Scholarly article on topic 'The center of the affine nilTemperley–Lieb algebra'

The center of the affine nilTemperley–Lieb algebra Academic research paper on "Physical sciences"

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Mathematische Zeitschrift
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Academic research paper on topic "The center of the affine nilTemperley–Lieb algebra"

Math. Z.

DOI 10.1007/s00209-016-1660-7

Mathematische Zeitschrift

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The center of the affine nilTemperley-Lieb algebra

Georgia Benkart1 • Joanna Meinel2

Received: 30 June 2015 / Accepted: 12 March 2016

© The Author(s) 2016. This article is published with open access at

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 nTLn over any ground field. Only two tools are used: a fine grading on nTLN and a representation of nTLn 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 nTLN 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

The authors thank Catharina Stroppel for many helpful discussions and express their gratitude to the Mathematical Sciences Research Institute (MSRI) where their joint research began. The authors thank the referee for very detailed remarks. The second author would like to thank Daniel Tubbenhauer for remarks on the embeddings. This work is part of the second author's Ph.D. project at the MPIM Bonn, supported by a grant of the Deutsche Telekom Stiftung.

B Joanna Meinel

Georgia Benkart

Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706, USA Mathematical Institute, University of Bonn, 53115 Bonn, Germany

Published online: 21 April 2016

1 Springer

nTL^ into nTL^+i 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 k, the affine nilTemperley-Lieb algebra nTLN is the unital associative Ik-algebra given by N generators ao,..., aN-i and nil relations af = 0 and aiai±iai = 0 for all i. Generators that are far apart commute, i.e. ajaj = ajaj for i — j = ±1 mod N. In these relations, the indices are interpreted modulo N so that the generators ao and aN—i are neighbours that do not commute. The subalgebra of nTLN generated by ai,..., aN—i is the (finite) nilTemperley-Lieb algebra nTLN, as in [i9]. The affine nilTemperley-Lieb algebra appears in many different settings, which we describe next.

i. nTLN is a quotient of the affine nilCoxeter algebra of type A n—i.

The affine nilCoxeter algebra Un of type An—i over a field k is the unital associative algebra generated by elements ui, 0 < i < N — i, satisfying the relations u2 = 0; UjUj = UjUi for i — j = ±i mod N; and uiui+iui = ui+iuiui+i for i < i < N — i, where the subscripts are read modulo N. The algebra nTLN is isomorphic to the quotient of Un obtained by imposing the additional relations uiui+iu = ui+iuiui+i = 0 for i < i < N — i. 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. [i4, i5]). The nilCoxeter algebra Un has generators ui, i < i < N — i, which satisfy the same relations as they do in Un .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, ii, i6, i7]). The definition of Un was inspired by the divided difference operators di on polynomials in variables x ={xi,..., xn } defined by

f (x) — f (ai x)

di (f) = -:

xi — xi + i

where the transposition ai fixes all the variables except for xi and xi+i, which it interchanges. The operators di 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 Un belongs to a two-parameter family of algebras having generators ui, i < i < N — i, which satisfy the relations uiuj = ujui for |i — j | > i and uiui+iui = ui+iuiui+i for i < i < N — 2 from above, together with the relation uf = aui + fi for all i, where a, fi are fixed parameters. In particular, the specialization a = fi = 0 yields the nilCoxeter algebra; a = 0, fi = i gives the standard presentation of the group algebra of the symmetric group kSn; and a = q — i,fi = q gives the Hecke algebra Hn (q) of type A.

Motivated by categorification results in [6], Khovanov [i0] introduced restriction and induction functors Fd and Fx corresponding to the natural inclusion of algebras U n Un+i on the direct sum C of the categories Cn of finite-dimensional Un-modules. These functors categorify the Weyl algebra of differential operators with polynomial coefficients in one variable and correspond to the Weyl algebra generators d and x (derivative and multiplication by x), which satisfy the relation dx — xd = i.

Brichard [5] used a diagram calculus on cylinders to determine the dimension of the center of Un and to describe a basis of the center for which the multiplication is trivial. In this diagram calculus on N strands, the generator ui corresponds to a crossing of the

strands i and i + 1. The nil relation uj = 0 is represented by demanding that any two strands may cross at most once; otherwise the diagram is identified with zero.

2. nTLN is a quotient of the negative part of the universal enveloping algebra of the affine Lie algebra s(n.

The negative part U— of the universal enveloping algebra U of the affine Lie algebra s(n has generators f ,0 < i < N — 1, which satisfy the Serre relations

fi fi+1 — 2fif+1 f + f+1 f2 = 0

= f+ fi — 2fi+1 ff+1 + f,f,2+1 and ff = fjfi for i — j = ±1 mod N

(all indices modulo N). Factoring U— by the ideal generated by the elements f2, 0 < i < N — 1, gives nTLN whenever the characteristic of k is different from 2.

3. nTLN acts on the small quantum cohomology ring of the Grassmannian.

As in [19, Sec. 2], (see also [12]), consider the cohomology ring H*(Gr(k, N)) with integer coefficients for the Grassmannian Gr(k, N) of k-dimensional subspaces of kN. It has a basis given by the Schubert classes [Ял], where X 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 X inside the northwest corner of a k x (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 Z[q]-module for an indeterminate q, the quantum cohomology ring of the Grassmannian is given by qH*(Gr(k, N)) = Z[q] H*(Gr(k, N)) together with a q-multiplication. The nTLN-action can be defined combinatorially on

qH*(Gr(k, N)) = spanZ[q] {(0, 1)-sequences of length N with k ones]

as described in the next item, and the multiplication of two Schubert classes [Ях] • [Я^] is equal to sX • [Я^] where sX is a certain Schur polynomial in the generators of nTLN as in[19, Cor. 8.3].

4. nTLN 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 nTLN. 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

span^[q] {fermionic particle configurations of k particles on a circle with N positions],

the generators ai of nTLN act by sending a particle lying at position i to position i + 1. Additionally, the particle configuration is multiplied by ±q when applying a0. The precise definition is given in Sect. 4, but here is a representative picture (Fig. 1).

5. nTLN appears as a subalgebra of the annihilation/creation algebra.

The finite nilTemperley-Lieb algebra is a subalgebra of the Clifford algebra having generators fâ | 0 < i < N — 1] and relations ^tÇj + fy& = 0, Ç*+ Ç*= 0, ÇiÇ* + Ç*Çi = Sij. The Clifford generators Ç (resp. Ç*) 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 ai ^ Ç^Çi. As discussed in [12, Sec. 8], the affine nilTemperley-Lieb algebra is a q-deformation of this construction.

6. nTLN is the associated graded algebra of the affine Temperley-Lieb algebra.

2 }•

•t 6

Fig. 1 N = 8: Application of a^a^a^ to the particle configuration (0, 1, 2, 5) gives (0, 1, 4, 6)

The affine Temperley-Lieb algebra TLn (S) has the usual commuting relations and the relations atat±iat = at and a2 = Sai for some parameter S e k instead of the nil relations (where again all indices are mod N). It is a filtered algebra with its Ith filtration space generated by all monomials of length < I. Since its associated graded algebra is nTLN for any value of S, elements of nTLN can be identified with reduced expressions in TLn (S).

The diagrammatic structure of TLn (S) 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 S. The relation at at±iat = at 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 atat±iat = 0 implies that strands which change the direction are identified with zero. Nevertheless, the diagram of a reduced expression in TLn may be considered as an element of nTLN. 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.

