CANONICAL BASES FOR HECKE ALGEBRA QUOTIENTS

R.M. Green and J. Losonczy

Dedicated to Professor R.W. Carter on the occasion of his 65th birthday

Department of Mathematics and Statistics Lancaster University Lancaster LA1 4YF England E-mail: r.m.green@lancaster.ac.uk

Department of Mathematics Long Island University Brookville, NY 11548 USA E-mail: losonczy@e-math.ams.org

Abstract. We establish the existence of an IC basis for the generalized Temperley– Lieb algebra associated to a Coxeter system of arbitrary type. We determine this basis explicitly in the case where the Coxeter system is simply laced and the algebra is finite dimensional.

To appear in Mathematical Research Letters

Introduction

An important construction in the theory of quantum groups and quantum al-

gebras is that of canonical bases. The original example of this construction is the

Kazhdan–Lusztig basis of the Hecke algebra associated to a Coxeter system, which

first appeared in [11]. Another well known example is the canonical basis for U^{+},

the “plus part” of the quantized enveloping algebra associated to a semisimple Lie

algebra over ℂ; this was discovered independently by Kashiwara [10] and Lusztig

[12]. In each of these examples, the basis which arises has many deep and beautiful

properties, some of which have geometric interpretations.

Typeset by AMS-TEX 2 R.M. GREEN AND J. LOSONCZY

The first author was supported in part by an award from the Nuffield Foundation.

The general theory of such bases is defined for an A-module (where A is the ring of Laurent polynomials ℤ[v, v^{−1}]) equipped with an involutive ℤ-linear map that sends v to v^{−1}. This theory was developed by Du in [2], where the bases arising are called IC bases. The letters “IC” stand for “intersection cohomology”; the name alludes to the fact that many of the natural examples have interpretations in terms of perverse sheaves. However, the existence and uniqueness of such a basis is not guaranteed.

The Temperley–Lieb algebra, a finite dimensional algebra arising in statistical mechanics [13] and knot theory [9], may be defined as a certain quotient of the Hecke algebra of a Coxeter system of type A. It is possible to generalize this construction to an arbitrary Coxeter system, obtaining the so-called “generalized Temperley–Lieb algebras” as quotients. In this paper, we show that IC bases exist for these Hecke algebra quotients associated to a Coxeter system of arbitrary type. We also determine the bases explicitly in the case of Coxeter systems of types A, D and E; it turns out that in each of these cases the basis coincides with a previously familiar basis for the corresponding generalized Temperley–Lieb algebra. The situation for non-simply-laced Coxeter systems turns out to be more complicated and surprising, as we will discuss.

It is tempting to think that IC bases for generalized Temperley–Lieb algebras may be obtained from the well-known Kazhdan–Lusztig bases for the Hecke algebra by projection to the quotient, but this is in fact far from clear. In general, the kernel of the canonical map from the Hecke algebra to the generalized Temperley–Lieb algebra is not spanned by the Kazhdan–Lusztig basis elements which it contains, even for Coxeter systems of low rank such as type D_{4}, and this causes complications. However, the situation in type A is relatively simple, as C.K. Fan and the first author have shown [4].

Our results give rise to some interesting problems concerning the general properties of IC bases for generalized Temperley–Lieb algebras; we mention a few of these in our remarks.

CANONICAL BASES FOR HECKE ALGEBRA QUOTIENTS 3

1. Generalized Temperley–Lieb algebras

Let X be a Coxeter graph, of arbitrary type, and let W (X) be the associated Coxeter group with distinguished set of generating involutions S(X). Denote by H(X) the Hecke algebra associated to W (X). (The reader is referred to [8, §7] for the basic theory of Hecke algebras arising from Coxeter systems.) The A-algebra H(X) has a basis consisting of elements T_{w}, with w ranging over W (X), that satisfy

T_{s}T_{w} = \{ \begin{matrix}T_{sw} & if ℓ(sw) > ℓ(w), \\ qT_{sw} + (q − 1)T_{w} & if ℓ(sw) < ℓ(w), \end{matrix}

where ℓ is the length function on the Coxeter group W (X), w ∈ W (X), and s ∈ S(X). The parameter q is equal to v^{2}, where A = ℤ[v, v^{−1}].

