The center of the affine nilTemperley–Lieb algebra

Abstract

We give a description of the center of the affine nilTemperley–Lieb algebra based on a certain grading of the algebra and on a faithful representation of it on fermionic particle configurations. We present a normal form for monomials, hence construct a basis of the algebra, and use this basis to show that the affine nilTemperley–Lieb algebra is finitely generated over its center. As an application, we obtain a natural embedding of the affine nilTemperley–Lieb algebra on N generators into the affine nilTemperley–Lieb algebra on $N+1$ generators.

1 Introduction

The main goal of this work is to describe the center of the affine nilTemperley–Lieb algebra $\text {n}\widehat{\text {TL}}_{N}$ over any ground field. Only two tools are used: a fine grading on $\text {n}\widehat{\text {TL}}_{N}$ and a representation of $\text {n}\widehat{\text {TL}}_{N}$ on fermionic particle configurations on a circle. It is essential that this graphical representation be faithful (see [12, Prop. 9.1]). We provide an alternative proof of that fact by constructing a basis for $\text {n}\widehat{\text {TL}}_{N}$ that is especially adapted to the problem. This basis has further advantages: It can be used to prove that the affine nilTemperley–Lieb algebra is finitely generated over its center. Also, it can be used to exhibit an explicit embedding of $\text {n}\widehat{\text {TL}}_{N}$ into $\text {n}\widehat{\text {TL}}_{N+1}$ defined on basis elements that otherwise would not be apparent, since the defining relations of these algebras are affine, and there is no embedding of the corresponding Coxeter graphs.

For a ground field $\mathbb {k}$, the affine nilTemperley–Lieb algebra $\text {n}\widehat{\text {TL}}_{N}$ is the unital associative $\mathbb {k}$-algebra given by N generators $a_0,\ldots ,a_{N-1}$ and nil relations $a_i^2=0$ and $a_ia_{i\pm 1}a_i=0$ for all i. Generators that are far apart commute, i.e. $a_ia_j=a_ja_i$ for $i-j \ne \pm 1 \text { mod }N$. In these relations, the indices are interpreted modulo N so that the generators $a_0$ and $a_{N-1}$ are neighbours that do not commute. The subalgebra of $\text {n}\widehat{\text {TL}}_{N}$ generated by $a_1,\ldots ,a_{N-1}$ is the (finite) nilTemperley–Lieb algebra $n TL _{N}$, as in [19]. The affine nilTemperley–Lieb algebra appears in many different settings, which we describe next.

• 1. $\text {n}\widehat{\text {TL}}_{N}$ is a quotient of the affine nilCoxeter algebra of type $\tilde{\textsf {A}}_{N-1}$.

  The affine nilCoxeter algebra $\widehat{\mathsf U}_N$ of type $\tilde{\textsf {A}}_{N-1}$ over a field $\mathbb {k}$ is the unital associative algebra generated by elements $u_i$, $0 \le i \le N-1$, satisfying the relations $u_i^2 = 0$; $u_i u_j = u_j u_i$ for $i - j \ne \pm 1\text { mod }N$; and $u_i u_{i+1}u_i = u_{i+1}u_i u_{i+1}$ for $1 \le i \le N-1$, where the subscripts are read modulo N. The algebra $\text {n}\widehat{\text {TL}}_{N}$ is isomorphic to the quotient of $\widehat{\mathsf U}_N$ obtained by imposing the additional relations $u_i u_{i+1}u_i = u_{i+1}u_i u_{i+1}=0$ for $1 \le i \le N-1$. The affine nilCoxeter algebra is closely connected with affine Schur functions, k-Schur functions, and the affine Stanley symmetric functions, which are related to reduced word decompositions in the affine symmetric group (see e.g. [14, 15]).

  The nilCoxeter algebra $\mathsf {U}_{N}$ has generators $u_i, 1\le i \le N-1$, which satisfy the same relations as they do in $\widehat{\mathsf {U}}_N$. It first appeared in work on the cohomology of flag varieties [3] and has played an essential role in studies on Schubert polynomials, Stanley symmetric functions, and the geometry of flag varieties (see for example [8, 11, 16, 17]). The definition of $\mathsf {U}_N$ was inspired by the divided difference operators $\partial _i$ on polynomials in variables $\mathbf {x} = \{x_1, \dots , x_{N}\}$ defined by

  $$\begin{aligned} \partial _i(f) = \frac{f(\mathbf x) - f(\sigma _i \mathbf {x})}{x_i - x_{i+1}}, \end{aligned}$$

  where the transposition $\sigma _i$ fixes all the variables except for $x_i$ and $x_{i+1}$, which it interchanges. The operators $\partial _i$ satisfy the nilCoxeter relations above, and applications of these relations enabled Fomin and Stanley [8] to recover known properties and establish new properties of Schubert polynomials.

  The algebra $\mathsf {U}_N$ belongs to a two-parameter family of algebras having generators $u_i$, $1 \le i \le N-1$, which satisfy the relations $u_i u_j = u_j u_i$ for $\vert i-j \vert > 1$ and $u_i u_{i+1}u_i = u_{i+1}u_i u_{i+1}$ for $1 \le i \le N-2$ from above, together with the relation $u_i^2= \alpha u_i + \beta$ for all i, where $\alpha , \beta$ are fixed parameters. In particular, the specialization $\alpha =\beta =0$ yields the nilCoxeter algebra; $\alpha =0$, $\beta =1$ gives the standard presentation of the group algebra of the symmetric group $\mathbb {k}\mathsf {S}_N$; and $\alpha = q-1, \beta = q$ gives the Hecke algebra $\mathsf {H}_N(q)$ of type $\textsf {A}$.

     Motivated by categorification results in [6], Khovanov [10] introduced restriction and induction functors $\mathsf {F}_D$ and $\mathsf {F}_{X}$ corresponding to the natural inclusion of algebras $\mathsf {U}_N \hookrightarrow \mathsf {U}_{N+1}$ on the direct sum $\mathcal C$ of the categories $\mathcal C_N$ of finite-dimensional $\mathsf {U}_N$-modules. These functors categorify the Weyl algebra of differential operators with polynomial coefficients in one variable and correspond to the Weyl algebra generators $\partial$ and x (derivative and multiplication by x), which satisfy the relation $\partial x - x\partial = 1$.

     Brichard [5] used a diagram calculus on cylinders to determine the dimension of the center of $\mathsf {U}_N$ and to describe a basis of the center for which the multiplication is trivial. In this diagram calculus on N strands, the generator $u_i$ corresponds to a crossing of the strands i and $i+1$. The nil relation $u_i^2=0$ is represented by demanding that any two strands may cross at most once; otherwise the diagram is identified with zero.

• 2. $\text {n}\widehat{\text {TL}}_{N}$ is a quotient of the negative part of the universal enveloping algebra of the affine Lie algebra .

  The negative part $U^-$ of the universal enveloping algebra U of the affine Lie algebra has generators $f_i$, $0 \le i \le N-1$, which satisfy the Serre relations

  $$\begin{aligned}&f_i^2f_{i+ 1} - 2 f_i f_{i+ 1}f_i + f_{i + 1}f_i^2 = 0\\&\qquad =f_{i+1}^2f_{i} - 2 f_{i+ 1} f_{i}f_{i+ 1}+ f_{i }f_{i+ 1}^2\ \text { and }\ f_i f_j = f_j f_i\text { for }i-j \ne \pm 1\text { mod }N \end{aligned}$$

  (all indices modulo N). Factoring $U^-$ by the ideal generated by the elements $f_i^2$, $0 \le i \le N-1$, gives $\text {n}\widehat{\text {TL}}_{N}$ whenever the characteristic of $\mathbb {k}$ is different from 2.

• 3. $\text {n}\widehat{\text {TL}}_{N}$ acts on the small quantum cohomology ring of the Grassmannian.

  As in [19, Sec. 2], (see also [12]), consider the cohomology ring $\text {H}^\bullet (\text {Gr}(k,N))$ with integer coefficients for the Grassmannian $\text {Gr}(k,N)$ of k-dimensional subspaces of $\mathbb {k}^N$. It has a basis given by the Schubert classes $[\Omega _\lambda ]$, where $\lambda$ runs over all partitions with k parts, the largest part having size $N-k$. By recording the k vertical and $N-k$ horizontal steps that identify the Young diagram of $\lambda$ inside the northwest corner of a $k \times (N-k)$ rectangle, such a partition corresponds to a (0, 1)-sequence of length N with k ones (resp. $N-k$ zeros) in the positions corresponding to the vertical (respectively horizontal) steps.

  As a ${\mathbb Z}[q]$-module for an indeterminate q, the quantum cohomology ring of the Grassmannian is given by $\text {qH}^\bullet (\text {Gr}(k,N))\ =\ {\mathbb Z}[q]\otimes _{\mathbb Z}\text {H}^\bullet (\text {Gr}(k,N))$ together with a q-multiplication. The $\text {n}\widehat{\text {TL}}_{N}$-action can be defined combinatorially on

  $$\begin{aligned} \text {qH}^\bullet (\text {Gr}(k,N))\ \cong \ span _{{\mathbb Z}[q]} \,\{(0,1)\text {-sequences of length }N\text { with }k\text { ones}\} \end{aligned}$$

  as described in the next item, and the multiplication of two Schubert classes $[\Omega _\lambda ]\cdot [\Omega _\mu ]$ is equal to $s_\lambda \cdot [\Omega _\mu ]$ where $s_\lambda$ is a certain Schur polynomial in the generators of $\text {n}\widehat{\text {TL}}_{N}$ as in [19, Cor. 8.3].

• 4. $\text {n}\widehat{\text {TL}}_{N}$ acts faithfully on fermionic particle configurations on a circle.

  This is the graphical representation from [12] (see also [19]), which we use in our description of the center of $\text {n}\widehat{\text {TL}}_{N}$. First, a (0, 1)-sequence with k ones is identified with a circular particle configuration having N positions, where the k particles are distributed at the position on the circle that corresponds to their position in the sequence, so that there is at most one particle at each position. On the space

  $$\begin{aligned} span _{\mathbb {k}[q]} \,\{\text {fermionic particle configurations of }k\text { particles on a circle with }N\text { positions}\}, \end{aligned}$$

  the generators $a_i$ of $\text {n}\widehat{\text {TL}}_{N}$ act by sending a particle lying at position i to position $i+1$. Additionally, the particle configuration is multiplied by $\pm q$ when applying $a_0$. The precise definition is given in Sect. 4, but here is a representative picture (Fig. 1).

• 5. $\text {n}\widehat{\text {TL}}_{N}$ appears as a subalgebra of the annihilation/creation algebra.