We proceed as follows: In Sect. 2, we introduce the notation used in this article. The ZN-grading of nTLN 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 nTLN-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 ±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 nTLN is the subalgebra


Cn = Cent(nTLN) = <1,t!,...,tN-1) = / , . N .. ,

(tktt | k = I)

where the generator tk = (— 1)k—1 X |i|=k a(i) is the sum of monomials a(l) corresponding to particle configurations given by increasing sequences i ={1 < ii < ••• < ik < N} of length k. The monomial a(î) sends particle configurations with n = k particles to 0 and acts on a particle configuration with k particles by projecting onto i and multiplying by (—1)k—1q. Hence, tk acts as multiplication by q on the configurations with k particles.

Our N — 1 central generators tk are essentially the N — 1 central elements constructed by Postnikov. Lemma 9.4 of [19] gives an alternative description of tk 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 nTLN. The above theorem shows that in fact these elements generate the entire center of nTLN. In Sect. 6, we establish that nTLN is finitely generated over its center. In Sect. 7, we define a monomial basis for nTLN 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 nTLN can be found in Sect. 8. With this basis at hand, we obtain inclusions nTLN C nTLN+1. The inclusions are not as obvious as those for the nilCoxeter algebra U n having underlying Coxeter graph of type An—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 < m < N — 1, there are unital algebra embeddings em : nTLN ^ nTLN+1 given by

ai ^ ai for 0 < i < m — 1, am ^ am+1am, ai ^ ai+1 form + 1 < i < N — 1.

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 aj of nTLN can be rewritten uniquely in the form

Of ... a™)... (a(n+1) ... a^n ... a")... (a« ... a^a, ... a,) with a(n) e {1, ao, a1,..., on—1} for all 1 < n < m, 1 < i < k, such that

(n+1) a G

m if an = 1.

{1, aj+1} if a\n) = aj.

The factors at1,..., atk are determtned by the property that the generator at— does not appear to the right of at t tn the origmal presentation of the monomtal. Alternatively, every nonzero monomtal Is untquely determtned by thefollowtng data from tts action on the graph-tcal representation:

• the tnput particle configuration wtth the mtntmal number of particles on whtch tt acts nontrivtally,

• the output parttcle configuratton,

• the power of q by whtch tt acts.

For the proof of this result, we recall a characterisation of the nonzero monomials in nTL^ from [9]. Then we prove faithfulness of the graphical representation of nTL^ 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 nTL^.

Our results hold over an arbitrary ground field 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 k-vector spaces and k-algebras with free R-modules and R-algebras, respectively. In particular, the affine nilTemperley-Lieb algebra over k is replaced by the R-algebra with the same generators and relations, and the polynomial ring 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 nTL^. However, for simplicity we have chosen to assume k is a field throughout the article.

2 Notation

Let k be any field, and assume N is a positive integer. The affine nilTemperley-Lieb algebra nTLw of rank N is the unital associative k-algebra generated by elements a0,...,aN -1 subject to the defining relations

af = 0 for all 0 < i < N — 1,

ajaj = ajaj for all i — j = ±1 mod N,

aiai+1ai = ai+1aiai+1 = 0 for all 0 < i < N — 1,

where all indices are taken modulo N, so in particular aN—1a0aN—1 = a0aN—a = 0. The finite nilTemperley-Lieb algebra nTLN, as defined in [19], is the subalgebra of nTL^ generated by a1,..., aN—1 (or in fact, by any N — 1 of the generators ai). We adopt the convention that nTL1 = k1. We fix the following notation for monomials in nTLN and nTLN: For an ordered index sequence j = (j1,..., jm) with 0 < j1,..., jm < N — 1, we define the ordered monomial a(j) = aj1 ... ajm. Unless otherwise specified, we use the letters i, j for indices from Z/N Z; in particular, we often identify the indices 0 and N. Throughout we will assume N > 3.

3 Gradings

One of the ingredients needed in Sect. 5 to study the center of nTLN 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 = (J)geG Ag 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 = ^geG ag be a central element of the graded algebra A = (J)Ag. We have for bh e Ah that

agbh = abh = bha = ^ bhag.

geG geG

Since this equality must hold in every graded component, we get agbh = bhag for all homogeneous elements bh. Now take any element b = ^ bh in A, then

gb = ^ a.gbh = ^ bhag = bag

heG heG

hence is central.

Since the defining relations are homogeneous, both nTL^ and nTLN have a Z-grading by the length of a monomial, i.e. all generators ai have Z-degree 1. This can be refined to a ZN-grading by assigning to the generator ai the degree Zi, the ith standard basis vector in ZN. In either grading, we say that the degree 0 part of an element in nTLN or nTLN is its constant term.

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

Remark 3.2 Why do we exclude the case of N < 2 from our considerations? For N = 1, 2, there are isomorphisms nTLN = nTLN+1, and in these cases the center is uninteresting. The algebra nTLi is 2-dimensional and commutative; while nTL2 has dimension 5, and its center can be computed by hand making use of Lemma 3.1 and can be shown to be the Ik-span of 1 , a0 a1 , a1 a0 .

Remark 3.3 The affine (or finite) Temperley-Lieb algebra, which has relations aiaj = ajai for i — j =±1 (mod N), aiai±1ai = ai, and af = Sai for some S e k, is a filtered algebra with respect to the length filtration. For this algebra, the Ith filtration space is generated by all monomials of length <£. Its associated graded algebra is nTLN (or nTLN). Thus, nTLN is infinite dimensional when N > 3, while nTLN has dimension equal to the N th Catalan

number n+tcno .

4 A faithful representation

The second ingredient we use to determine the center is a faithful representation of nTLN • Here we recall the definition of the representation from [12] and describe its graphical realization, which is very convenient to work with.

Fixabasis V1,...,VN of kN. Consider the vector space V = @ (k[g ] ® /\k kN I. It has

k=0 ^ '

a standard k[q ]-basis consisting of wedges

v(i) := vi1 A ••• A vik for all (strictly) increasing sequences i ={1 < ¿1 < ••• < ik < N j

for all 0 < k < N, where the basis element of k = /\0 kN is denoted v(0). Throughout the rest of the paper, all tensor products are taken over k, and we omit the tensor symbol in k[g ]-linear combinations of wedges.

Remark 4.1 The indices of the vectors vj should be interpreted modulo N. We make no distinction between v0 and vn and often use the two interchangeably.

It is helpful to visualize the basis elements v(l) as particle configurations having 0 < k < 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) = v1 A v5 A v6 (Fig. 2). The vector v(0)

Fig. 2 The element v\ A V5 A vg in the graphical realization

•t 6

corresponds to the configuration with no particles. Then V is the k[q]-span of such circular particle configurations.

There is an action of the affine nilTemperley-Lieb algebra nTLN defined on the basis vectors v(l) of V as follows:

Definition 4.2 For 1 < j < N — 1,

aj v(i) =

Vi1 A ■■■ A vie-1 A vj+i A vie+1 A ■■■ A vik, if ii = j for some I, 0, otherwise.

For the action of ao, note that vn appears in the basis element v(i) if and only if it occurs in the last position, i.e. Vfk = vn, and define

ao v(i) =

(-1)k 1q ■ vi A vii A ■■■ A vik_i, if ik = N, 0, otherwise.

The sign appears in aov(l) because of the equality

q ■ vi1 A ■■■ A vik-1 A vi = (- 1)k-1q ■ vi A vi1 A ■■■ A vik-1.