We define J(X) to be the ideal of H(X) generated by all elements

∑_{w∈⟨s_{i},s_{j}⟩} T_{w},

where (s_{i}, s_{j}) runs over all pairs of elements of S(X) that correspond to adjacent nodes in the Coxeter graph. (If the nodes corresponding to (s_{i}, s_{j}) are connected by a bond of infinite strength, we omit the corresponding relation.)

Definition 1.1. The generalized Temperley–Lieb algebra, T L(X), is defined to be the quotient A-algebra H(X)/J(X).

The algebra T L(X) has a basis which arises naturally from the T -basis of H(X). To define it, we introduce some standard combinatoric notions.

Definition 1.2. Let s_{i} and s_{j} be elements of S(X), the set of generating involutions, such that s_{i} and s_{j} correspond to adjacent nodes in the Coxeter graph. Let w_{ij} be the longest element in ⟨s_{i}, s_{j}⟩.

We call an element w ∈ W (X) complex if it can be written as x_{1}w_{ij}x_{2}, where x_{1}, x_{2} ∈ W (X) and ℓ(x_{1}w_{ij}x_{2}) = ℓ(x_{1}) + ℓ(w_{ij}) + ℓ(x_{2}).

Denote by W_{c}(X) the set of all elements of W (X) which are not complex.

Let t_{w} denote the image of the basis element T_{w} ∈ H(X) in the quotient T L(X).

Theorem 1.3 (Graham). The set {t_{w} : w ∈ W_{c}} is an A-basis for the algebra T L(X).

Proof. This is [5, Theorem 6.2]. □

4 R.M. GREEN AND J. LOSONCZY

We will call the basis of Theorem 1.3 the “t-basis” of the algebra T L(X). It plays an important rôle in the sequel, since the canonical basis will be defined in terms of it.

Lemma 1.4. The algebra T L(X) has a ℤ-linear automorphism of order 2 which sends v to v^{−1} and t_{w} to t_{w^{−1}}^{−1}.

Note. The statement of the lemma makes sense: it is well known that the elements T_{w} ∈ H(W) are invertible, from which it follows that the elements t_{w} are invertible also.

Proof. It is known from [11] that the Hecke algebra H(X) over ℤ[v, v^{−1}] has a ℤlinear automorphism of order 2 which exchanges v and v^{−1} and sends T_{w} to T_{w^{−1}}^{−1}. It is therefore enough to show that this automorphism of H(X) fixes the ideal J(X).

Given a finite Coxeter group W′ with longest element w_{0}, it is a standard result that, for any w_{1} ∈ W′, there exists an element w_{2} ∈ W′ such that w_{0} = w_{1}w_{2} and ℓ(w_{0}) = ℓ(w_{1}) + ℓ(w_{2}). It follows from this fact that

( ∑_{w∈⟨s_{i},s_{j}⟩} T_{w}) T_{w_{ij}}^{−1} = ( ∑_{w∈⟨s_{i},s_{j}⟩} T_{w^{−1}}^{−1} ) ,

so that the right hand side of the above equation lies in the ideal J(X). We deduce that the automorphism of H(X) given above fixes the ideal J(X) setwise, and thus induces an automorphism of T L(X) which has the required properties. □

Lemma 1.5. Let w ∈ W (not necessarily in W_{c}). Then

t_{w} = ∑_{{x∈Wc \atop x≤w}} D_{x,w}t_{x},

where D_{x,w} ∈ ℤ[q]. Furthermore, D_{w,w} = 1 if w ∈ W_{c}.

Proof. We proceed by induction on ℓ(w). The proposition is trivial if w ∈ W_{c}, which covers the cases where ℓ(w) ≤ 1, as well as the last assertion in the statement.

