. J. Am. Math. Soc. 4(2), 365–421 (1991) MATHMathSciNetCrossRefGoogle Scholar
- 14.Marsden, J., Weinstein, A.: Reduction of symplectic manifolds with symmetry. Rep. Math. Phys. 5(1), 121–130 (1974) MATHMathSciNetCrossRefGoogle Scholar
- 15.Misra, K., Miwa, T.: Crystal base for the basic representation of \(U_{q}(\widehat {\mathfrak{sl}}_{n})\). Commun. Math. Phys. 134(1), 79–88 (1990) MATHMathSciNetCrossRefGoogle Scholar
- 16.Mumford, D.: The Red Book of Varieties and Schemes. Lecture Notes in Mathematics, vol. 1358. Springer, Berlin (1999). Second, expanded edition, with contributions by Enrico Arbarello MATHCrossRefGoogle Scholar
- 17.Nakajima, H.: Instantons on ALE spaces, quiver varieties, and Kac–Moody algebras. Duke Math. J. 76(2), 365–416 (1994) MATHMathSciNetCrossRefGoogle Scholar
- 18.Nakajima, H.: Homology of moduli spaces of instantons on ALE spaces. I. J. Differ. Geom. 40(1), 105–127 (1994) MATHMathSciNetGoogle Scholar
- 19.Nakajima, H.: Quiver varieties and Kac–Moody algebras. Duke Math. J. 91(3), 515–560 (1998) MATHMathSciNetCrossRefGoogle Scholar
- 20.Nakajima, H.: Lectures on Hilbert Schemes of Points on Surfaces. University Lecture Series, vol. 18. Am. Math. Soc., Providence (1999) MATHGoogle Scholar
- 21.Nakajima, H.: Quiver varieties and tensor products. Invent. Math. 146(2), 399–449 (2001). arXiv:math/0103008v2 MATHMathSciNetCrossRefGoogle Scholar
- 22.Nakajima, H.: t-analogs of q-characters of quantum affine algebras of type A n, D n. In: Combinatorial and Geometric Representation Theory, Seoul, 2001. Contemp. Math., vol. 325, pp. 141–160. Am. Math. Soc., Providence (2003). arXiv:math/0204184v1 CrossRefGoogle Scholar
- 23.Nakajima, H., Yoshioka, K.: Algebraic Structures and Moduli Spaces. CRM Proc. Lecture Notes, vol. 38, pp. 31–101. Am. Math. Soc., Providence (2004). arXiv:math/0311058v1. Lectures on instanton counting Google Scholar
- 24.Nakajima, H., Yoshioka, K.: Instanton counting on blowup. I. 4-dimensional pure gauge theory. Invent. Math. 162(2), 313–355 (2005). arXiv:math/0306198v2 MATHMathSciNetCrossRefGoogle Scholar
- 25.Saito, Y.: Crystal bases and quiver varieties. Math. Ann. 324(4), 675–688 (2002). arXiv:math/0111232v1 MATHMathSciNetCrossRefGoogle Scholar
- 26.Savage, A.: Geometric and combinatorial realizations of crystal graphs. Algebr. Represent. Theory 9(2), 161–199 (2006). arXiv:math/0310314v4 MATHMathSciNetCrossRefGoogle Scholar
- 27.Tingley, P.: Monomial crystals and partition crystals. SIGMA 6, 035 (2010). arXiv:0909.2242v3 MathSciNetGoogle Scholar