Remark 4.3 It follows that aj v(i) = 0 if the sequence i contains j + 1 or if it does not contain j. In other words, aj acts by replacing vj by vj+1. If this creates a wedge expression with two factors equal to vj+1, the result is zero. Thus, for any monomial a(j) there is a unique increasing sequence j = {1 < ji < ••• < jk < N} with k minimal on which the monomial acts nontrivially.

In the graphical description, aj moves a particle clockwise from position j to position j + 1, and one records 'passing position 0' by multiplying by ±q as illustrated by the particle configurations in Fig. 3.

It is easy to verify that the defining relations for nTLN hold for this action, assuming that N > 3. Hence we obtain

Fig. 3 Examples for the action of nTLN °n a particle configuration. a ag(v! A v5 A vg) = v1 A v5 A v-j, b a7a1a6(v1 A v5 A vg) = v2 A v5 A vo, c ao(v5 A vo) = — q ■ v1 A v5

Lemma 4.4 (a) Definition 4.2 gives a representation of nTLN on V. (b) The number of wedges (i.e., the number of particles) remains constant under the action of the generators ai, so that V = 0 k==0 (k[q ] ® Ak kN) is a direct sum decomposition of Vas an nTLN-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 nTLN on V when N > 3.

From now on, we will identify elements of nTLN with their action on the particle configurations of the graphical representation.

Remark 4.6 The spaces k[q ] ® A0 kN and k[q ] ® AN kN are trivial summands in V on which every generator ai acts as 0, and so they may be ignored when proving Theorem 4.5.

For a standard basis element v(l) of 1 < k < N — 1 wedges corresponding to an increasing sequence i ={1 < i'1 < ••• < ik < N j, the next lemma defines a certain monomial a (i) that projects v(l) onto (— 1)k—1q v(l) and sends v(i') to zero for i' = i. 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 v(i) = v1 A v5 A v6. With a(15 6) = (a0a7)-(a4a3a2)-(a1a5a6) weobtaina(15 6) v1Av5Av6 = (—1)2qv Av5Av6, which looks as follows in the graphical description (Fig. 4).

The factor a1a5a6 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^a6, 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 a4a3a2. The particle now at position 7 can be moved by a0a7 to position 1. Hence, the result of applying (a0a7) ■ (a4a3a2) ■ (a^a6) is the same particle configuration as the original one. However, the answer must be multiplied by ±q, since applying a0a7 involves crossing the zero position once. To determine the sign, note from Definition 4.2 that (a0a7) ■ (a4a3a2) ■ (a1a5a6)(v1 A v5 A v6) = q ■ v5 A v6 A v1 = (— 1)2q ■ v1 A v5 A v6, so the sign is +.

Now we describe the general procedure:

Fig. 4 The action of a(l 5 6) on the particle configuration

v1 A v5 A v6

Lemma 4.8 Assume d(i) is a particle configuration, where i ={1 < ii < ••• < ik < N} is an increasing sequence and 1 < k < N — 1. Then there exists an index i such that ii + 1 < ¿¿+i (or ik + 1 < ii), i.e. the sequence has a 'gap' between ii and ii+1. Split the sequence i into the two parts {'1 < ••• < ii} and {ii+1 < ••• < ik}. Set

a(I) : = (ai 1 — 1 ai 1 —2 • • • aik+2aik + 1) • (a«s+1-1a«s+1-2 • • • ais +2ais + 1)

• (ai£+1 aii+2 • • • aik_1 aik) • (ai1 ai2 • • • aii-1 aii), (*)

where the indices are modulo N in the factor (ai1_1ai1_2 • • • aik+2aik+1). Then

a(Dvd') = f(_1)k—1q •V(i) if l =^

0 for all i' = i (of any length),

and a(i) has ZN-degree (1, 1).

Proof The assertions can be seen using the graphical realization of V. The terms in the second line of equation (*) move a particle at site ij e i one step forward to ij + 1 for each j, while the terms in the first line send the particle from ij + 1 to the original position of ij+1.

Consider first a^Ml^ By applying (ait+1 a^+2 • • • aik—1 aik) • (ai1 ai2 • • • aie—1 a^), every particle is first moved clockwise by one position. By our choice of the index ¡¿, we avoid mapping the whole particle configuration to zero. After that step, every particle is moved by one of the factors (ais+1_1ais+1 _2 • • • ais +2ais+1) to the original position of its successor in the sequence i, so the particle configuration remains the same. One of the particles has passed the zero position, so we have to multiply by ±q. Definition 4.2 tells us the appropriate sign is (_1)k_1.

Now consider a(i)v(l') for i' = l The monomial (aie+1 ai+2 • • • aik_1 aik) • (ai1 ai2 • • • aie_1 aif) expects a particle at each of the sites , so if any of these positions is empty

in v(i'), the result of applying a(i) is zero. If the positions i1,^^,ik are already filled, and there is an additional particle somewhere, multiplication by (aie+1_1aie+1_2 • • • aic+2aic+i) will cause two particles to be at the same position, hence the result is again zero.

Since every aj appears in a(i) exactly once, the monomial a(i) has ZN-degree (1, □

Example 4.9 In the previous example, N = 8, i = (1, 5, 6), and we may assume the two subsequences are (1) and (5, 6). Then the terms in the second line of (*) are (a5a6) • (a1) = a1a5a6. The term corresponding to j = 1 in the product on the first line of (*) is a4a3a2, and the expression corresponding to j = 2 is empty, hence taken to be 1. The first factor on the first line is aoaj. Thus, for i = (1, 5, 6), a(i) = (aoaj) • (a4a3a2) • (a^a6), as in Example 4.7. If the gap between 6 and 0 is used instead, the right-hand factor of the second line is a1a5a6 and the left-hand factor is 1. The factors in the first line remain the same, and so one obtains the same expression for a(i).

Remark 4.10 Because V is a faithful module, a(i) is, as an element in nTLN (i.e. up to reordering according to the defining relations), uniquely determined by the increasing sequence l One can read off i from a(i) as follows: In the defining equation (*) of a(i), the factors in the first line are pairwise commuting. The underlying subsequence

(i's+1 _1, i's+1 _2, •••, is+2, is +1) corresponding to the factor ais+1 _1ais+1_2 • • • ais +2ais + 1

of a(i) is a decreasing sequence. After all such decreasing sequences are removed from a(i), what remains is a product of generators a j with an increasing subsequence of indices or a

product of two such subsequences corresponding to the factors in the second line. This is i. Given any monomial a (r) of ZN-degree (1,..., 1), one can rewrite it using the relations in nTLN so that it is of the form a(l) for some increasing sequence i. Then v(i) is the unique standard basis element upon which a(r) = a(i) acts by multiplication by ±q.

5 Description of the center

In this section, we give an explicit description of the center Cn of nTLN. We start with the following initial characterisation of the central elements:

Lemma 5.1 Any central element c in nTLN with constant term 0 is a linear combination of

monomials a(j) = aj1.....ajm where every generator ai, 0 < i < N — 1, appears at least

once. In particular, a homogeneous nonconstant central element c has Z-degree at least N.