Now consider the case where w = su, ℓ(s) = 1 and ℓ(u) = ℓ(w) − 1. Then, by induction,

t_{w} = t_{s}t_{u} = ∑_{{x∈Wc \atop x≤u}} D_{x,u}t_{s}t_{x}.

It follows from standard properties of H(X) that

t_{s}t_{x} = \{ \begin{matrix}t_{sx} & if ℓ(sx) > ℓ(x), \\ qt_{sx} + (q − 1)t_{x} & if ℓ(sx) < ℓ(x). \end{matrix}

It is easy to see that sx ≤ w and x ≤ w for each basis element t_{x} appearing in the sum. If sx < w then, by induction, t_{sx} is a ℤ[q]-linear combination of basis elements t_{z} (z ∈ W_{c}) where z ≤ sx ≤ w.

The only remaining case is when sx = w, w ∉ W_{c}. This forces x = u and thus u ∈ W_{c}, so there is only one term in the sum. Since w is complex, it can be written as x_{1}w_{ij}x_{2} as in Definition 1.2. We now use the fact that in T L(X) we have

t_{w_{ij}} = − ∑_{y<w_{ij}} t_{y}.

This enables t_{w} = t_{x_{1}}t_{w_{ij}}t_{x_{2}} to be expressed as a ℤ[q]-linear combination of elements t_{y}, where y < w. By induction, these elements t_{y} are ℤ[q]-linear combinations of basis elements t_{z} (z ∈ W_{c}) where z ≤ y ≤ w.

The lemma follows. □

2. IC bases

We now recall the basic properties of IC bases from [2, §1].

Let A := ℤ[v, v^{−1}], where v is an indeterminate, and let A^{−} = ℤ[v^{−1}] and A^{+} = ℤ[v]. Let ¯ be the involution on the ring A which satisfies v̄ = v^{−1}.

Let M be a free A-module with basis {m_{i}}_{i∈I} and an involutive ℤ-linear map ¯ : M ⟶ M such that \overline{am} = ām̄ for any m ∈ M and a ∈ A. For each i ∈ I, let r_{i} be an integer.

Let L be the free A^{−}-submodule with basis {m′_{i}}_{i∈I}, where m′_{i} := v^{r_{i}}m_{i}. Let π : L ⟶ L/v^{−1}L be the canonical projection.

Definition 2.1. If there exists a unique basis \{c_{i}\}_{i∈I} for L such that \overline{c_{i}} = c_{i} and π(c_{i}) = π(m′_{i}), then the basis {c_{i}}_{i∈I} is called an IC basis of M with respect to the triple ({m_{i}}_{i∈I}, ¯ , L).

Note that the existence of an IC basis for M depends only on the original basis {m_{i}}_{i∈I}, the map ¯ and the integers r_{i}.

6 R.M. GREEN AND J. LOSONCZY

Theorem 2.2 (Du). Let (I, ≤) be a poset such that the sets {i : i ∈ I, i ≤ j} are finite for all j ∈ I. Suppose that

\overline{m′_{j}} = ∑_{{i∈I \atop i≤j}} a_{ij}m′_{i}

with a_{ij} ∈ A such that a_{ii} = 1 for all i ∈ I. Then an IC basis of M with respect to ({m_{i}}, ¯ , L) exists. Proof. This comes from [2, Theorem 1.2, Remark 1.2.1 (1)]. □

The importance of this result for us is that the t-basis of the algebra T L(X) fits naturally into this setup, as we now explain.

Fix a Coxeter graph X. Let I = I(X) be the set {w : w ∈ W_{c}}. We make I into a poset (I, ≤) by restricting the Bruhat–Chevalley order on the Coxeter group W = W (X) to the subset I. (Standard properties of this order imply that the poset (I, ≤) has the finiteness property required by Theorem 2.2.) We take r_{w} = −ℓ(w) and m_{w} = t_{w}, so that m′_{w} = v^{−ℓ(w)}t_{w}, and we take ¯ to be the automorphism of T L(X) defined in Lemma 1.4.