  The finite nilTemperley–Lieb algebra is a subalgebra of the Clifford algebra having generators $\{\xi _i,\xi _i^{*}\ |\ 0\le i\le N-1\}$ and relations $\xi _i\xi _j+\xi _j\xi _i=0$, $\xi _i^{*}\xi _j^{*}+\xi _j^{*}\xi _i^{*}=0$, $\xi _i\xi _j^{*}+\xi _j^{*}\xi _i=\delta _{ij}$. The Clifford generators $\xi _i$ (resp. $\xi _i^{*}$) act on the fermionic particle configurations by annihilation (resp. creation) of a particle at position i. The finite nilTemperley–Lieb algebra appears inside the Clifford algebra via $a_i\mapsto \xi _{i+1}^{*}\xi _i$. As discussed in [12, Sec. 8], the affine nilTemperley–Lieb algebra is a q-deformation of this construction.

• 6. $\text {n}\widehat{\text {TL}}_{N}$ is the associated graded algebra of the affine Temperley–Lieb algebra.

  The affine Temperley–Lieb algebra $\widehat{ TL }_{N}(\delta )$ has the usual commuting relations and the relations $a_ia_{i\pm 1}a_i=a_i$ and $a_i^2=\delta a_i$ for some parameter $\delta \in \mathbb {k}$ instead of the nil relations (where again all indices are mod N). It is a filtered algebra with its $\ell$th filtration space generated by all monomials of length $\le \ell$. Since its associated graded algebra is $\text {n}\widehat{\text {TL}}_{N}$ for any value of $\delta$, elements of $\text {n}\widehat{\text {TL}}_{N}$ can be identified with reduced expressions in $\widehat{ TL }_{N}(\delta )$.

  The diagrammatic structure of $\widehat{ TL }_{N}(\delta )$ is given by the same pictures as for the Temperley–Lieb algebra, but now the diagrams are wrapped around the cylinder (see e.g. [7, 13]). The top and bottom of the cylinder each have N nodes. Monomials in the affine Temperley–Lieb algebra are represented by diagrams of N non-crossing strands, each connecting a pair of those 2N nodes. Multiplication of two monomials is realized by stacking the cylinders one on top of the other, and connecting and smoothing the strands. Whenever the strands form a circle, this is removed from the diagram at the expense of multiplying by the parameter $\delta$. The relation $a_ia_{i\pm 1}a_i=a_i$ corresponds to the isotopy between a strand that changes direction and a strand that is pulled straight.

  In contrast, this diagrammatic realization for the affine nilTemperley–Lieb algebra would not respect isotopy: The relation $a_ia_{i\pm 1}a_i=0$ implies that strands which change the direction are identified with zero. Nevertheless, the diagram of a reduced expression in $\widehat{ TL }_{N}$ may be considered as an element of $\text {n}\widehat{\text {TL}}_{N}$. Such a diagram consists of a number (possibly 0) of arcs that connect two nodes on the top of the cylinder, the same number of arcs connecting two nodes on the bottom, and arcs that connect a top node and a bottom one. The latter arcs wrap around the cylinder either all in a strictly clockwise direction or all in a strictly counterclockwise way. Since the multiplication of two such diagrams may give zero, we will not use this diagrammatic realization here.

Fig. 1

$N=8$: Application of $a_3a_2a_5$ to the particle configuration (0, 1, 2, 5) gives (0, 1, 4, 6)

We proceed as follows: In Sect. 2, we introduce the notation used in this article. The ${\mathbb Z}^N$-grading of $\text {n}\widehat{\text {TL}}_{N}$ is given is Sect. 3, and its importance for the description of the center is discussed. In Sect. 4, we give a detailed definition of the $\text {n}\widehat{\text {TL}}_{N}$-action on particle configurations on a circle. We also define special monomials that serve as the projections onto a single particle configuration (up to multiplication by $\pm q$). Theorem 4.5 of that section recalls [12, Prop. 9.1] stating that the representation is faithful. In [12], this fact is deduced from the finite nilTemperley–Lieb algebra case, as treated in [4] and [2, Prop. 2.4.1]. We give a complete, self-contained proof in Sect. 8. Our proof is elementary and relies on the construction of a basis. Section 5 contains the main result (Theorem 5.5) of this article:

Theorem

The center of $\text {n}\widehat{\text {TL}}_{N}$ is the subalgebra

where the generator is the sum of monomials corresponding to particle configurations given by increasing sequences of length k. The monomial sends particle configurations with $n \ne k$ particles to 0 and acts on a particle configuration with k particles by projecting onto and multiplying by $(-1)^{k-1}q$. Hence, $\mathbf t _k$ acts as multiplication by q on the configurations with k particles.

Our $N-1$ central generators $\mathbf t _k$ are essentially the $N-1$ central elements constructed by Postnikov. Lemma 9.4 of [19] gives an alternative description of $\mathbf t _k$ as product of the kth elementary symmetric polynomial (with factors cyclically ordered) with the $(N-k)$th complete homogeneous symmetric polynomial (with factors reverse cyclically ordered) in the noncommuting generators of $\text {n}\widehat{\text {TL}}_{N}$. The above theorem shows that in fact these elements generate the entire center of $\text {n}\widehat{\text {TL}}_{N}$. In Sect. 6, we establish that $\text {n}\widehat{\text {TL}}_{N}$ is finitely generated over its center. In Sect. 7, we define a monomial basis for $\text {n}\widehat{\text {TL}}_{N}$ indexed by pairs of particle configurations together with a natural number indicating how often the particles have been moved around the circle. A proof that this is indeed a basis of $\text {n}\widehat{\text {TL}}_{N}$ can be found in Sect. 8. With this basis at hand, we obtain inclusions $\text {n}\widehat{\text {TL}}_{N}\subset \text {n}\widehat{\text {TL}}_{N+1}$. The inclusions are not as obvious as those for the nilCoxeter algebra having underlying Coxeter graph of type $\textsf {A}_{N-1}$, since one cannot deduce them from embeddings of the affine Coxeter graphs. Our result, Theorem 7.1, reads as follows:

Theorem

For all $0\le m \le N-1$, there are unital algebra embeddings $\varepsilon _m :\ \text {n}\widehat{\text {TL}}_{N}\rightarrow \text {n}\widehat{\text {TL}}_{N+1}$ given by

$$\begin{aligned} a_i \mapsto a_i\ \text { for } \, 0\le i\le m-1,\qquad a_{m}\ \mapsto \ a_{m+1} a_{m},\quad a_i&\mapsto \ a_{i+1} \ \text { for} \, m+1\le i\le N-1. \end{aligned}$$

In Sect. 8, we show how to construct the monomial basis, namely by using a normal form algorithm that reorders the factors of a nonzero monomial. Our basis is reminiscent of the Jones normal form for reduced expressions of monomials in the Temperley–Lieb algebra, as discussed in [20], and is characterised in Theorem 8.6 as follows: (See also Theorem 7.5 which gives a different description.)

Theorem

(Normal form) Every nonzero monomial in the generators $a_j$ of $\text {n}\widehat{\text {TL}}_{N}$ can be rewritten uniquely in the form

$$\begin{aligned} (a^{(m)}_{i_1}\ldots a^{(m)}_{i_k})\ldots (a^{(n+1)}_{i_1}\ldots a^{(n+1)}_{i_k})(a^{(n)}_{i_1}\ldots a^{(n)}_{i_k})\ldots (a^{(1)}_{i_1}\ldots a^{(1)}_{i_k})(a_{i_1}\ldots a_{i_k}) \end{aligned}$$

with $a^{(n)}_{i_\ell }\in \{1,a_0,a_1,\ldots ,a_{N-1}\}\text { for all }1\le n\le m,\ 1\le \ell \le k$, such that

$$\begin{aligned} a^{(n+1)}_{i_\ell }\in {\left\{ \begin{array}{ll}\{1\}&{}\quad \text {if }\quad a^{(n)}_{i_\ell }=1,\ \\ \{1,a_{j+1}\}&{}\quad \text {if }\quad a^{(n)}_{i_\ell }=a_j.\end{array}\right. } \end{aligned}$$

The factors $a_{i_1},\ldots ,a_{i_k}$ are determined by the property that the generator $a_{i_\ell -1}$ does not appear to the right of $a_{i_\ell }$ in the original presentation of the monomial. Alternatively, every nonzero monomial is uniquely determined by the following data from its action on the graphical representation:

• the input particle configuration with the minimal number of particles on which it acts nontrivially,

• the output particle configuration,

• the power of q by which it acts.

For the proof of this result, we recall a characterisation of the nonzero monomials in $\text {n}\widehat{\text {TL}}_{N}$ from [9]. Then we prove faithfulness of the graphical representation of $\text {n}\widehat{\text {TL}}_{N}$ by describing explicitly the matrices representing our basis elements. Al Harbat [1] has recently described a normal form for fully commutative elements of the affine Temperley–Lieb algebra, which gives a different normal form when passing to $\text {n}\widehat{\text {TL}}_{N}$.

Our results hold over an arbitrary ground field $\mathbb {k}$, even one of characteristic 2, simply by ignoring signs in that case. In fact, our arguments work for any associative commutative unital ground ring R by replacing $\mathbb {k}$-vector spaces and $\mathbb {k}$-algebras with free R-modules and R-algebras, respectively. In particular, the affine nilTemperley–Lieb algebra over $\mathbb {k}$ is replaced by the R-algebra with the same generators and relations, and the polynomial ring $\mathbb {k}[q]$ is replaced by R[q]. We can even drop the assumption that the ring R is commutative if we slightly modify the statements about the center. This is possible because our arguments mainly rely on investigating monomials in the generators of $\text {n}\widehat{\text {TL}}_{N}$. However, for simplicity we have chosen to assume $\mathbb {k}$ is a field throughout the article.

2 Notation

Let $\mathbb {k}$ be any field, and assume N is a positive integer. The affine nilTemperley–Lieb algebra $\text {n}\widehat{\text {TL}}_{N}$ of rank N is the unital associative $\mathbb {k}$-algebra generated by elements $a_0,\ldots ,a_{N-1}$ subject to the defining relations

where all indices are taken modulo N, so in particular $a_{N-1}a_0a_{N-1}=a_0a_{N-1}a_0=0$. The finite nilTemperley–Lieb algebra $n TL _{N}$, as defined in [19], is the subalgebra of $\text {n}\widehat{\text {TL}}_{N}$ generated by $a_1,\ldots ,a_{N-1}$ (or in fact, by any $N-1$ of the generators $a_i$). We adopt the convention that $n TL _{1} =\mathbb {k}1$. We fix the following notation for monomials in $\text {n}\widehat{\text {TL}}_{N}$ and $n TL _{N}$: For an ordered index sequence with $0\le j_1,\ldots ,j_m\le N-1$, we define the ordered monomial . Unless otherwise specified, we use the letters i, j for indices from ${\mathbb Z}/N{\mathbb Z}$; in particular, we often identify the indices 0 and N.

Throughout we will assume $N\ge 3$.