Proof Assume c = ^j cja(j), where cj e k for all j. By Lemma 3.1, we can assume c is

a homogeneous central element with respect to the ZN-grading. By our assumption, c e k. For all i , we need to show that ai occurs in each monomial a( j ) appearing in c. Without loss of generality, we show this for i = 0. If some summand is missing a0, then no summand contains a0 because c is homogeneous. Hence a0a(j) = 0 and a(j)a0 = 0 for all j with cj = 0, and since a0c = ca0, none of the a(j) can contain the factor a1 either, as otherwise the factor a0 cannot pass through c from left to right (so also aN—1 cannot be contained in the a( j)). Proceeding inductively, we see that all a( j) must be a constant, contrary to our assumption. □

The next proposition states that on the standard wedge basis vector v(i) of V, any central element acts via multiplication by a polynomial pk e k[q ] that only depends on the length k = |i| of the increasing sequence i = {1 < i'1 < ••• < ik < N j. In other words, the decomposition of V into the summands k[q] ® /\k kN is a decomposition with respect to different central characters (apart from the two trivial summands for k e {0, N j).

Proposition 5.2 For any central element c e nTLN and all increasing sequences i with fixed length k, there is some element pk e k[q] such that cv(I) = pk v(i).

Proof We may assume c is a nonconstant ZN-homogeneous central element of nTLN. For k e {0, N j, the action of a generator ai on a monomial of length k is 0, so pk = 0 for such values of k. Now consider 1 < k < N — 1, and suppose that i ={1 < i'1 < ••• < ik < N j 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 ai. Hence cv(I) = X|i'|=k ci' v(i') for some polynomials ci' e k[q]. We want to prove that ci' = 0 for all i' = i.

We have shown in Lemma 4.8 that to each increasing sequence j C {1,..., N j there corresponds a monomial a(j) e nTLN that allows us to select a single basis vector:

a(j)v(l) =

(—1) qv(j) if i = j, 0 otherwise.

Thus, for j = i, we see that

0 = c(a(j)v(l)) = a(J)(cv(l)) = a(j) ( £ ci' v(l') i = cj (-1)k-1qv(j),

Ii' |=k

implying cj = 0 for j = i. Hence, we may assume for each increasing sequence i that cv(i) = pi v(i) for some polynomial pi e k[q ].

Now it is left to show that pi = pi for all i' with |i'| = |i| = k. It is enough to verify this for i, i' that differ in exactly one entry, i.e. is = i, i's = i + 1, and ii = ii for all i = s, for some 1 < s < k and i e Z/NZ. If 1 < i < N — 1, we have

pi v(i') = cv(i') = c(ai v(i)) = ai (cv(i)) = ai (pi v(i)) = pi v(i'),

and if i = 0, we get

(_1)k_1qpi' v(i') = (_1)k—1qcv(l') = c(a0v(l)) = a»(cv(l)) = a»(pi v(i)) = (_1)k_1qpi v(i')•

Hence, pi' = pi, and this common polynomial is the desired polynomial pk. □

Corollary 5.3 Any central element in nTLN with constant term 0 acts on a standard basis vector v(i) e Vas multiplication by an element of qk[q ].

Proof According to Lemma 5.1, each summand of such a central element must contain the factor a0, and a0 acts on a wedge product by 0 or multiplication by ±q. □

Now we are ready to introduce nontrivial central elements in nTLN. For each 1 < k < N _ 1, set

tk := (_1)k_1X a(i), (1)

where the monomials a(i) correspond to increasing sequences i ={1 < i'1 < ••• < ik < N} of length k as defined in Lemma 4.8.

Example 5.4 In nTL3:

t1 = a2a1 a0 + a0a2a1 + a1 a0a2, t2 = —a0aa — a1a2a0 — a2a0a1.

In nTL4:

t1 = a3a2aa + a0a3a2a1 + a1a0a3a2 + a2a1a0a3,

t2 = —a0a3aa — a0a2a1a3 — a3a2a0a1 — a1a0a2a3 — a1a3a0a2 — a2a1a3a0 t3 = a0a1a2a3 + a1a2a3a0 + a2a3a0a1 + a3a0a!a2-

In the graphical realization of V, tk 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 tk (and it is multiplied with (_

1)2(k—1)q =

q, since a particle has been moved through position 0). All the tk have ZN-degree equal to 1) and Z-degree equal to N. Any monomial whose ZN-degree is 1) occurs as

a summand in some central element (after possibly reordering the factors), and the number of summands of tk equals (k) = dim(/\k kN); see Remark 4.10.

Theorem 5.5 1. The tk are central for all 1 < k < N — 1, and the center of nTLN is generated by 1 and the tk, 1 < k < N — 1.

2. The subalgebra generated by tk is isomorphic to the polynomial ring k[q ] for all 1 < k < N — 1. Moreover ttfi = 0 for all k = I. Hence the center ofnTLN is the subalgebra


Cn = k 0 tik[ti ]0---0tW-lk[tN-1] =

(tkh | k = I)

Proof 1. The action of tk on V is the projection onto the nTLN-submodule k[q ] ® /\k kN followed by multiplication by q. This commutes with the action of every other element of nTLN. Since V is a faithful module, tk commutes with any element of nTLN. As we have seen in Proposition 5.2, any central element c without constant term acts on the summand k[q] ® /\k kN via multiplication by some polynomial pck e qk[q]. Once again using the faithfulness of V, we get that c = ^1 pck (tk). 2. Recall that k[q ]®/\ k kN isafree k[q ]-moduleofrank N. Since tk acts by multiplication with q on that module, the subalgebra of nTTLN generated by tk must be isomorphic to the polynomial ring k[q ]. Since a (j )a(i) = 0 for all j = i, we get tk ti = 0 for k = I, as they consist of pairwise distinct summands. □

Theorem 5.5 enables us to describe the k-algebra EndnfLN (W) of nTLN-endomorphisms of the space of nontrivial particle configurations W := (J)11 ^k[q] ® /\k kN^ C V. We first observe that on W multiplication by q is given by the action of a central element in Cn , therefore it is justified to speak about k[q ]-linearity of a nTLN-endomorphism of W.

Lemma 5.6 EndnT£N (W) C Endk[q](W), hence any nTLN-module endomorphism y of W is k[q]-linear.

Proof Observe that ^N—1 tk e nTLN acts by multiplication by q on every element in W. Therefore multiplication by q commutes with the application of every y e End^r^ (W).

Proposition 5.7 The endomorphism algebra EndnTLN (W) is isomorphic to a direct sum of N — 1 polynomial algebras k[TJ 0 ••• 0 I[Tn—1].

Proof The proof is very similar to that of Proposition 5.2. First we show that y(v(i)) is a k[q ]-linear multiple of v(i) for any y e EndnTLw (W) and any increasing sequence i. This statement holds if and only if ±qy(v(i)) e k[q] v(i). Indeed, by Lemmas 4.8 and 5.6 we get

±qy(v(i)) = y(±qv(i)) = y(a(l)v(i)) = a(i)y(v(i)) e k[q] v(i).