Maintaining this notation, we have the following central result. Theorem 2.3. Let X be an arbitrary Coxeter graph. There exists an IC basis for the algebra T L(X) with respect to ({m_{i}}, ¯ , L). Proof. It is enough to show that the formula for \overline{m′_{w}} (where w ∈ W_{c}) given in Theorem 2.2 can be satisfied for suitable elements a_{ij} ∈ A. We proceed by induction on ℓ(w), the case ℓ(w) = 0 being trivial (w = e and \overline{m_{e}} = m_{e}).

To deal with the inductive step, we suppose that w = su, where ℓ(s) = 1 and ℓ(u) = ℓ(w) − 1. It is clear from the definition of W_{c} that u ∈ W_{c}.

Using the fact that m′_{s} = v^{−1}t_{s}, it is easily verified that

\overline{m′_{s}} = m′_{s} − (v − v^{−1})m′_{e},

where m′_{e} is the identity in T L(X). Thus

\overline{m′_{w}} = \overline{m′_{s}}\overline{m′_{u}} = (m′_{s} − (v − v^{−1})m′_{e})\overline{m′_{u}}.

By induction,

\overline{m′_{u}} = ∑_{{z∈Wc \atop z≤u}} a_{zu}m′_{z}.

For each z appearing in the sum, we have z ≤ u ≤ w and also sz ≤ w by standard properties of the Bruhat–Chevalley order. It therefore follows from Lemma 1.5 that

\overline{m′_{w}} = ∑_{{y∈Wc \atop y≤w}} a_{yw}m′_{y}

for suitable a_{yw} ∈ A.

It remains to show that a_{ww} = 1. Considering lengths and using Lemma 1.5, we find that the only term in the expression for \overline{m′_{s}}\overline{m′_{u}} which can contribute to the coefficient of m′_{w} arises from the product m′_{s} ×a_{uu}m′_{u}. It is clear that m′_{s}m′_{u} = m′_{w}, and we have a_{uu} = 1 by induction, completing the proof. □

Remark 2.4.

(1) The structure constants associated to the IC basis of T L(X) lie in ℕ[v, v^{−1}] if X is of type A, D or E_{n}. (This will follow from Theorem 3.6.) We do not know an example where this positivity property fails for the IC basis, and the property certainly holds for some non-simply-laced types, such as T L(H_{n}). In [7, §4.1], the first author constructed a basis of T L(H_{n}) with structure constants in ℕ[v, v^{−1}] using morphisms in the category of decorated tangles. (This category was introduced in [6].) It turns out that the basis of T L(H_{n}) in [7] is precisely the IC basis of T L(H_{n}) arising from Theorem 2.3. This means that the results of this paper give an elementary characterisation of the basis in [7]. We do not supply the details here, but we hope to do so in a forthcoming paper.

(2) An interesting problem to consider is that of identifying the Coxeter graphs X for which the IC basis of T L(X) is equal to the image of the set C of Kazhdan– Lusztig basis elements C′_{w} ∈ H(X) indexed by w ∈ W_{c}. Coxeter graphs X of type A have this property, by [4, Theorem 3.8.2] and our Theorem 3.6. It seems likely that the set C projects to the IC basis in type D, as well. This problem is closely related to the question of whether an element v^{−ℓ(w)}T_{w} ∈ H(X) (w ∉ W_{c}) necessarily lies in the lattice L after passing to T L(X). One can also express the problem in terms of a degree bound on the polynomials D_{x,w} of Lemma 1.5. A significant partial result would be to know that v^{−ℓ(w)}T_{w} ∈ H(X) projects into L when w = su, s ∈ S(X), u ∈ W_{c} and w ∉ W_{c}. This would allow an inductive construction of the IC basis.