3 Gradings

One of the ingredients needed in Sect. 5 to study the center of $\text {n}\widehat{\text {TL}}_{N}$ is a grading on the algebra.

Gradings faciliate the computation of the center of an algebra, as the following standard result reduces the work to determining homogeneous central elements.

Lemma 3.1

If $A=\bigoplus _{g\in G} A_g$ is an algebra graded by some abelian group G, then the center of A is homogeneous, i.e. it inherits the grading.

Proof

Let $a=\sum _{g\in G}a_g$ be a central element of the graded algebra $A=\bigoplus _{g\in G} A_g$. We have for $b_h\in A_h$ that

$$\begin{aligned} \sum \limits _{g\in G} a_g b_h = a b_h = b_h a= \sum \limits _{g\in G} b_h a_g. \end{aligned}$$

Since this equality must hold in every graded component, we get $a_g b_h = b_h a_g$ for all homogeneous elements $b_h$. Now take any element $b=\sum \limits _{h\in G}b_h$ in A, then

$$\begin{aligned} a_g b = \sum \limits _{h\in G}a_g b_h = \sum \limits _{h\in G}b_h a_g= b a_g, \end{aligned}$$

hence $a_g$ is central. $\square$

Since the defining relations are homogeneous, both $\text {n}\widehat{\text {TL}}_{N}$ and $n TL _{N}$ have a ${\mathbb Z}$-grading by the length of a monomial, i.e. all generators $a_i$ have ${\mathbb Z}$-degree 1. This can be refined to a ${\mathbb Z}^N$-grading by assigning to the generator $a_i$ the degree $\zeta _i$, the ith standard basis vector in ${\mathbb Z}^N$. In either grading, we say that the degree 0 part of an element in $\text {n}\widehat{\text {TL}}_{N}$ or $n TL _{N}$ is its constant term.

The ${\mathbb Z}^N$-grading is finer than the ${\mathbb Z}$-grading in the sense that any ${\mathbb Z}$-graded component of degree different from 0 decomposes into a sum of ${\mathbb Z}^N$-graded components of strictly smaller dimension.

Remark 3.2

Why do we exclude the case of $N\le 2$ from our considerations? For $N = 1,2$, there are isomorphisms $\text {n}\widehat{\text {TL}}_{N}\cong n TL _{N+1}$, and in these cases the center is uninteresting. The algebra $\text {n}\widehat{\text {TL}}_{1}$ is 2-dimensional and commutative; while $\text {n}\widehat{\text {TL}}_{2}$ has dimension 5, and its center can be computed by hand making use of Lemma 3.1 and can be shown to be the $\mathbb {k}$-span of $1, a_0a_1, a_1a_0$.

Remark 3.3

The affine (or finite) Temperley–Lieb algebra, which has relations $a_i a_j = a_j a_i$ for $i-j \ne \pm 1\ (\text {mod }N)$, $a_ia_{i\pm 1}a_i=a_i$, and $a_i^2=\delta a_i$ for some $\delta \in \mathbb {k}$, is a filtered algebra with respect to the length filtration. For this algebra, the $\ell$th filtration space is generated by all monomials of length ${\le }\ell$. Its associated graded algebra is $\text {n}\widehat{\text {TL}}_{N}$ (or $n TL _{N}$). Thus, $\text {n}\widehat{\text {TL}}_{N}$ is infinite dimensional when $N \ge 3$, while $n TL _{N}$ has dimension equal to the Nth Catalan number $\frac{1}{N+1}{{2N} \atopwithdelims ()N}$.

4 A faithful representation

The second ingredient we use to determine the center is a faithful representation of $\text {n}\widehat{\text {TL}}_{N}$. Here we recall the definition of the representation from [12] and describe its graphical realization, which is very convenient to work with.

Fix a basis $v_1,\ldots , v_N$ of $\mathbb {k}^N$. Consider the vector space $\textsf {V}=\bigoplus \limits _{k=0}^N\left( \mathbb {k}[q] \otimes \bigwedge ^k\mathbb {k}^N\right)$. It has a standard $\mathbb {k}[q]$-basis consisting of wedges

for all $0\le k\le N$, where the basis element of $\mathbb {k}=\bigwedge ^0\mathbb {k}^N$ is denoted $v(\emptyset )$. Throughout the rest of the paper, all tensor products are taken over $\mathbb {k}$, and we omit the tensor symbol in $\mathbb {k}[q]$-linear combinations of wedges.

Remark 4.1

The indices of the vectors $v_j$ should be interpreted modulo N. We make no distinction between $v_0$ and $v_N$ and often use the two interchangeably.

It is helpful to visualize the basis elements as particle configurations having $0\le k\le N$ particles arranged on a circle with N positions, where there is at most one particle at each site, as pictured below for $N=8$ and $v(1,5,6)=v_1\wedge v_5\wedge v_6$ (Fig. 2). The vector $v(\emptyset )$ corresponds to the configuration with no particles. Then $\textsf {V}$ is the $\mathbb {k}[q]$-span of such circular particle configurations.

Fig. 2

The element $v_1\wedge v_5\wedge v_6$ in the graphical realization

There is an action of the affine nilTemperley–Lieb algebra $\text {n}\widehat{\text {TL}}_{N}$ defined on the basis vectors of $\textsf {V}$ as follows:

Definition 4.2

For $1\le j\le N-1$,

For the action of $a_0$, note that $v_N$ appears in the basis element if and only if it occurs in the last position, i.e. $v_{i_k}=v_N$, and define

The sign appears in because of the equality

$$\begin{aligned} q\cdot v_{i_1}\wedge \cdots \wedge v_{i_{k-1}}\wedge v_{1}\ =\ (-1)^{k-1}q\cdot v_1\wedge v_{i_1}\wedge \cdots \wedge v_{i_{k-1}}. \end{aligned}$$

Remark 4.3

It follows that if the sequence contains or if it does not contain j. In other words, acts by replacing by . If this creates a wedge expression with two factors equal to , the result is zero. Thus, for any monomial there is a unique increasing sequence with k minimal on which the monomial acts nontrivially.

In the graphical description, $a_j$ moves a particle clockwise from position j to position $j+1$, and one records ‘passing position 0’ by multiplying by $\pm q$ as illustrated by the particle configurations in Fig. 3.

Fig. 3

Examples for the action of $\text {n}\widehat{\text {TL}}_{N}$ on a particle configuration. a $a_6(v_1\wedge v_5\wedge v_6)=v_1\wedge v_5\wedge v_7$, b $a_7a_1a_6(v_1\wedge v_5\wedge v_6)=v_2\wedge v_5\wedge v_0$, c $a_0(v_5\wedge v_0) = -q\cdot v_1\wedge v_5$

It is easy to verify that the defining relations for $\text {n}\widehat{\text {TL}}_{N}$ hold for this action, assuming that $N\ge 3$. Hence we obtain

Lemma 4.4

• (a) Definition 4.2 gives a representation of $\text {n}\widehat{\text {TL}}_{N}$ on $\textsf {V}$.

• (b) The number of wedges (i.e., the number of particles) remains constant under the action of the generators $a_i$, so that $\textsf {V}={\bigoplus _{k=0}^N\left( \mathbb {k}[q] \otimes \bigwedge ^k\mathbb {k}^N\right) }$ is a direct sum decomposition of $\textsf {V}$ as an $\text {n}\widehat{\text {TL}}_{N}$-module.

The following crucial statement is taken from [2, Prop. 2.4.1] and [12, Prop. 9.1.(2)]. We will give a detailed proof adapted to our notation in Sect. 8.

Theorem 4.5

The action from Definition 4.2 gives a faithful representation of $\text {n}\widehat{\text {TL}}_{N}$ on $\textsf {V}$ when $N \ge 3$.

Fig. 4

The action of $a(\widehat{1\ 5\ 6})$ on the particle configuration $v_1\wedge v_5\wedge v_6$

From now on, we will identify elements of $\text {n}\widehat{\text {TL}}_{N}$ with their action on the particle configurations of the graphical representation.

Remark 4.6

The spaces $\mathbb {k}[q]\otimes \bigwedge ^0\mathbb {k}^N$ and $\mathbb {k}[q] \otimes \bigwedge ^N\mathbb {k}^N$ are trivial summands in $\textsf {V}$ on which every generator $a_i$ acts as 0, and so they may be ignored when proving Theorem 4.5.

For a standard basis element of $1 \le k \le N-1$ wedges corresponding to an increasing sequence , the next lemma defines a certain monomial that projects and sends to zero for . Before stating the result, we give an example to demonstrate in the graphical description how this projector will be defined.

Example 4.7

Let $N=8$, and consider the particle configuration With $a(\widehat{1\ 5\ 6})\ =\ (a_0 a_7)\cdot (a_4 a_3 a_2)\cdot (a_1 a_5 a_6)$ we obtain $a(\widehat{1\ 5\ 6})\cdot \ v_1\wedge v_5\wedge v_6\ =\ (-1)^2 q\cdot v_1\wedge v_5\wedge v_6$, which looks as follows in the graphical description (Fig. 4).

The factor $a_1a_5a_6$ moves every particle one step forward clockwise. It is critical that we start by moving the particle at position 6 before moving the particle at position 5, as otherwise the result would be zero. But since there is a ‘gap’ at position 7, we can move the particle from site 6 to 7, and afterwards the particle from site 5 to 6, without obtaining zero. The assumption that $k<N$ ensures such a gap always exists.

After applying $a_1a_5a_6$, the particles are at positions 2, 6, and 7. The particle previously at position 5 is now at position 6, which is where we want a particle to be. The particle currently at position 2 can be moved to position 5 by applying the product $a_4 a_3 a_2$. The particle now at position 7 can be moved by $a_0 a_7$ to position 1. Hence, the result of applying $(a_0 a_7)\cdot (a_4 a_3 a_2) \cdot (a_1 a_5 a_6)$ is the same particle configuration as the original one. However, the answer must be multiplied by $\pm q$, since applying $a_0 a_7$ involves crossing the zero position once. To determine the sign, note from Definition 4.2 that $(a_0 a_7) \cdot (a_4 a_3 a_2) \cdot (a_1 a_5 a_6)(v_1 \wedge v_5 \wedge v_6) = q \cdot v_5 \wedge v_6 \wedge v_1 = (-1)^2 q \cdot v_1 \wedge v_5 \wedge v_6$, so the sign is $+$.

Now we describe the general procedure:

Lemma 4.8

Assume is a particle configuration, where is an increasing sequence and $1 \le k \le N-1$. Then there exists an index $\ell$ such that $i_\ell +1<i_{\ell +1}$ (or $i_{k}+1<i_1$), i.e. the sequence has a ‘gap’ between $i_\ell$ and $i_{\ell +1}$. Split the sequence into the two parts $\{i_1<\cdots < i_\ell \}$ and $\{i_{\ell +1}<\cdots <i_k\}$. Set

(*)

where the indices are modulo N in the factor $(a_{i_{1}-1}a_{i_{1}-2}\ldots a_{i_k+2}a_{i_k+1})$. Then

and has -degree $(1,1,\ldots ,1)$.