Therefore, we can write y(v(i)) = pi ■ v(i) for some polynomial pi e k[q]. Note that this implies

/N—1 \ N—1

EndnTLN ( 0 (k[q] ® A'k^ ) ^0 (EndnT^k[q] ^kkN)) .

k=1 k=1

What remains is to show that these polynomials only depend on the number of particles in i, in other words there exists pk e k[q] so that pi = pk for all i with |i| = k. Again it suffices to show this for two sequences i, i' of length k that differ in exactly one entry. So say is = i, i's = i + 1, and ii = ifor all I = s, for some 1 < s < k and i e Z/NZ. When 1 < i < N — 1,

pi v(i') = y(v(i')) = y(ai v(i)) = aiy(v(i)) = ai (pi v(i)) = pi v(i'),

and when i = 0,

(_ 1)k—1qpi v(i') = (_ 1)k—1qV(v(i)) = V(a»v(i)) = a»v(v(i)) = a»(pi v(i)) = (_1)k—1qpi v(i')

Hence we can write y = ^pknk where n is the projection onto k[q ] ® /\k kN, and we get that

EndnTLN (k[q] ® AkkN) = k[Tk]

where Tk denotes the multiplication action of the central element tk, which is indeed a nTLN-module endomorphism of W. Thus, EndnTLN (W) is isomorphic to a direct sum of polynomial algebras as claimed. □

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 k \ {0}. In this case, we obtain that the summands f\k kN are simple modules and Endn^( ®f=_11 Ak kN) = kN—1. For q = 0, the situation is more complicated: If q is specialized to zero, the generator a0 acts by zero on the module. The action of nTLN factorizes over nTLN, and the module /\k kN is no longer simple. Instead it has a one-dimensional head spanned by the particle configuration v(1,^^,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, • ••, N ),but in general there are more endomorphisms. For example, in A4 k8, 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 EndnTL^ A4 k8^ 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 > 3; however, the following finiteness result holds:

Theorem 6.1 The algebra nTLN is finitely generated over its center.

Proof Given an arbitrary monomial a(j) e nTLN, we first factor it as a(j') • a(j(0)) in the following way: Take the minimal particle configuration j = {1 < j < ••• < jk < N} on which the monomial a(j) acts nontrivially; see Remark 4.3. The monomial a(j) 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 a( j) 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 a( j) = a( j') • a (j(0)), where j(0) is a sequence obtained by permuting j ,•••, jk so that the particle at position jr is moved one step clockwise by the action of ajr for all 1 < r < k. The remaining particle moves are carried out by a(j').

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

Next, we want to find an expression of the form

a(j) = afin • tn • a(j0)),

where afin is a monomial of some subalgebra i nTLN of nTLN, t^ is in the center of nTLN, and a (j(0)) is the above factor. Here

inTLN = (a0, •••, ai_1, ai+1, •••, aN_1 > (2)

denotes a copy of the finite nilTemperley-Lieb algebra nTLN sitting in nTLN. To accomplish this, we have to subdivide the action of a( j) on the particle configuration j = {j < ••• < jk} one more time. There are two cases:

1. There is an index i not appearing in j'. In this case, a(j') is an element of inTLN and we are done.

2. All indices appear at least n > 1 times in j'. Let us investigate the action of a(j') on the particle configuration v(i) = a(j(0))v(j), where i = {j'1 + 1,---, jk + 1}. Note that i is the minimal particle configuration for a(j'). Each of the particles in i is moved by a( j') to the position of the next particle in the sequence i, 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 a(j') = a(j") • a(l)n, with the monomial a(i) from Lemma 4.8. For maximal n, the remaining factor a( j") is an element ofinTLN for some i. Observe that a(i)na(j(0)) = tnka(j(0)), which follows immediately from the definition of tk and Lemma 4.8.

Therefore, we have shown that

a(j) = a(j) • a(/0)) = a^ • a(i)n • a(/0)) = afin • tnk • a(j(0)),

where n = 0 in the first case. Since there are only finitely many monomials in 0nTLN, 1nTLN, •••, N—1nTLN and only finitely many monomials a(j(0)) such that every index 0, 1,---, N — 1 occurs at most once in the sequence a(j(0)), the affine nilTemperley-Lieb algebra is indeed finitely generated over its center. □

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 nTLN into nTL^ coming from the N different embeddings of the Coxeter graph An_1 into An_1. Next we construct N embeddings of nTLN into nTLN +1. They correspond to the subdivision of an edge of An_1 by inserting a vertex on the edge to obtain An .

Theorem 7.1 Let N > 3. For any number 0 < m < N — 1, there is a unital embedding of algebras em : nTLN ^ nTLN+1 given by

a¡ for 0 < i < m — 1,

am+iam for i = m, (3)

at+1 for m + 1 < i < N — 1.

Lemma 7.2 For N > 3, the map em from uTLn to uTLn+1 given by (3) is an algebra homomorphism.

Fig. 5 £5(nTL7) c nTL8:The action of £5(00050504) = ¡¡00170160504 on the particle configuration v (4)

Proof Due to the circular nature of the relations, it suffices to check this for £0. This amounts to showing the following, since all the other relations are readily apparent. To avoid confusion, we indicate generators of nTLN +1 in these calculations by <i:

^0)^0) = <J1 (<50<J1<J0) = 0, <J2(<J1<J0)<J2 = (<J2<J1<J2< = 0, an(<J1<J0)<jN =a1(an<J0<JN) = 0,

(<J1<J0)<J2(<J1<J0) = (<J1<J2X<J0<J1<J0) = 0, (<J1<J0)<JN (<J1<J0) = (<J1<J0<J1)(<JN<J0) = 0.

Remark 7.3 How should one visualize the action of £m (nTLN) C nTLN +1 on the particle configurations on a circle with N + 1 positions? Except for am, all generators of nTLN are mapped to corresponding generators of nTLN +1. They will act as before, by moving a particle one step clockwise around the circle. Since am is mapped by £m to the product <Jm+1<Jm in nTLN+1, it will move a particle from m to m + 2 as depicted in Fig. 5. In other words, the elements in £m (nTLN) do not move a particle to or from position m + 1.

Next we introduce a basis of nTLN that will enable us to see directly that these homo-morphisms are embeddings. The basis has a simple description in terms of the graphical representation V from Sect. 4. For any two particle configurations with 1 < k < N — 1 particles corresponding to the increasing sequences i = {1 < i'1 < ••• < ik < N} and j = {1 < ]1 < ••• < jk < N}, there is a monomial in nTLN moving particles at the positions j to the positions i. We require that every particle from j 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 i = j, take eij to be the monomial such that the power of q in eijv(j) = ±qiv(i) is minimal (under the assumption that every particle from j must be moved). By faithfulness of the graphical representation, eij is uniquely determined. For i = j, we have eii = a(i), the special monomial defined in Sect. 4, hence eiiv(i) = ±qv(i). Observe that one can write tk = X|i|=k eii, where the sum runs over all possible increasing sequences i of length k, and that t^eij is a monomial, since all but one summand vanish for k = |i|.

Remark 7.4 The condition that eij move all particles from j by at least one step guarantees that it acts as zero on all particle configurations with fewer particles than |i| = |j|.

For example, when N = 7,

e(2)(1) = a1, e(0,2)(0,1) = 06a5a4a3a1a2a0a1.

(Note that a1 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 ej, the minimality condition implies that ej = afin ■ 1 ■ a( j(0)), where if j ={j1 < ••• < jk}, then j(0) is a sequence obtained by permuting the elements of j.

Theorem 7.5 The set of monomials

{1} U {tieij | I e Z>0, 1 <|i| = |j|= k < N — 1}

defines a k-basis of the affine nilTemperley-Lieb algebra nTLN.

Proof First, observe that tfeij is indeed a monomial since |i| = k. We show that the elements tfeij act k-linearly independently on the graphical representation V = ©N=0 (k[q] ® Ak kN). By Remark 7.4, the monomial ej acts by zero on summands

k[q] ® Ak kN for k' < |i|. On k[q] ® A'1' kN, the matrix representing the action of tfeLJ relative to the standard basis has exactly one nonzero entry, and this one distinguishes all monomials with the same minimal number of particles |i| = | j|. From these two observations, the linear independence follows. On the other hand, given any nonzero monomial in nTLN, there exists a minimal particle configuration j on which it acts nontrivially. Recording the image particle configuration i and the power of q, we conclude that there is some f so that the element tfeij acts on V in the same way as the given monomial does. Due to the faithfulness of this representation (see Theorem 4.5), the proposition follows. □

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 f. Up to an index shift in the output configuration i and a shift of the natural number f, the labelling sets agree, and both bases actually coincide.