8 R.M. GREEN AND J. LOSONCZY

3. The ADE case

In §3, we restrict ourselves to the case where the Coxeter graph is of type A, D or E. However, we allow the graphs of type E to be of arbitrary rank. This means that a graph of type E_{n} (n ≥ 6) may consist of a straight line of nodes numbered 1, 2, …, n − 1 together with a node numbered 0 which is connected only to node 3. (Contrast this with the more familiar definition of type E which is the same but requires n ≤ 8.)

It is known [5, Theorem 7.1] that the algebras T L(E_{n}) are finite dimensional for all values of n. In fact, for simply laced X, the algebra T L(X) is finite dimensional if and only if X is of type A, type D or type E_{n} for some n. If X satisfies these hypotheses, we say it is of type ADE. Definition 3.1. If s ∈ S(X), we define b_{s} to be the element v^{−1}t_{s} + v^{−1}t_{e}.

If w ∈ W_{c} and w = s_{1}s_{2} ⋯ s_{r} (reduced), we define

b_{w} := b_{s_{1}}b_{s_{2}} ⋯ b_{s_{r}}.

This definition may appear to depend on the reduced expression chosen for w, but in fact it does not, because b_{s} and b_{s′} commute whenever s and s′ commute, and every reduced expression for w can be obtained from any other by a sequence of commutation moves (see [3]).

It is well known (and follows easily from Theorem 1.3 and Lemma 1.5) that the set {b_{w} : w ∈ W_{c}} is a basis for T L(X). We call this the monomial basis.

We are particularly interested in the case where the Coxeter graph X is connected and simply laced (all bonds have strength 2 or 3). Lemma 3.2. If X is connected and simply laced then the algebra T L(X) is generated as an associative, unital algebra by the elements b_{s} (one for each node of X) and defining relations

b_{s}^{2} = q_{c}b_{s},
b_{s}b_{t} = b_{t}b_{s} if s and t are not connected
b_{s}b_{t}b_{s} = b_{s} if s and t are connected

where q_{c} := v + v^{−1}. Proof. This is a standard result from [3]. □

CANONICAL BASES FOR HECKE ALGEBRA QUOTIENTS 9

Our aim in §3 is to prove that in types A, D and E, the monomial basis is the IC basis. The following is a key ingredient of the proof that we will give.

Lemma 3.3. Let X be a Coxeter graph of type ADE. Let w ∈ W_{c}(X) and let s_{1}s_{2} ⋯ s_{r} be a reduced expression for w. Then for any 1 ≤ i_{1} < i_{2} < ⋯ < i_{k} ≤ r (k < r), we have b_{s_{i_{1}}} b_{s_{i_{2}}} ⋯ b_{s_{i_{k}}} = q_{c}^{m}b_{x} for some x ∈ W_{c}(X), where m ≤ r − k − 1 and q_{c} := v + v^{−1}.

Proof. This result is a special case of [5, Lemma 9.13]. Let b be an arbitrary monomial in the generators b_{s}. The relations of T L(X) ensure that b is equal to q_{c}^{m}b_{x} for some nonnegative integer m and some x ∈ W_{c}. Then [5, Lemma 9.13] shows that the removal of one generator from the monomial b results in an expression equal to q_{c}^{m′}b_{x′}, where x′ ∈ W_{c} and m′ ≤ m + 1. Furthermore, if b = b_{w} for some w ∈ W_{c}, we have m′ = m. This means that the maximum exponent of q_{c} which could occur after r − k generators have been removed from b_{w}, as in the statement, is r − k − 1. □

Remark 3.4. It is possible to prove Lemma 3.3 without appealing to the results of [5] by using a combinatoric argument based on tom Dieck’s graphical calculus [1], and in fact the latter approach establishes the lemma for a slightly wider class of Coxeter systems, including type Ẽ_{7}. However, we do not pursue this here.

In the following lemma we use the standard notation ε_{x} := (−1)^{ℓ(x)}.

Lemma 3.5. Let w ∈ W_{c}. Then

v^{−ℓ(w)}t_{w} = ε_{w} ∑_{{x∈Wc \atop x≤w}} ε_{x}Q̃_{x,w}b_{x},

where Q̃_{w,w} = 1 and Q̃_{x,w} ∈ v^{−1}A^{−} if x < w.