Proof

The assertions can be seen using the graphical realization of $\textsf {V}$. The terms in the second line of equation (*) move a particle at site one step forward to $i_j+1$ for each j, while the terms in the first line send the particle from $i_j+1$ to the original position of $i_{j+1}$.

Consider first By applying , every particle is first moved clockwise by one position. By our choice of the index $i_\ell$, we avoid mapping the whole particle configuration to zero. After that step, every particle is moved by one of the factors $(a_{i_{s+1}-1}a_{i_{s+1}-2}\ldots a_{i_s+2}a_{i_s+1})$ to the original position of its successor in the sequence , so the particle configuration remains the same. One of the particles has passed the zero position, so we have to multiply by $\pm q$. Definition 4.2 tells us the appropriate sign is $(-1)^{k-1}$.

Now consider for The monomial $(a_{i_{\ell +1}}a_{i_{\ell +2}}\ldots a_{i_{k-1}}a_{i_{k}})\cdot (a_{i_1}a_{i_{2}}\ldots a_{i_{\ell -1}}a_{i_\ell })$ expects a particle at each of the sites $i_1,\ldots , i_k$, so if any of these positions is empty in , the result of applying is zero. If the positions are already filled, and there is an additional particle somewhere, multiplication by $(a_{i_{\ell +1}-1}a_{i_{\ell +1}-2}\ldots a_{i_\ell +2}a_{i_\ell +1})$ will cause two particles to be at the same position, hence the result is again zero.

Since every $a_j$ appears in exactly once, the monomial has -degree $(1,1, \ldots , 1)$. $\square$

Example 4.9

In the previous example, $N = 8$, , and we may assume the two subsequences are (1) and (5, 6). Then the terms in the second line of (*) are $(a_5 a_6)\cdot (a_1) = a_1 a_5 a_6$. The term corresponding to $j = 1$ in the product on the first line of (*) is $a_4 a_3 a_2$, and the expression corresponding to $j=2$ is empty, hence taken to be 1. The first factor on the first line is $a_0 a_7$. Thus, for , , as in Example 4.7. If the gap between 6 and 0 is used instead, the right-hand factor of the second line is and the left-hand factor is 1. The factors in the first line remain the same, and so one obtains the same expression for .

Remark 4.10

Because is a faithful module, is, as an element in (i.e. up to reordering according to the defining relations), uniquely determined by the increasing sequence One can read off from as follows: In the defining equation (*) of the factors in the first line are pairwise commuting. The underlying subsequence corresponding to the factor $a_{i_{s+1}-1}a_{i_{s+1}-2}\ldots a_{i_s+2}a_{i_s+1}$ of is a decreasing sequence. After all such decreasing sequences are removed from what remains is a product of generators with an increasing subsequence of indices or a product of two such subsequences corresponding to the factors in the second line. This is . Given any monomial of ${\mathbb Z}^N$-degree $(1,\ldots ,1)$, one can rewrite it using the relations in so that it is of the form for some increasing sequence . Then is the unique standard basis element upon which acts by multiplication by $\pm q$.

5 Description of the center

In this section, we give an explicit description of the center ${\textsf {C}}_N$ of $\text {n}\widehat{\text {TL}}_{N}$. We start with the following initial characterisation of the central elements:

Lemma 5.1

Any central element c in $\text {n}\widehat{\text {TL}}_{N}$ with constant term 0 is a linear combination of monomials where every generator $a_i$, $0\le i\le N-1$, appears at least once. In particular, a homogeneous nonconstant central element c has ${\mathbb Z}$-degree at least N.

Proof

Assume , where for all . By Lemma 3.1, we can assume c is a homogeneous central element with respect to the ${\mathbb Z}^N$-grading. By our assumption, $c\notin \mathbb {k}$. For all i, we need to show that $a_i$ occurs in each monomial appearing in c. Without loss of generality, we show this for $i=0$. If some summand is missing $a_0$, then no summand contains $a_0$ because c is homogeneous. Hence and for all with , and since $a_0 c= c a_0$, none of the can contain the factor $a_1$ either, as otherwise the factor $a_0$ cannot pass through c from left to right (so also $a_{N-1}$ cannot be contained in the ). Proceeding inductively, we see that all must be a constant, contrary to our assumption. $\square$

The next proposition states that on the standard wedge basis vector of $\textsf {V}$, any central element acts via multiplication by a polynomial $p_k\in \mathbb {k}[q]$ that only depends on the length of the increasing sequence . In other words, the decomposition of $\textsf {V}$ into the summands $\mathbb {k}[q] \otimes \bigwedge ^k\mathbb {k}^N$ is a decomposition with respect to different central characters (apart from the two trivial summands for $k\in \{0,N\}$).

Proposition 5.2

For any central element $c\in \text {n}\widehat{\text {TL}}_{N}$ and all increasing sequences with fixed length k, there is some element $p_k\in \mathbb {k}[q]$ such that .

Proof

We may assume c is a nonconstant ${\mathbb Z}^N$-homogeneous central element of $\text {n}\widehat{\text {TL}}_{N}$. For $k\in \{0,N\}$, the action of a generator $a_i$ on a monomial of length k is 0, so $p_k=0$ for such values of k. Now consider $1\le k\le N-1$, and suppose that is an increasing sequence of length k. According to Lemma 4.4 (b), the number of wedges in a vector remains constant under the action of the $a_i$. Hence for some polynomials . We want to prove that for all .

We have shown in Lemma 4.8 that to each increasing sequence there corresponds a monomial that allows us to select a single basis vector:

Thus, for we see that

implying for . Hence, we may assume for each increasing sequence that for some polynomial .

Now it is left to show that for all with . It is enough to verify this for , that differ in exactly one entry, i.e. $i_s=i$, $i_s^{\prime }=i+1$, and $i_\ell =i_\ell ^{\prime }$ for all $\ell \ne s$, for some $1\le s\le k$ and $i\in {\mathbb Z}/N{\mathbb Z}$. If $1\le i\le N-1$, we have

and if $i=0$, we get

Hence, , and this common polynomial is the desired polynomial $p_k$. $\square$

Corollary 5.3

Any central element in $\text {n}\widehat{\text {TL}}_{N}$ with constant term 0 acts on a standard basis vector as multiplication by an element of $q\mathbb {k}[q]$.

Proof

According to Lemma 5.1, each summand of such a central element must contain the factor $a_0$, and $a_0$ acts on a wedge product by 0 or multiplication by $\pm q$. $\square$

Now we are ready to introduce nontrivial central elements in $\text {n}\widehat{\text {TL}}_{N}$. For each $1\le k\le N-1$, set

(1)

where the monomials correspond to increasing sequences of length k as defined in Lemma 4.8.

Example 5.4

In $\text {n}\widehat{\text {TL}}_{3}$:

$$\begin{aligned} \mathbf t _1&=\ a_2 a_1 a_0 + a_0 a_2 a_1 + a_1 a_0 a_2,\\ \mathbf t _2&=\ -a_0a_1a_2 - a_1a_2a_0 - a_2 a_0 a_1. \end{aligned}$$

In $\text {n}\widehat{\text {TL}}_{4}$:

$$\begin{aligned} \mathbf t _1&=\ a_3a_2 a_1 a_0 + a_0 a_3a_2 a_1 + a_1 a_0a_3 a_2 + a_2 a_1 a_0 a_3,\\ \mathbf t _2&=\ - a_0a_3a_1a_2 -a_0a_2a_1a_3 - a_3a_2a_0a_1 - a_1a_0a_2a_3 - a_1a_3a_0a_2 - a_2a_1a_3a_0 \\ \mathbf t _3&=\ a_0a_1a_2a_3 + a_1a_2a_3a_0 + a_2a_3 a_0 a_1 + a_3a_0a_1 a_2. \end{aligned}$$

In the graphical realization of $\textsf {V}$, $\mathbf t _k$ acts by annihilating all particle configurations whose number of particles is different from k. For particle configurations having k particles, every particle is moved clockwise to the original site of the next particle. Hence, the particle configuration itself remains fixed by the action of $\mathbf t _k$ (and it is multiplied with $(-1)^{2(k-1)}q =q$, since a particle has been moved through position 0). All the $\mathbf t _k$ have ${\mathbb Z}^N$-degree equal to $(1,\ldots ,1)$ and ${\mathbb Z}$-degree equal to N. Any monomial whose ${\mathbb Z}^N$-degree is $(1,\ldots ,1)$ occurs as a summand in some central element (after possibly reordering the factors), and the number of summands of $\mathbf t _k$ equals $\left( {\begin{array}{c}N\\ k\end{array}}\right) = dim (\bigwedge ^k\mathbb {k}^N)$; see Remark 4.10.

Theorem 5.5

• 1. The $\mathbf t _k$ are central for all $1\le k\le N-1$, and the center of $\text {n}\widehat{\text {TL}}_{N}$ is generated by 1 and the $\mathbf t _k$, $1\le k\le N-1$.

• 2. The subalgebra generated by $\mathbf t _k$ is isomorphic to the polynomial ring $\mathbb {k}[q]$ for all $1\le k\le N-1$. Moreover $\mathbf t _k\mathbf t _\ell =0$ for all $k\ne \ell$. Hence the center of $\text {n}\widehat{\text {TL}}_{N}$ is the subalgebra

  $$\begin{aligned} {\textsf {C}}_N\ =\ \mathbb {k}\oplus \mathbf t _1\mathbb {k}[\mathbf t _1]\oplus \cdots \oplus \mathbf t _{N-1}\mathbb {k}[\mathbf t _{N-1}]\ \cong \ \frac{\mathbb {k}[\mathbf t _1,\ldots ,\mathbf t _{N-1}]}{(\mathbf t _k\mathbf t _\ell \mid k\ne \ell )}. \end{aligned}$$

Proof

• 1. The action of $\mathbf t _k$ on $\textsf {V}$ is the projection onto the $\text {n}\widehat{\text {TL}}_{N}$-submodule $\mathbb {k}[q] \otimes \bigwedge ^k\mathbb {k}^N$ followed by multiplication by q. This commutes with the action of every other element of $\text {n}\widehat{\text {TL}}_{N}$. Since $\textsf {V}$ is a faithful module, $\mathbf t _k$ commutes with any element of $\text {n}\widehat{\text {TL}}_{N}$. As we have seen in Proposition 5.2, any central element c without constant term acts on the summand $\mathbb {k}[q] \otimes \bigwedge ^k\mathbb {k}^N$ via multiplication by some polynomial $p^c_{k}\in q\mathbb {k}[q]$. Once again using the faithfulness of $\textsf {V}$, we get that $c = \sum _{k=1}^{N-1} p_k^c(\mathbf t _k)$.