Proof (Theorem 7.1) We have already proven in Lemma 7.2 that em is an algebra homomor-phism. Using Remark 7.3, observe that the monomial eij e nTLN is mapped to a monomial ei j e nTLN+1 (tilde again indicates in nTLN+1), where the new index sets are obtained by i ^ i for0 < i < m and i ^ i + 1form + 1 < i < N —1. The injectivity follows since basis

elements ^^|K|=k exx^ • eij of nTLN are mapped to basis elements ^^|K'|=k eK'K') • £i'j'

of nTLN+1. □

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. aiai±1ai = ai. Interestingly, the relation a2 = Sai is respected for S = 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 > 3, V is a faithful nTLN-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 nTLn . 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 nTTLN and describe a labeling set. (3) The final step is the description of the action of a monomial on 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 nTL^. They correspond to fully commutative elements in TLn, see [9].

Lemma 8.1 The monomial a(j ) = 0 if and only if for any two neighbouring appearances ofai in a(j ) there are exactly one ai+i and one ai—i in between, apart from possible factors a£ for I = i — 1, i, i + 1 (indices to be understood modulo N).

According to this result, two consecutive ai have to enclose ai+i and ai—1, i.e. ai ... ai ±i ... ai ., with the dots being possible products of a/s with I = i ± 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 a( j ) is zero if and only if we can bring two neighbouring factors ai together so that we obtain either af ('square') or aiai±1ai ('braid'). But expressions of the form ai ... ai±1 ... ai... ai cannot be resolved this way by commutativity relations. On the other hand, if there are two neighbouring factors ai with either none or only one of the terms ai±1 in between, we get after commutations either a(2 or aiai±1ai. If there are at least two factors ai+1 (or ai—1) in between the two ai, 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 ai +k in the kth step of the argument. Since we always consider the space in between two neighbouring factors ai, ai+1,..., ai+k, none of the previous ai, ai+1,..., ai +k—1 occurs between the two neighbouring ai+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 araf±< which is zero for any m > 0. □

Definition 8.2 For any i e {0, 1,..., N — 1}, we define a (clockwise) order ^ on the set {0, 1,..., N — 1} starting at i by

i < i + 1 ^ ... ^ i + N _ 1. 8.2 Step 1: A normal form

Given an arbitrary nonzero monomial a(j) in nTLN, reorder its factors according to the following algorithm (as usual, the indices are considered modulo N):

1. Find all factors ai in a( j) with no ai _i to their right. We denote them by ai1,..., atk, ordered according to their appearance in a( j); in other words, a(j) is of the form

a(j) = ...ah . .. ah......ak.

2. Move the ai1 ,•••, aik to the far right, without changing their internal order,

a(j) = a(j) • (ai1 ai2 •••aik) = a(j) • aj»)

for j(0) = (¿1, •••, ik) and some sequence j' = (j with ¿1, • ••, ik removed). This is possible because

(a) by assumption, there is no ai _1 to the right of an ai in this list;

(b) if for some i, ai+1 occurs to the right of some ai, then either ai • ••ai+1 • • •ai would occur as a subword without ai _1 in between, hence a( j) = 0, or else ai+1 does not have ai to its right, so it is one of the a^ , • • , aik itself, and will be moved to the far right of a( j), too;

(c) ai commutes with all af for f = i — 1, i + 1.

3. Repeat for a( j') until we get

a(j) = ajm)) • a(j(m—1)).....a(f>) • a(f>)

for sequences j(m), • • • , j(1) obtained successively the same way as described above. Notice:

• Inside a sequence j(n), every index occurs at most once. If two consecutive indices

occur within j(n), they are increasingly ordered using the order 4 from Definition 8.2. _

• For two consecutive sequences j (n+1), j(n) and for every index i(n+1> occurring in j("+1), we can find some index in j(n) such that i(n+1 = i(n'> + 1.

• From that property, it also follows that the length of j (n+1) is less or equal than the length of j(n).

4. Reorder the factors a(j(m)), •••, a(j(1)), a(j(0)) internally:

(a) Start with a (j(0)). There is some 0 < 1 < N — 1 which does not occur in j(0), but 1 — 1 occurs. For example, this is satisfied by 1 = ik + 1, as ik occurs in j(0) and is to the right of every other factor of a(j). Choose the largest such 1 (with respect to the usual order). Then we can move 1 — 1 to the very right of the sequence j(0), because 1 is not present, and 1 — 2 may only occur to the left of 1 — 1 due to the construction of j(0). We proceed in the same way with those indices 1 — 2,1 — 3,•••,í — (N — 1) that appear in j(0). The result is a reordering of the sequence j(0) so that it is increasing

from left to right with respect to 4.

(b) Repeat with all other factors a(j(1)), a(j(2)), • ••, a(j(m)) taking as the initial right-hand index of the sequence 1, 1 + 1 ,••, 1+m — 1 respectively, and reordering within

each a( j(n)) so that the indices are increasing from left to right with respect to ' +n. Throughout, the index 1 is the one from step (4a).

Example 8.3 As an example for nTL7, suppose a(j) = a(6 421354206132 5). (We omit the commas to simplify the notation.)

Find all aj without aj _i to their right:

Move them to the far right, and do not

change their internal order: Repeat:

With the right-hand indices of the

a( j(n)), n > 0, arranged according to

1 1 ~ 1 ~ ; + m — 1 ^ ... + 1 = 6

from left to right, reorder the factors in

each a( j(n)) increasingly with respect

to from left to right:

a(6 421354206132 5) a(642 1 3 542063) • a(1 2 5)

a(6 4235412063) • a(1 2 5)

a(6 4 2 3 5 4 1 0) • a(2 6 3) • a(1 2 5) a(6 423541Q) • a(2 63) • a(1 25)

a(6 4 2 5 1) • a(3 4 0) • a(2 6 3) • a(1 2 5) a(6 4 2 51) • a(3 4 0) • a(2 6 3) • a(1 2 5)

a(6 2) • a(4 5 1) • a(3 4 0) • a(2 6 3) • a(1 2 5) a(6 2) • a(4 5 1) • a(3 4 0) • a(2 3 6) • a(1 2 5)

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

Example 8.4 Let N = 6.

(5 123041502314502314 2) = (1)(5 0 2)(3 4 5 1)(2 3 4 0)(1 2 3 5)(0 1 2 4)

= (1 502 3451 2340 1235 012 4).

Lemma 8.5 Let a( j) be a nonzero monomial in nTLN, where we use as always the notation from Section 2. Let a(j(m)), a(j(m—1)), ..., a(j(1)), a(j(0)) be the monomials constructed by the algorithm above.

1. The equality aj) = aj (m))a(j(m—1)) ••• a(j(1))a(j(0)) holds in nTLN.

2. Given any two representatives a(j), a(j#) of the same element in nTLN, the above algorithm creates the same representative a(j (m))a(j(m—1)) ••• a(j (1))a(j(0)) for both a(j) and a(j#).

Proof 1. The algorithm never interchanges the order of two factors ai, ai ±1 with consecutive indices within a(j). Hence, the reordering of the factors of a(j) uses only the commutativity relation aiaj = ajai for i — j = ±1 mod N of nTLN.