Proof. Fix w ∈ W_{c} and fix a reduced expression w = s_{1}s_{2} ⋯ s_{r}. We have

\begin{aligned}v^{−ℓ(w)}t_{w} & = (v^{−1}t_{s_{1}})(v^{−1}t_{s_{2}}) ⋯ (v^{−1}t_{s_{r}}) \\ & = (b_{s_{1}} − v^{−1})(b_{s_{2}} − v^{−1}) ⋯ (b_{s_{r}} − v^{−1}). \end{aligned}

By expanding the last product and using Lemma 3.2 and the subexpression characterisation of the Bruhat–Chevalley order, we can see that all the b_{x} occurring in the sum satisfy x ∈ W_{c} and x ≤ w.

More precisely, the product expands to a sum of terms

v^{k−ℓ(w)}b_{s_{i_{1}}} b_{s_{i_{2}}} ⋯ b_{s_{i_{k}}} .

If k < ℓ(w), Lemma 3.3 shows that this is equal to

v^{k−ℓ(w)}q_{c}^{m}b_{x},

where x ∈ W_{c} and m ≤ ℓ(w) − k − 1. It follows that the coefficient of b_{x}, and hence Q̃_{x,w}, lies in v^{−1}A^{−} in this case. The other possibility is that k = ℓ(w), which produces the basis element b_{w} with coefficient 1 and no other basis elements. It follows that Q̃_{w,w} = 1, as required. □

We are ready to prove the main result of §3.

Theorem 3.6. Let X be a Coxeter graph of type ADE. Then the IC basis for T L(X) defined in Theorem 2.3 is precisely the monomial basis {b_{w} : w ∈ W_{c}}.

Proof. It is easily checked that the generators b_{s} are fixed by the involution ¯ , from which it follows that any monomial in these generators also has this property, so that \overline{b_{w}} = b_{w}.

Lemma 3.5 shows that the transfer matrix (ε_{w}ε_{x}Q̃_{x,w}) from the monomial basis to the m′_{i}-basis is upper unitriangular with respect to a total refinement of the partial order ≤, and all the entries above the diagonal lie in v^{−1}A^{−}. Elementary linear algebra shows that the inverse of this matrix, (P̃_{x,w}), is also upper unitriangular with all the entries above the diagonal in v^{−1}A^{−}. This shows that

b_{w} = ∑_{{x∈Wc \atop x≤w}} P̃_{x,w}(v^{−ℓ(x)}t_{x}),

and therefore that π(b_{w}) = π(v^{−ℓ(w)}t_{w}).

We have now shown that the monomial basis is the IC basis with respect to the triple ({m_{i}}_{i∈I}, ¯ , L), as claimed. □

Remark 3.7.

(1) It is not true that the monomial basis of T L(X) equals the IC basis for all simply laced X. For example, take X to be the Coxeter graph of type Ã_{3} consisting of four nodes connected by four edges in the shape of a square. If we number the nodes 1, 2, 3, 4 around the square, the element b_{1}b_{3}b_{2}b_{4}b_{1}b_{3} is not an IC basis

element. Similar remarks hold for X of type Ã_{l}, where l > 3 is odd.

If X is non-simply-laced, the monomial and IC bases do not agree: any element

in W_{c} of the form w = ss′s (where s, s′ ∈ S(X)) has the property that b_{s}b_{s′}b_{s} is

not an IC basis element.

(2) Another problem to consider is that of determining the precise relationship be-

tween the elements P̃_{x,w} in the proof of Theorem 3.6 and the Kazhdan–Lusztig

polynomials P_{x,w} of [11]. It is natural to be curious about this relationship,

because our elements P̃_{x,w} play a rôle analogous to that of v^{ℓ(x)−ℓ(w)}P_{x,w} in

[11]. Similarly, the elements Q̃_{x,w} are analogous to the inverse Kazhdan–Lusztig

polynomials.

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