• 2. Recall that $\mathbb {k}[q] \otimes \bigwedge ^k\mathbb {k}^N$ is a free $\mathbb {k}[q]$-module of rank ${N \atopwithdelims ()k}$. Since $\mathbf t _k$ acts by multiplication with q on that module, the subalgebra of $\text {n}\widehat{\text {TL}}_{N}$ generated by $\mathbf t _k$ must be isomorphic to the polynomial ring $\mathbb {k}[q]$. Since for all we get $\mathbf t _k \mathbf t _\ell =0$ for $k \ne \ell$, as they consist of pairwise distinct summands.$\square$

Theorem 5.5 enables us to describe the $\mathbb {k}$-algebra $End _{\text {n}\widehat{\text {TL}}_{N}}(\textsf {W})$ of $\text {n}\widehat{\text {TL}}_{N}$-endomorphisms of the space of nontrivial particle configurations $\textsf {W}:=\bigoplus _{k=1}^{N-1}\left( \mathbb {k}[q] \otimes \bigwedge ^k\mathbb {k}^N\right) \subset \ \textsf {V}$. We first observe that on $\textsf {W}$ multiplication by q is given by the action of a central element in ${\textsf {C}}_N$, therefore it is justified to speak about $\mathbb {k}[q]$-linearity of a $\text {n}\widehat{\text {TL}}_{N}$-endomorphism of $\textsf {W}$.

Lemma 5.6

$End _{\text {n}\widehat{\text {TL}}_{N}}(\textsf {W})\subset End _{\mathbb {k}[q]}(\textsf {W})$, hence any $\text {n}\widehat{\text {TL}}_{N}$-module endomorphism $\varphi$ of $\textsf {W}$ is $\mathbb {k}[q]$-linear.

Proof

Observe that $\sum _{k=1}^{N-1}\mathbf t _k\in \text {n}\widehat{\text {TL}}_{N}$ acts by multiplication by q on every element in $\textsf {W}$. Therefore multiplication by q commutes with the application of every $\varphi \in End _{\text {n}\widehat{\text {TL}}_{N}}(\textsf {W})$.

$\square$

Proposition 5.7

The endomorphism algebra $End _{\text {n}\widehat{\text {TL}}_{N}}(\textsf {W})$ is isomorphic to a direct sum of $N-1$ polynomial algebras $\mathbb {k}[T_1]\oplus \cdots \oplus \mathbb {k}[T_{N-1}]$.

Proof

The proof is very similar to that of Proposition 5.2. First we show that is a $\mathbb {k}[q]$-linear multiple of for any $\varphi \in End _{\text {n}\widehat{\text {TL}}_{N}}(\textsf {W})$ and any increasing sequence . This statement holds if and only if . Indeed, by Lemmas 4.8 and 5.6 we get

Therefore, we can write for some polynomial . Note that this implies

$$\begin{aligned} End _{\text {n}\widehat{\text {TL}}_{N}}\left( \bigoplus \limits _{k=1}^{N-1}\Big (\mathbb {k}[q] \otimes \textstyle {\bigwedge ^k}\mathbb {k}^N\Big )\right) \ =\ \bigoplus \limits _{k=1}^{N-1}\left( End _{\text {n}\widehat{\text {TL}}_{N}}\Big (\mathbb {k}[q] \otimes \textstyle {\bigwedge ^k}\mathbb {k}^N\Big )\right) . \end{aligned}$$

What remains is to show that these polynomials only depend on the number of particles in , in other words there exists $p_k \in \mathbb {k}[q]$ so that for all with . Again it suffices to show this for two sequences , of length k that differ in exactly one entry. So say $i_s=i$, $i_s^{\prime }=i+1$, and $i_\ell =i_\ell ^{\prime }$ for all $\ell \ne s$, for some $1\le s\le k$ and $i\in {\mathbb Z}/N{\mathbb Z}$. When $1\le i\le N-1$,

and when $i = 0$,

Hence we can write $\varphi = \sum _{k=1}^{N-1} p_k \pi _k$ where $\pi _k$ is the projection onto $\mathbb {k}[q]\otimes \bigwedge ^k\mathbb {k}^N$, and we get that

$$\begin{aligned} End _{\text {n}\widehat{\text {TL}}_{N}}\left( \mathbb {k}[q] \otimes \textstyle { \bigwedge ^k}\mathbb {k}^N\right) \ =\ \mathbb {k}[T_k], \end{aligned}$$

where $T_k$ denotes the multiplication action of the central element $\mathbf t _k$, which is indeed a $\text {n}\widehat{\text {TL}}_{N}$-module endomorphism of $\textsf {W}$. Thus, $End _{\text {n}\widehat{\text {TL}}_{N}}(\textsf {W})$ is isomorphic to a direct sum of polynomial algebras as claimed. $\square$

Remark 5.8

The arguments in the proof of Proposition 5.7 remain valid even if we specialize the indeterminate q to some element in $\mathbb {k}\setminus \{0\}$. In this case, we obtain that the summands $\bigwedge ^k\mathbb {k}^N$ are simple modules and $End _{\text {n}\widehat{\text {TL}}_{N}}\big (\bigoplus _{k=1}^{N-1}\bigwedge ^k\mathbb {k}^N\big )\ \cong \ \mathbb {k}^{N-1}$. For $q=0$, the situation is more complicated: If q is specialized to zero, the generator $a_0$ acts by zero on the module. The action of $\text {n}\widehat{\text {TL}}_{N}$ factorizes over $n TL _{N}$, and the module $\bigwedge ^k\mathbb {k}^N$ is no longer simple. Instead it has a one-dimensional head spanned by the particle configuration $v(1,\ldots ,k)$, and any endomorphism is given by choosing an image of this top configuration. It is always possible to map it to itself and to the one-dimensional socle spanned by $v(N-k,\ldots ,N)$, but in general there are more endomorphisms. For example, in $\bigwedge ^4\mathbb {k}^8$, the image of v(1, 2, 3, 4) may be any linear combination of v(1, 2, 3, 4), v(2, 3, 4, 8), v(3, 4, 7, 8), v(4, 6, 7, 8) and v(5, 6, 7, 8), so that $End _{\text {n}\widehat{\text {TL}}_{8}}\Big (\bigwedge ^4\mathbb {k}^8\Big )$ is 5-dimensional.

6 The affine nilTemperley–Lieb algebra is finitely generated over its center

The affine nilTemperley–Lieb algebra is infinite dimensional when $N \ge 3$; however, the following finiteness result holds:

Theorem 6.1

The algebra $\text {n}\widehat{\text {TL}}_{N}$ is finitely generated over its center.

Proof

Given an arbitrary monomial , we first factor it as in the following way: Take the minimal particle configuration on which the monomial acts nontrivially; see Remark 4.3. The monomial moves all of the particles by at least one step, because the particle configuration was assumed to be minimal. Using the faithfulness of the representation, we know that we may reorder the monomial so that first each particle is moved one step clockwise, and afterwards the remaining particle moves are carried out. Hence, we may choose some factorization , where is a sequence obtained by permuting $j_1, \ldots , j_k$ so that the particle at position $j_r$ is moved one step clockwise by the action of $a_{j_r}$ for all $1\le r\le k$. The remaining particle moves are carried out by .

In Sect. 8, this decomposition is explicitly constructed (not using the faithful representation).

Next, we want to find an expression of the form

where $a_\text {fin}$ is a monomial of some subalgebra ${}^i n TL _{N}$ of $\text {n}\widehat{\text {TL}}_{N}$, $\mathbf t _k^n$ is in the center of $\text {n}\widehat{\text {TL}}_{N}$, and is the above factor. Here

$$\begin{aligned} {}^i n TL _{N} = \langle a_0,\ldots ,a_{i-1},a_{i+1},\ldots ,a_{N-1}\rangle \end{aligned}$$

(2)

denotes a copy of the finite nilTemperley–Lieb algebra $n TL _{N}$ sitting in $\text {n}\widehat{\text {TL}}_{N}$. To accomplish this, we have to subdivide the action of on the particle configuration one more time. There are two cases:

• 1. There is an index i not appearing in . In this case, is an element of ${}^i n TL _{N}$ and we are done.

• 2. All indices appear at least $n\ge 1$ times in . Let us investigate the action of on the particle configuration , where . Note that is the minimal particle configuration for . Each of the particles in is moved by to the position of the next particle in the sequence , because there is no index missing (a missing index is equivalent to a particle being stopped before reaching the position of its successor), before possibly continuing to move along the circle. Again invoking the faithfulness of the representation, we can rewrite , with the monomial from Lemma 4.8. For maximal n, the remaining factor is an element of ${}^i n TL _{N}$ for some i. Observe that , which follows immediately from the definition of $\mathbf t _k$ and Lemma 4.8.

Therefore, we have shown that

where $n =0$ in the first case. Since there are only finitely many monomials in ${}^0 n TL _{N},{}^1 n TL _{N},\ldots ,{}^{N-1} n TL _{N}$ and only finitely many monomials such that every index $0,1,\ldots ,N-1$ occurs at most once in the sequence , the affine nilTemperley–Lieb algebra is indeed finitely generated over its center. $\square$

Remark 6.2

The affine nilTemperley–Lieb algebra is not free over its center (see [18]).

7 Embeddings of affine nilTemperley–Lieb algebras

In the proof of Theorem 6.1, we have used the N obvious embeddings of $n TL _{N}$ into $\text {n}\widehat{\text {TL}}_{N}$ coming from the N different embeddings of the Coxeter graph into . Next we construct N embeddings of $\text {n}\widehat{\text {TL}}_{N}$ into $\text {n}\widehat{\text {TL}}_{N+1}$. They correspond to the subdivision of an edge of by inserting a vertex on the edge to obtain .

Theorem 7.1

Let $N\ge 3$. For any number $0\le m \le N-1$, there is a unital embedding of algebras $\varepsilon _m:\ \text {n}\widehat{\text {TL}}_{N}\ \rightarrow \ \text {n}\widehat{\text {TL}}_{N+1}$ given by

$$\begin{aligned} a_i\ \mapsto \ {\left\{ \begin{array}{ll} a_i &{}\quad \text {for }\ 0\le i\le m-1,\\ a_{m+1} a_{m}&{}\quad \text {for }\ i=m,\\ a_{i+1}&{}\quad \text {for }\ m+1\le i\le N-1. \end{array}\right. } \end{aligned}$$

(3)

Lemma 7.2

For $N \ge 3$, the map $\varepsilon _m$ from $\text {n}\widehat{\text {TL}}_{N}$ to $\text {n}\widehat{\text {TL}}_{N+1}$ given by (3) is an algebra homomorphism.