2. Two monomials a( j), a( j#) in nTTLN are equal if and only if they only differ by applications of commutativity relations aiaj = ajai for i — j = ±1 mod N, hence, if and only if they contain the same number of factors ai for each i and the relative position of each ai and ai±1 is the same. Since the outcome of the algorithm depends only on the relative positions of consecutive indices, the resulting decomposition aj (m))aj(m—1)) ••• aj(1))aj(0)) is the same.

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 j is nonempty.

Theorem 8.6 Assume N > 3.

1. The algorithm in Step 1 above provides a normal form for nonzero monomials a(j) in the generators a ofnTLN, or equivalently for nonzero fully commutative monomials in TLn, so that

a(j) = (a^ • • • a(f) • • • (<+" • • • a^(a^ ... a«) • • • (a™ • • • a^)(an ... alk),

where a!?^ e {1, a0, aN_1} for all 1 < n < m, 1 < f < k, and

{1} if a^ = 1,

{1, aj+1} ifa<in[) = aj•

The factors ai1, •••, aik are determined by the property that the generator aif_1 does not appear to the right ofaif 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 1 is the largest value in {0, 1,--, N — 1} such that 1 — 1 e {i'1,---, ik}, but 1 e {1 ,•••, ik}.

2. The set {a(j) in normal form} U {1} is a k-basis ofnTLN.

8.3 Step 2: Labelling of basis elements

Definition 8.7 Given a(j) = a(j(m))a(j im_1>) ••• a(j i1>)a( j(0)) in normal form, we call jthe f th block of j, and a string of indices of maximal length of the form is e j(0), is +1 e j(1), is + 2 e j(2), • • • (modulo N) the sth strand of j. We use the notation [•••, is + 1, is] for the strands.

Example 8.8 Let N = 6, and consider Example 8.4 once again, where j = (1 502 3451 2340 1235 012 4),

The blocks are j_(0) = (0124), j_(1) = (1235), j® = (2340), j_(3) = (3451), j_(4) = (502), and j® = (1). 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 a(j) e nTLN determines two sets ij1, iout and an integer f j e Z>0 as follows:

i™ = {i e{0, 1,•••,N — 1}| no i — 1 to the right of i in j} i°ut = {i e{0, 1,•••,N — 1}| no i + 1 to the left of i in j} f j = the number of zeros in j •

These are well defined because, as in the proof of Lemma 8.5, any element of nTLN is uniquely determined by the number of factors ai and the relative position of each ai and ai ±1, for all i. The set ij1 equals the underlying set of j(0) in the normal form from the

a( + ) e it

algorithm above. All strands of j begin with an element in i j and end with an element from

The goal of this subsection is to show Proposition 8.9 The mapping

^ : {a(j ) g nTL^ in normal form} ^ Pn x Pn x Z>o (4)

a(j) ^ (j Ij\lj), is injective, where Pn is the power set of {0, 1,..., N — 1}.

Remark 8.10 The map ^ is defined so that in the graphical description of the representation V of nTLN, the set ij equals the set of positions where a(j) expects a particle to be. The

set iout equals the set of positions where a(j) moves the particles from j but each one is translated by 1, that is,

a(j ) applied to a particle at i g i j gives a particle at j + 1 for some j g iout.

The map ^ is far from being surjective. An obvious constraint is that Ij = |iout|, and furthermore, for some pairs (if, i°jut), one can only obtain sufficiently large values £ j.

To ease the presentation, we start by proving injectivity of the restriction xfo of ^ to those monomials a( j ) in normal form whose first element i1 of j(0) is 0. The proof itself will amount to counting indices.

Proposition 8.11 The map

^0 : {a(j ) g nTLN in normal form, with i1 = 0} ^ Pn x Pn x Z>o, a(j) ^ (j j,£l)

is injective.

Before beginning the proof of this result, we note that for monomials a( j) with i1 = 0, the inequality ik < N — 1 must hold in j since i1 = 0 implies that i1 — 1 = N — 1 is not an element of i™. Consequently, the ordering of the indices in i™ agrees with the natural ordering of Z, so we can regard (i™, <) as a subset of (Z, <) and replace the modular index sequence j by an integral index sequence jZ such that j Z( mod N) = j as follows:

Definition 8.12 Assume j = j(m).....j• j(0) is a normal form sequence with j(0) =

{0 = ix < ••• < ik < N — 1} and j(n) = (ihj + n,..., ihk(n) + n) ç (ix + n,..., ik + n), where indices in j(n) are modulo N and 1 < k (n) < k for all 1 < n < m. The integral normal form sequence for j is

jZ = (j(m))Z.....(j(1))Z • j(0) where (j:= (^ + n,..., km + n) g zkn

for n = 1,..., m.

Example 8.13 We continue Example 8.4 with N = 6.

If j = (1 502 3451 2340 1235 012 4), then jZ = (7 5 6 8 3 4 5 7 2 3 4 6 1 2 3 5 0 1 2 4).

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

Lemma 8.14 Let j Z be the integral normal form sequence for j and let [is ,•••, is + ns ] for s = 1,•••, k be the strands of jZ. Assume i'1 = 0. Then

(a) n 1 = i'1 + n1 < i'2 + n2 < ••• < ik + nk;

(b) ik + nk < i'1 + n1 + N = n1 + N.

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

Proof (Proposition 8.11) Since the sequence j will be fixed throughout the proof, we will

drop the subscript j on if i<out, fj .To show the injectivity of ^0, we consider the factorization — j j —

xfo = Y o f o a given by

^0 : a(j) —U a(iZ) —U ((iin)z,(iout)z) (iin, iout, f),

where (iin)Z = im and (iout)z = {i e jZ | no i + 1 to the left of i} similar to the definition of iout. The map a replaces indices in Z/NZ by indices in Z as in Definition 8.12 above. The map f is given by reading off (iout)z and (im)z from jZ. The map y sends the pair ((iin)z, (iout)z) to a triple consisting of the respective images im, iout modulo N of the pair and the integer f = 1 + ^ fr where fr = N J for each jr e (iout)z. The summand 1 corresponds to 0 = i'1; all other occurrences of 0 are counted by ^ fr. Now we check injectivity.

The map a is clearly injective since jZ u jZ ( mod N) is a left inverse map. To see that f is injective, we need to know that jZ can be uniquely reconstructed from ((iin)z, (iout)z). Observe that jZ is determined by knowing all the 'strands' is, is + 1, is + 2,•••,is + ns for 1 < s < k, hence by assigning an element is + ns e (iout)z to each is e (iln)z. But itfollows fromLemma 8.14(a) that i'1 + n1 must be the smallest element of (iout)z, i'2 + n2 the second smallest, etc., so that the element is + ns is assigned to the sth element in im, that is, to is.