Proof

Due to the circular nature of the relations, it suffices to check this for $\varepsilon _0$. This amounts to showing the following, since all the other relations are readily apparent. To avoid confusion, we indicate generators of $\text {n}\widehat{\text {TL}}_{N+1}$ in these calculations by $\tilde{a}_i$:

$$\begin{aligned}&(\tilde{a}_1\tilde{a}_0)(\tilde{a}_1\tilde{a}_0) = \tilde{a}_1(\tilde{a}_0\tilde{a}_1\tilde{a}_0) = 0, \quad \tilde{a}_2(\tilde{a}_1\tilde{a}_0)\tilde{a}_2 = (\tilde{a}_2\tilde{a}_1\tilde{a}_2)\tilde{a}_0 = 0, \quad \tilde{a}_N(\tilde{a}_1\tilde{a}_0)\tilde{a}_N = \tilde{a}_1(\tilde{a}_N\tilde{a}_0\tilde{a}_N) = 0,\\&(\tilde{a}_1\tilde{a}_0)\tilde{a}_2(\tilde{a}_1\tilde{a}_0) = (\tilde{a}_1 \tilde{a}_2)(\tilde{a}_0\tilde{a}_1\tilde{a}_0) = 0, \quad (\tilde{a}_1\tilde{a}_0)\tilde{a}_N(\tilde{a}_1\tilde{a}_0) = (\tilde{a}_1\tilde{a}_0\tilde{a}_1)(\tilde{a}_N\tilde{a}_0) = 0. \end{aligned}$$

Remark 7.3

How should one visualize the action of $\varepsilon _m(\text {n}\widehat{\text {TL}}_{N}) \subset \text {n}\widehat{\text {TL}}_{N+1}$ on the particle configurations on a circle with $N+1$ positions? Except for $a_m$, all generators of $\text {n}\widehat{\text {TL}}_{N}$ are mapped to corresponding generators of $\text {n}\widehat{\text {TL}}_{N+1}$. They will act as before, by moving a particle one step clockwise around the circle. Since $a_m$ is mapped by $\varepsilon _m$ to the product $\tilde{a}_{m+1}\tilde{a}_m$ in $\text {n}\widehat{\text {TL}}_{N+1}$, it will move a particle from m to $m+2$ as depicted in Fig. 5. In other words, the elements in $\varepsilon _m(\text {n}\widehat{\text {TL}}_{N})$ do not move a particle to or from position $m+1$.

Fig. 5

$\varepsilon _5(\text {n}\widehat{\text {TL}}_{7})\subset \text {n}\widehat{\text {TL}}_{8}$: The action of $\varepsilon _5(a_0a_6a_5a_4) =\tilde{a}_0\tilde{a}_7\tilde{a}_6\tilde{a_5}\tilde{a}_4$ on the particle configuration v(4)

Next we introduce a basis of $\text {n}\widehat{\text {TL}}_{N}$ that will enable us to see directly that these homomorphisms are embeddings. The basis has a simple description in terms of the graphical representation $\textsf {V}$ from Sect. 4. For any two particle configurations with $1\le k\le N-1$ particles corresponding to the increasing sequences and , there is a monomial in $\text {n}\widehat{\text {TL}}_{N}$ moving particles at the positions to the positions . We require that every particle from be moved by at least one step, but we do not prescribe explicitly which of the j’s is mapped to which of the i’s. For , take to be the monomial such that the power of q in is minimal (under the assumption that every particle from must be moved). By faithfulness of the graphical representation, is uniquely determined. For , we have , the special monomial defined in Sect. 4, hence Observe that one can write , where the sum runs over all possible increasing sequences of length k, and that is a monomial, since all but one summand vanish for .

Remark 7.4

The condition that move all particles from by at least one step guarantees that it acts as zero on all particle configurations with fewer particles than .

For example, when $N=7$,

$$\begin{aligned} e_{(2)(1)} = a_1,\quad e_{(0,2)(0,1)} = a_6a_5a_4a_3a_1a_2a_0a_1. \end{aligned}$$

(Note that $a_1$ moves v(0, 1) to v(0, 2), but this does not satisfy the requisite property that all the particles must be moved by at least one step.) If we apply the factorization of monomials from Theorem 6.1 to , the minimality condition implies that , where if , then is a sequence obtained by permuting the elements of .

Theorem 7.5

The set of monomials

defines a $\mathbb {k}$-basis of the affine nilTemperley–Lieb algebra $\text {n}\widehat{\text {TL}}_{N}$.

Proof

First, observe that is indeed a monomial since . We show that the elements act $\mathbb {k}$-linearly independently on the graphical representation $\textsf {V}=\bigoplus _{k=0}^N \left( \mathbb {k}[q] \otimes \bigwedge ^k \mathbb {k}^N\right)$. By Remark 7.4, the monomial acts by zero on summands $\mathbb {k}[q] \otimes \bigwedge ^{k^{\prime }} \mathbb {k}^N$ for . On , the matrix representing the action of relative to the standard basis has exactly one nonzero entry, and this one distinguishes all monomials with the same minimal number of particles . From these two observations, the linear independence follows. On the other hand, given any nonzero monomial in $\text {n}\widehat{\text {TL}}_{N}$, there exists a minimal particle configuration on which it acts nontrivially. Recording the image particle configuration and the power of q, we conclude that there is some $\ell$ so that the element acts on $\textsf {V}$ in the same way as the given monomial does. Due to the faithfulness of this representation (see Theorem 4.5), the proposition follows. $\square$

In Sect. 8, a basis is constructed using a different approach (without relying on the faithful representation). Both bases are labelled by pairs of particle configurations (pairs of increasing sequences) together with a natural number $\ell$. Up to an index shift in the output configuration and a shift of the natural number $\ell$, the labelling sets agree, and both bases actually coincide.

Proof

(Theorem 7.1) We have already proven in Lemma 7.2 that $\varepsilon _m$ is an algebra homomorphism. Using Remark 7.3, observe that the monomial is mapped to a monomial (tilde again indicates in $\text {n}\widehat{\text {TL}}_{N+1}$), where the new index sets are obtained by $i\mapsto i$ for $0\le i\le m$ and $i\mapsto {i+1}$ for $m+1\le i\le N-1$. The injectivity follows since basis elements of $\text {n}\widehat{\text {TL}}_{N}$ are mapped to basis elements of $\text {n}\widehat{\text {TL}}_{N+1}$. $\square$

Remark 7.6

It is possible to verify this theorem on generators and relations in the language of Sect. 8 without using the graphical description.

Remark 7.7

Observe that these embeddings work specifically for the affine nilTemperley–Lieb algebras but fail for the ordinary Temperley–Lieb algebras. The relation that fails to hold is the braid relation for Temperley–Lieb algebras, i.e. $a_i a_{i\pm 1}a_i = a_i$. Interestingly, the relation $a_i^2=\delta a_i$ is respected for $\delta =1$.

8 A normal form and the faithfulness of the graphical representation

In this section, we prove Theorem 4.5 which we recall here:

Proposition

For $N\ge 3$, $\textsf {V}$ is a faithful $\text {n}\widehat{\text {TL}}_{N}$-module with respect to the action described in Definition 4.2.

For the proof, we will explicitly prove the linear independence of the matrices representing the monomials in $\text {n}\widehat{\text {TL}}_{N}$. We proceed in three steps: (1) First, we define a normal form for the monomials. (2) Next, we find a bijection between the monomials and certain pairs of particle configurations together with a power of q. In other words, we find a basis for $\text {n}\widehat{\text {TL}}_{N}$ and describe a labeling set. (3) The final step is the description of the action of a monomial on $\textsf {V}$ using its matrix realization. The matrices representing the monomials have a distinguished nonzero entry that is given in terms of the particle configurations and the power of q from the bijection, and for most matrices, this is the only nonzero entry. From this description it will quickly follow that all these matrices are linearly independent.

8.1 Some useful facts

The following lemma characterises nonzero monomials in $\text {n}\widehat{\text {TL}}_{N}$. They correspond to fully commutative elements in $\widehat{ TL }_{N}$, see [9].

Lemma 8.1

The monomial if and only if for any two neighbouring appearances of $a_i$ in there are exactly one $a_{i+1}$ and one $a_{i-1}$ in between, apart from possible factors $a_\ell$ for $\ell \ne i-1,i,i+1$ (indices to be understood modulo N).

According to this result, two consecutive $a_i$ have to enclose $a_{i+1}$ and $a_{i-1}$, i.e. $a_i\ldots a_{i\pm 1}\ldots a_{i\mp 1}\ldots a_i$, with the dots being possible products of $a_\ell$’s with $\ell \ne i\pm 1,i$. This lemma is a special case of [9, Lem. 2.6]; here is a quick proof for the convenience of the reader.

Proof

The monomial is zero if and only if we can bring two neighbouring factors $a_i$ together so that we obtain either $a_i^2$ (‘square’) or $a_ia_{i\pm 1}a_i$ (‘braid’). But expressions of the form $a_i\ldots a_{i\pm 1}\ldots a_{i\mp 1}\ldots a_i$ cannot be resolved this way by commutativity relations. On the other hand, if there are two neighbouring factors $a_i$ with either none or only one of the terms $a_{i\pm 1}$ in between, we get after commutations either $a_i^2$ or $a_ia_{i\pm 1}a_i$. If there are at least two factors $a_{i+1}$ (or $a_{i-1}$) in between the two $a_i$, one can repeat the argument: Either we can create a square or a braid, or we have at least two factors of the same kind in between. In the case of a square or a braid we are done; otherwise we pick two neighbouring $a_{i+k}$ in the kth step of the argument. Since we always consider the space in between two neighbouring factors $a_i,a_{i+1},\ldots ,a_{i+k}$, none of the previous $a_i,a_{i+1},\ldots ,a_{i+k-1}$ occurs between the two neighbouring $a_{i+k}$. Unless we found a square or a braid in an earlier step, we end up in step $N-1$ with a subexpression of the form $a_{r}a_{r\pm 1}^ma_{r}$ which is zero for any $m\ge 0$. $\square$

Definition 8.2

For any $i\in \{0,1,\ldots ,N-1\}$, we define a (clockwise) order $\mathop {\prec }\limits ^{i}$ on the set $\{0,1,\ldots ,N-1\}$ starting at i by

$$\begin{aligned} i\,\mathop {\prec }\limits ^{i}\,i+1\,\mathop {\prec }\limits ^{i}\, \ldots \,\mathop {\prec }\limits ^{i} i +N-1. \end{aligned}$$

8.2 Step 1: A normal form

Given an arbitrary nonzero monomial in $\text {n}\widehat{\text {TL}}_{N}$, reorder its factors according to the following algorithm (as usual, the indices are considered modulo N):

• 1. Find all factors $a_i$ in with no $a_{i-1}$ to their right. We denote them by $a_{i_1},\ldots ,a_{i_k}$, ordered according to their appearance in ; in other words, is of the form

  

• 2. Move the $a_{i_1},\ldots ,a_{i_k}$ to the far right, without changing their internal order,

  

  for and some sequence . This is possible because

  • (a) by assumption, there is no $a_{i-1}$ to the right of an $a_i$ in this list;

  • (b) if for some i, $a_{i+1}$ occurs to the right of some $a_i$, then either $a_i\ldots a_{i+1}\ldots a_i$ would occur as a subword without $a_{i-1}$ in between, hence , or else $a_{i+1}$ does not have $a_i$ to its right, so it is one of the $a_{i_1},\ldots ,a_{i_k}$ itself, and will be moved to the far right of , too;

  • (c) $a_i$ commutes with all $a_\ell$ for $\ell \ne i-1, i+1$.