Now to see that y is injective, we need to recover ((im)z, (iout)z) in a unique way from (i11, iout, f). Write iin = {0 = i1 < ••• < ik < N — 1}, and set (iin)z:=iin. By Lemma 8.14(a), we know that (iout)z is of the form (i'1 + n1 < ••• < ik + nk), and since the elements of iout have to be equal to the elements of (iout)Z modulo N, we can write ir + nr = Nfr + dr for fr = L^+r1 J and some dr e i . Comparing fr and fs for r < s, we have

Nfr < Nfr + dr = ir + nr < is + ns = Nfs + ds < N(fs + 1)>

So fr < fs + 1, i.e. fr < fs. Similarly, we obtain from (b) of Lemma 8.14 that fk < f 1 + 1. As a result,

Nfk < N fk + dk = ik + nk < i1 + n1 + N = N(f 1 + 1) + d1 < N(f 1 + 2),

i.e. fk < f 1 + 2. Together we have f 1 = ••• = fs < fs+1 = ••• = f 1 + 1for some 1 < s < k (where we treat the case s = k by f1 = ••• = fk). Set f:=f 1. Then

ir + nr = N f + dr for 1 < r < s,

ir + nr = N (f~+ 1) + ds for s + 1 < r < k•

As a first consequence,

f = 1 + ^ fr = 1 + kf + (k — s),

which determines i = L^-1J, and hence all lr, as well as the index s. Using Lemma 8.14, we determine that

is+1 + ns+i < ••• < ik + nk < ii + ni + N < ••• < is + ns + N,

and so

N (£ + 1) + d+1 < ••• < N (£ + 1) + dk < N (£ + 1) + di < ••• < N (£ + 1) + ds.

Therefore, ds +1 < ••• < dk < d1 < ••• < ds, which fixes the choice of dr for all r. We conclude that given (iin, iout, £), we can reconstruct (iout)z by setting ir + nr :=N ir + dr. This completes the proof of Proposition 8.11. □

Proof (Lemma 8.14) (a) Let jZ be a nonempty integral normal form sequence with 0 = ii < ••• < ik < N — 1 and strands [i r, ir + nr ] for 1 < r < k. Assume that there is some index 1 < t < k — 1 such that it + nt > it+1 + nt+1. Since it < it+1, we have nt > nt+1. So

jZ = ... (... it +nt ...) ... (... it + nt+1 it+1 + nt+1 ...) ....

the ntth bracket the nt+1th bracket

From it + nt+1 < it+1 + nt+1 < it + nt it follows that there is some integer nt+1 < p < nt such that it+1 + nt+1 = it + p appears in the strand [it,. ..,it + nt ], i.e.

jZ = ... (... it +nt ...) ... (... it + p ...) ... (... it + nt+1 it+1 + nt+1 ...) ...

the nt th bracket the pth bracket the nt+1th bracket

with it + p = it+1 + nt+1. But by the definition of the strands, there is no it+1 + nt+1 + 1 appearing to the left of it+1 + nt+1. Due to Lemma 8.1, we know that (even modulo N) there is no repetition of it+1 + nt+1 to the left. Thus it + p = it+1 + nt+1 is not possible, and we obtain i1 + n1 < ¿2 + n2 < ••• < ik + nk.

For (b) of Lemma 8.14, assume ik + nk > i1 + n1 + N. It is true generally that N > ik, so we get ik + nk > i1 + n1 + N > ik + n1. Hence i1 + n1 + N = ik + b for some n1 < b < nk, i.e. i1 + n1 + N appears in the strand [ik, . ..,ik + nk ] and we have

jZ = ... (... + n^ ... (... ik + b ...) ... (i1 + n1 ... ik + nj) ....

the nk th bracket the bth bracket the n1th bracket

Here it may be that the nk th bracket and the bth bracket coincide, but in any case, we find that ik + b = i1 + n1 + N = i1 + «1 mod N, and so ik + b appears to the left of i1 + n1.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 j = jZ mod N there is no i1 + n1 mod N to the left of i1 + n1 allowed, which leads to a contradiction. Hence ik + nk < i1 + n1 + N must hold. □

Having established that ty is injective when restricted to sequences with i1 = 0, we now show the injectivity of ty in general.

Proof (Proposition 8.9) We have the following disjoint decompositions according to the smallest value ii in j (0) for j :

{a(j) in normal form} = ]J{a(j) in normal form, with ii = i j

{j if, £j)} = y« j if, ij_) i ii = i e j

= JJ : {a(j) in normal form, with ii = i j

^ {(j if,j | ii = i e ij}) .

By Proposition 8.11, the map : a( j) ^ (i1jn, iout, lj) restricted to those a(j) with ii = 0 is injective. We argue next that by an index shift this result is true for all other . Now it follows from Proposition 8.ii that the map

j0 : {a(j) e nTLN in normal form, with ii = 0} ^ {(j if, f ) | ii = 0 e iinj is injective, where lj counts the occurences of N — i in j. Recall that i j = lr +1 and lr is the number of zeros in the rth strand [ir,..., ir +nr ] of j mod N.

Now observe that we can obtain i j from i j as

iL = f — |{dr e j1 | dr > N — i}| + I {ir e iin | ir > N — i}| + i, which follows from a computation using lj = ^r f and

f = the number of N — i in the rth strand [ir,..., ir + nr ] mod N

= 1 ir + + i L N J 1 ir + nr + i L N J - 1 if ir < N - i if ir > N - i

= ' i Nlr +dr+i L N i Nir +dr+i L N J J - if ir < N -1 if ir > N - i i

'ir + 1 if ir < N - i and dr + i > N

lr if ir < N - i and dr + i < N

ir if ir > N - i and dr + i > N

lr - 1 if ir > N - i and dr + i < N

We obtain ^ by first shifting the indices of j by subtracting i from each index, j — (i,...,i), then applying x/tq, and finally shifting the indices from I™ and lout by adding i to each. Hence, is injective for each i, and ^ is injective because the unions are disjoint. □

8.4 Step 3: Description and linear independence of the matrices

Recal1 that the standard k-basis of the representation V = 0n=q (k[?] ^k is given

{qe ■ vil A ••• A vik | i e Z>o, 1 < ii < ••• < ik < N}

where (i\, ..., ik) 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 a(j) e nTLN as a 2N x 2N-matrix with entries in k[q]. Since V decomposes as a nTLN-module into submodules k[q] ® /\k kN for k = 0, 1,..., N, the matrix of a(j) is block diagonal with N + 1 blocks Ao, Ai,..., An, where Ao = An = (0) corresponding to the trivial representation.

/0 0 0

a( j ) =

The block Ak is a (k) x (k)-matrix, with entries from k[q ] indexed by all possible particle configurations whose number of particles equal to k.

Now fix a nonzero monomial a ( j ) in normal form that is specified by the triple (i j1, iout ,lj )

defined in Step 2. Let k = |ij^. All blocks A,,..., Ak-i are zero since a(j) expects at least k particles. For r > k there might be nonzero blocks Ar. Such nonzero blocks appear unless the particles from ij are moved around the whole circle with no position left out, in which case there are no surplus particles allowed. This occurs if a(j) contains at least every other generator ai, at +2,....

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

( Ak )iin,iout

= ±q.

From this we see first that all matrices representing monomials a( j) in normal form with llj11 = N — 1 are k-linearly independent: They have only one nonzero entry which is equal

to ±qij- at position (ij1, iout). Furthermore, if all matrices representing monomials a(j) in normal form with |i1-1| > k are k-linearly independent, then also all matrices representing monomials a(j) in normal form with llj1! > k — 1 are k-linearly independent. This follows because the additional monomials a(j) with ii-1! = k — 1 have nonzero entries

(Ak—1)iin iout = ±qin the (k — 1)th block which is zero for all a(j) with ii1-1! > k. So by

L, L —

induction, all matrices representing monomials a( j) in normal form are 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 nTLN on V is faithful, because according to Theorem 8.6, {a(j) in normal form} U {1} is a k-basis of nTLN.

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

Acknowledgments Open access funding provided by Max Planck Society Max Planck Institute for Mathematics.

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