• 3. Repeat for until we get

  

  for sequences obtained successively the same way as described above. Notice:

  • Inside a sequence , every index occurs at most once. If two consecutive indices occur within , they are increasingly ordered using the order $\mathop {\prec }\limits ^{i_k}$ from Definition 8.2.

  • For two consecutive sequences , and for every index $i^{(n+1)}_r$ occurring in , we can find some index $i^{(n)}_s$ in such that $i^{(n+1)}_r= i^{(n)}_{s}+1$.

  • From that property, it also follows that the length of is less or equal than the length of .

• 4. Reorder the factors internally:

  • (a) Start with . There is some which does not occur in , but occurs. For example, this is satisfied by , as $i_k$ occurs in and is to the right of every other factor of . Choose the largest such (with respect to the usual order). Then we can move to the very right of the sequence , because is not present, and may only occur to the left of due to the construction of . We proceed in the same way with those indices that appear in . The result is a reordering of the sequence so that it is increasing from left to right with respect to .

  • (b) Repeat with all other factors taking as the initial right-hand index of the sequence respectively, and reordering within each so that the indices are increasing from left to right with respect to . Throughout, the index is the one from step (4a).

Example 8.3

As an example for $\text {n}\widehat{\text {TL}}_{7}$, suppose . (We omit the commas to simplify the notation.)

Find all $a_i$ without $a_{i-1}$ to their right:	$a(6\ 4\ 2\ 1\ 3\ 5\ 4\ 2\ 0\ 6\ \underline{\underline{1}}\ 3\ \underline{\underline{2}}\ \underline{\underline{5}})$
Move them to the far right, and do not change their internal order:	$a(6\ 4\ 2\ 1\ 3\ 5\ 4\ 2\ 0\ 6\ 3)\cdot a(1\ 2\ 5)$
Repeat:	$a(6\ 4\ 2\ 3\ 5\ 4\ 1\ \underline{\underline{2}}\ 0\ \underline{\underline{6}}\ \underline{\underline{3}})\cdot a(1\ 2\ 5)$
	$a(6\ 4\ 2\ 3\ 5\ 4\ 1\ 0)\cdot a(2\ 6\ 3)\cdot a(1\ 2\ 5)$
	$a(6\ 4\ 2\ \underline{\underline{3}}\ 5\ \underline{\underline{4}}\ 1\ \underline{\underline{0}})\cdot a(2\ 6\ 3)\cdot a(1\ 2\ 5)$
	$a(6\ 4\ 2\ 5\ 1)\cdot a(3\ 4\ 0)\cdot a(2\ 6\ 3)\cdot a(1\ 2\ 5)$
	$a(6\ \underline{4}\ 2\ \underline{\underline{5}}\ \underline{\underline{1}})\cdot a(3\ 4\ 0)\cdot a(2\ 6\ 3)\cdot a(1\ 2\ 5)$
	$a(6\ 2)\cdot a(4\ 5\ 1)\cdot a(3\ 4\ 0)\cdot a(2\ 6\ 3)\cdot a(1\ 2\ 5)$
With the right-hand indices of the , $n \ge 0$, arranged according to from left to right, reorder the factors in each increasingly with respect to from left to right:	$a(6\ 2)\cdot a(4\ 5\ 1)\cdot a(3\ 4\ 0)\cdot a(2\ 3\ 6)\cdot a(1\ 2\ 5)$

As a shorthand notation, in the following we often identify the index sequence with (and manipulate according to the same relations as ) as demonstrated in the following example.

Example 8.4

Let $N=6$.

Lemma 8.5

Let be a nonzero monomial in $\text {n}\widehat{\text {TL}}_{N}$, where we use as always the notation from Section 2. Let be the monomials constructed by the algorithm above.

• 1. The equality holds in $\text {n}\widehat{\text {TL}}_{N}$.

• 2. Given any two representatives , of the same element in $\text {n}\widehat{\text {TL}}_{N}$, the above algorithm creates the same representative for both and .

Proof

• 1. The algorithm never interchanges the order of two factors $a_i$, $a_{i\pm 1}$ with consecutive indices within . Hence, the reordering of the factors of uses only the commutativity relation $a_i a_j = a_j a_i$ for $i-j \ne \pm 1\text { mod }N$ of $\text {n}\widehat{\text {TL}}_{N}$.

• 2. Two monomials , in $\text {n}\widehat{\text {TL}}_{N}$ are equal if and only if they only differ by applications of commutativity relations $a_i a_j = a_j a_i$ for $i-j \ne \pm 1\text { mod }N$, hence, if and only if they contain the same number of factors $a_i$ for each i and the relative position of each $a_i$ and $a_{i\pm 1}$ is the same. Since the outcome of the algorithm depends only on the relative positions of consecutive indices, the resulting decomposition is the same.

$\square$

We have shown the following. In stating this result and subsequently, whenever we refer to monomials in normal form, we assume the monomial is nonzero and nonconstant, in particular the sequence is nonempty.

Theorem 8.6

Assume $N \ge 3$.

• 1. The algorithm in Step 1 above provides a normal form for nonzero monomials in the generators $a_i$ of $\text {n}\widehat{\text {TL}}_{N}$, or equivalently for nonzero fully commutative monomials in $\widehat{ TL }_{N}$, so that

  

  where $a^{(n)}_{i_\ell }\in \{1,a_0,a_1,\ldots ,a_{N-1}\}\text { for all }1\le n\le m,\ 1\le \ell \le k$, and

  $$\begin{aligned} a^{(n+1)}_{i_\ell }\in {\left\{ \begin{array}{ll}\{1\} &{}\quad \text {if }a^{(n)}_{i_\ell }=1,\\ \{1,a_{j+1}\} &{}\quad \text {if }a^{(n)}_{i_\ell }=a_j.\end{array}\right. } \end{aligned}$$

  The factors $a_{i_1},\ldots ,a_{i_k}$ are determined by the property that the generator $a_{i_\ell -1}$ does not appear to the right of $a_{i_\ell }$ in the original presentation of the monomial. The internal ordering of the factors is increasing with respect to the relation , as in Step (4a) of the normal form algorithm, where is the largest value in $\{0,1,\ldots , N-1\}$ such that , but .

• 2. The set is a $\mathbb {k}$-basis of $\text {n}\widehat{\text {TL}}_{N}$.

8.3 Step 2: Labelling of basis elements

Definition 8.7

Given in normal form, we call the $\ell$th block of , and a string of indices of maximal length of the form (modulo N) the sth strand of . We use the notation $[\ldots ,i_s+1,i_s]$ for the strands.

Example 8.8

Let $N=6$, and consider Example 8.4 once again, where

The blocks are , , , , , and . The strands are [3210], [54321], [105432] and [21054]. In particular, strands (and blocks) can have different lengths, but the longest strand has length $m=6$.

Each monomial determines two sets and an integer as follows:

These are well defined because, as in the proof of Lemma 8.5, any element of $\text {n}\widehat{\text {TL}}_{N}$ is uniquely determined by the number of factors $a_i$ and the relative position of each $a_i$ and $a_{i\pm 1}$, for all i. The set equals the underlying set of in the normal form from the algorithm above. All strands of begin with an element in and end with an element from .

The goal of this subsection is to show

Proposition 8.9

The mapping

(4)

is injective, where $\mathcal{P}_N$ is the power set of $\{0,1,\ldots ,N-1\}$.

Remark 8.10

The map $\psi$ is defined so that in the graphical description of the representation $\textsf {V}$ of $\text {n}\widehat{\text {TL}}_{N}$, the set equals the set of positions where expects a particle to be. The set equals the set of positions where moves the particles from , but each one is translated by 1, that is,

The map $\psi$ is far from being surjective. An obvious constraint is that , and furthermore, for some pairs , one can only obtain sufficiently large values .

To ease the presentation, we start by proving injectivity of the restriction $\psi _0$ of $\psi$ to those monomials in normal form whose first element $i_1$ of is 0. The proof itself will amount to counting indices.

Proposition 8.11

The map

is injective.

Before beginning the proof of this result, we note that for monomials with $i_1=0$, the inequality $i_k<N-1$ must hold in , since $i_1 = 0$ implies that $i_1-1=N-1$ is not an element of . Consequently, the ordering of the indices in agrees with the natural ordering of ${\mathbb Z}$, so we can regard as a subset of $({\mathbb Z},<)$ and replace the modular index sequence by an integral index sequence such that as follows:

Definition 8.12

Assume is a normal form sequence with and , where indices in are modulo N and $1\le k(n)\le k$ for all $1\le n\le m$. The integral normal form sequence for is

for $n = 1,\dots ,m$.

Example 8.13

We continue Example 8.4 with $N=6$.

Our proof of Proposition 8.11 will hinge upon the following technical lemma.

Lemma 8.14

Let be the integral normal form sequence for and let $[i_s,\ldots , i_s+n_s]$ for $s=1,\dots ,k$ be the strands of . Assume $i_1 = 0$. Then

• (a) $n_1 =i_1+n_1< i_2+n_2< \cdots < i_k+n_k;$

• (b) $i_k+n_k< i_1+n_1+N = n_1+N$.

We postpone the proof of this result and proceed directly to proving the proposition.

Proof

(Proposition 8.11) Since the sequence will be fixed throughout the proof, we will drop the subscript on . To show the injectivity of $\psi _0$, we consider the factorization $\psi _0=\gamma \circ \beta \circ \alpha$ given by

where and similar to the definition of . The map $\alpha$ replaces indices in ${\mathbb Z}/N{\mathbb Z}$ by indices in ${\mathbb Z}$ as in Definition 8.12 above. The map $\beta$ is given by reading off and from . The map $\gamma$ sends the pair to a triple consisting of the respective images modulo N of the pair and the integer $\ell =1+\sum \ell _r$ where $\ell _r=\lfloor \frac{j_r}{N}\rfloor$ for each . The summand 1 corresponds to $0=i_1$; all other occurrences of 0 are counted by $\sum \ell _r$. Now we check injectivity.

The map $\alpha$ is clearly injective since is a left inverse map.

To see that $\beta$ is injective, we need to know that can be uniquely reconstructed from . Observe that is determined by knowing all the ‘strands’ $i_s, i_s+1, i_s+2,\ldots , i_s+n_s$ for $1\le s\le k$, hence by assigning an element to each . But it follows from Lemma 8.14 (a) that $i_1+n_1$ must be the smallest element of , $i_2+n_2$ the second smallest, etc., so that the element $i_s+n_s$ is assigned to the sth element in , that is, to $i_s$.

Now to see that $\gamma$ is injective, we need to recover in a unique way from . Write , and set . By Lemma 8.14 (a), we know that is of the form $(i_1+n_1<\cdots <i_k+n_k)$, and since the elements of have to be equal to the elements of modulo N, we can write $i_r+n_r=N\ell _r+d_r$ for $\ell _r=\lfloor \frac{i_r+n_r}{N}\rfloor$ and some . Comparing $\ell _r$ and $\ell _s$ for $r<s$, we have

$$\begin{aligned} N\ell _r\ \le \ N \ell _r+d_r\ =\ i_r+n_r\ <\ i_s+n_s\ =\ N\ell _s+d_s\ \le \ N(\ell _s+1). \end{aligned}$$

So $\ell _r< \ell _s+1$, i.e. $\ell _r\le \ell _s$. Similarly, we obtain from (b) of Lemma 8.14 that $\ell _k\le \ell _1+1$.

As a result,

$$\begin{aligned} N \ell _k\ \le \ N\, \ell _k+d_k \ =\ i_k+n_k\ <\ i_1+n_1+N\ =\ N(\ell _1+1)+d_1\ \le \ N(\ell _1+2), \end{aligned}$$

i.e. $\ell _k<\ell _1+2$. Together we have $\ell _1=\cdots =\ell _s <\ell _{s+1}=\cdots =\ell _1+1$ for some $1<s\le k$ (where we treat the case $s=k$ by $\ell _1=\cdots =\ell _k$). Set $\tilde{\ell }{:=}\ell _1$. Then

$$\begin{aligned} i_r + n_r&= N\,\tilde{\ell }+d_r\qquad \qquad \text {for }1\le r\le s,\\ i_r + n_r&= N(\tilde{\ell }+1)+d_s \quad \text {for }s+1\le r\le k. \end{aligned}$$

As a first consequence,

$$\begin{aligned} \ell \ =\ 1+\sum _r \ell _r\ =\ 1+k \tilde{\ell }+(k-s), \end{aligned}$$

which determines $\tilde{\ell }= \lfloor \frac{\ell -1}{k}\rfloor$, and hence all $\ell _r$, as well as the index s. Using Lemma 8.14, we determine that

$$\begin{aligned} i_{s+1}+n_{s+1}< \cdots< i_k+n_k< i_1+n_1+N< \cdots < i_s+n_s+N, \end{aligned}$$

and so

$$\begin{aligned} N\,(\tilde{\ell }+1)+d_{s+1}< \cdots< N\,(\tilde{\ell }+1)+d_k< N\,(\tilde{\ell }+1)+d_1< \cdots < N\,(\tilde{\ell }+1)+d_s. \end{aligned}$$

Therefore, $d_{s+1}<\cdots<d_k<d_1<\cdots <d_s$, which fixes the choice of $d_r$ for all r. We conclude that given , we can reconstruct by setting $i_r+n_r{:=}N\, \ell _r+d_r$. This completes the proof of Proposition 8.11. $\square$

Proof

(Lemma 8.14) (a) Let be a nonempty integral normal form sequence with $0=i_1<\cdots <i_k\le N-1$ and strands $[i_r,\ldots ,i_r+n_r]$ for $1\le r\le k$. Assume that there is some index $1\le t\le k-1$ such that $i_t+n_t\ \ge \ i_{t+1}+n_{t+1}$. Since $i_t<i_{t+1}$, we have $n_t> n_{t+1}$. So

From $i_t +n_{t+1}\ <\ i_{t+1}+n_{t+1}\ \le \ i_t+n_t$ it follows that there is some integer $n_{t+1}<p\le n_t$ such that $i_{t+1}+n_{t+1}=i_t+p$ appears in the strand $[i_t,\ldots ,i_t+n_t]$, i.e.

with $i_t+p=i_{t+1}+n_{t+1}$. But by the definition of the strands, there is no $i_{t+1}+n_{t+1}+1$ appearing to the left of $i_{t+1}+n_{t+1}$. Due to Lemma 8.1, we know that (even modulo N) there is no repetition of $i_{t+1}+n_{t+1}$ to the left. Thus $i_t+p=i_{t+1}+n_{t+1}$ is not possible, and we obtain $i_1+n_1< i_2+n_2< \cdots < i_k+n_k$.

For (b) of Lemma 8.14, assume $i_k+n_k\ge i_1+n_1+N$. It is true generally that $N>i_k$, so we get $i_k+n_k \ge i_1+n_1+N>i_k+n_1$. Hence $i_1+n_1+N=i_k+b$ for some $n_1<b\le n_k$, i.e. $i_1+n_1+N$ appears in the strand $[i_k,\ldots , i_k+n_k]$ and we have

Here it may be that the $n_k$th bracket and the bth bracket coincide, but in any case, we find that $i_k+b\ =\ i_1+n_1+N\ = \ i_1+n_1\text { mod } N$, and so $i_k+b$ appears to the left of $i_1+n_1$. By the definition of the strands, there is no $i_1+n_1+1$ to the left of $i_1+n_1$, and from Lemma 8.1 we deduce that in there is no $i_1+n_1\text { mod }N$ to the left of $i_1+n_1$ allowed, which leads to a contradiction. Hence $i_k+n_k < i_1+n_1+N$ must hold. $\square$

Having established that $\psi$ is injective when restricted to sequences with $i_1 = 0$, we now show the injectivity of $\psi$ in general.

Proof

(Proposition 8.9) We have the following disjoint decompositions according to the smallest value $i_1$ in :

By Proposition 8.11, the map restricted to those with $i_1=0$ is injective. We argue next that by an index shift this result is true for all other $\psi _i$.

Now it follows from Proposition 8.11 that the map

is injective, where counts the occurences of $N-i$ in . Recall that

Now observe that we can obtain from as

which follows from a computation using and

$$\begin{aligned} \widehat{\ell _r}&= \text { the number of} ~N-i~\text {in the }r\text {th strand } \ [i_r,\ldots ,i_r+n_r]\text { mod }N \\&=\ {\left\{ \begin{array}{ll}\lfloor \frac{i_r+n_r+i}{N}\rfloor &{}\quad \text {if }i_r\le N-i\\ \lfloor \frac{i_r+n_r+i}{N}\rfloor -1 &{}\quad \text {if }i_r> N-i\end{array}\right. }\\&=\ {\left\{ \begin{array}{ll}\lfloor \frac{N \ell _r+d_r+i}{N}\rfloor &{}\quad \text {if }i_r\le N-i\\ \lfloor \frac{N \ell _r+d_r+i}{N}\rfloor -1 &{}\quad \text {if }i_r> N-i\end{array}\right. }\\&=\ {\left\{ \begin{array}{ll}\ell _r+1 &{}\quad \text {if }i_r\le N-i\text { and } d_r+i\ge N\\ \ell _r &{}\quad \text {if }i_r\le N-i \text { and } d_r +i< N\\ \ell _r &{}\quad \text {if }i_r> N-i\text { and }d_r+i\ge N\\ \ell _r-1 &{}\quad \text {if }i_r> N-i \text { and } d_r+i< N.\end{array}\right. } \end{aligned}$$

We obtain $\psi _i$ by first shifting the indices of by subtracting i from each index, , then applying $\widehat{\psi }_0$, and finally shifting the indices from and by adding i to each. Hence, $\psi _i$ is injective for each i, and $\psi$ is injective because the unions are disjoint. $\square$

8.4 Step 3: Description and linear independence of the matrices

Recall that the standard $\mathbb {k}$-basis of the representation $\textsf {V}=\bigoplus _{k=0}^N\left( \mathbb {k}[q] \otimes \bigwedge ^k\mathbb {k}^N\right)$ is given by

$$\begin{aligned} \{ q^\ell \cdot v_{i_1}\wedge \cdots \wedge v_{i_k}\mid \ell \in {\mathbb Z}_{\ge 0},\ 1\le i_1<\cdots <i_k\le N\} \end{aligned}$$

where $(i_1,\ldots ,i_k)$ is identified with the particle configuration having particles in those positions in the graphical description. Now we describe with respect to this basis the matrix representing a nonzero monomial as a $2^N\times 2^N$-matrix with entries in $\mathbb {k}[q]$. Since $\textsf {V}$ decomposes as a $\text {n}\widehat{\text {TL}}_{N}$-module into submodules $\mathbb {k}[q] \otimes \bigwedge ^k\mathbb {k}^N$ for $k = 0,1,\ldots , N$, the matrix of is block diagonal with $N+1$ blocks $A_0,A_1,\ldots ,A_N$, where $A_0=A_N=(0)$ corresponding to the trivial representation.

The block $A_k$ is a $\left( {\begin{array}{c}N\\ k\end{array}}\right) \times \left( {\begin{array}{c}N\\ k\end{array}}\right)$-matrix, with entries from $\mathbb {k}[q]$ indexed by all possible particle configurations whose number of particles equal to k.

Now fix a nonzero monomial in normal form that is specified by the triple defined in Step 2. Let . All blocks $A_1,\ldots ,A_{k-1}$ are zero since expects at least k particles. For $r>k$ there might be nonzero blocks $A_r$. Such nonzero blocks appear unless the particles from are moved around the whole circle with no position left out, in which case there are no surplus particles allowed. This occurs if contains at least every other generator $a_i,a_{i+2},\ldots$.

More importantly, the block $A_k$ has precisely one nonzero entry, and this is given by

From this we see first that all matrices representing monomials in normal form with are $\mathbb {k}$-linearly independent: They have only one nonzero entry which is equal to at position . Furthermore, if all matrices representing monomials in normal form with are $\mathbb {k}$-linearly independent, then also all matrices representing monomials in normal form with are $\mathbb {k}$-linearly independent. This follows because the additional monomials with have nonzero entries in the $(k-1)$th block which is zero for all with . So by induction, all matrices representing monomials in normal form are $\mathbb {k}$-linearly independent. Since all of them have a zero entry in the upper left (and lower right) corner, we may add the identity matrix to the linearly independent set of matrices, and it remains linearly independent. So the representation of $\text {n}\widehat{\text {TL}}_{N}$ on $\textsf {V}$ is faithful, because according to Theorem 8.6, is a $\mathbb {k}$-basis of .

Section 8 has given a normal form for each monomial and has provided an alternate proof of the faithfulness of the representation of by elementary arguments.