P -kernels, IC bases and Kazhdan-Lusztig polynomials

Francesco Brenti^{1} Dipartimento di Matematica Universitá di Roma “Tor Vergata” Via della Ricerca Scientifica, 1 00133 Roma, Italy brenti@mat.uniroma2.it

Abstract

In [18] Stanley introduced the concept of a P -kernel for any locally finite partially ordered set P. In [10] Du introduced, for any set P, the concept of an IC basis. The purpose of this paper is to show that, under some mild hypotheses, these two concepts are equivalent and, given a Coxeter group W partially ordered by Bruhat order, to characterize among all possible W -kernels the one corresponding to the Kazhdan-Lusztig basis of the Hecke algebra of W. Finally, we show that this W -kernel factorizes as a product of other W -kernels, and that these provide a solution to the Yang-Baxter equations for W.

1 Introduction

In his 1992 paper [18] Stanley introduced, for any locally finite poset P, the concept of a P -kernel. This concept includes as special cases several interesting objects, including Kazhdan-Lusztig polynomials, and the local intersection homology Poincaré polynomials of toric varieties. Furthermore, several results on these polynomials actually generalize to the more general setting of P -kernels (see, e.g., [18, Part II], [5], [6]). In an unrelated development, Du in 1994 introduced in [10], for any set P, the concept of an IC basis. IC bases include as special cases many interesting bases such as Kazhdan-Lusztig bases of Hecke algebras of Coxeter groups and of q-Schur algebras, as well as canonical bases of quantized enveloping algebras, of quantum linear groups, and of generalized Temperley-Lieb algebras.

1Partially supported by MURST.

The purpose of this paper is to show that, under some mild hypotheses, P - kernels and IC bases are equivalent, and to characterize, given a Coxeter group W partially ordered by Bruhat order, among all W -kernels the one that corresponds to the Kazhdan-Lusztig basis of the Hecke algebra of W (we call this the Kazhdan-Lusztig kernel of W). We also show that this W -kernel factorizes as a product of other W -kernels, that these provide a solution to the Yang-Baxter equations for W, and we compute explicitly the IC bases corresponding to them.

The organization of the paper is as follows. In the next section we collect some definitions and results that we need in the sequel. In §3 we show the equivalence, given a locally finite poset P, of P -kernels and IC bases. In §4 we give a characterization, among all possible W -kernels, of the Kazhdan-Lusztig kernel of a Coxeter group W. In §5 we show that the Kazhdan-Lusztig kernel factorizes as a product of other W -kernels, we compute explicitly the IC bases corresponding to these factors, and show that they provide a solution to the Yang-Baxter equations for W, thereby generalizing to any Coxeter group a special case of the main result of [7].

2 Definitions, Notation, and Preliminaries

In this section we collect some definitions, notation and results that will be used in the rest of this work. We let P =^{def} {1, 2, 3, …} , N=^{def}P ∪{0}, Z be the ring of integers, Q be the field of rational numbers, and R be the field of real numbers. By a directed graph we mean a pair D = (V, E) where V is a set and E ⊆ V^{2}. We call the elements of V vertices and those of E directed edges. A directed path in D is a sequence Γ = (a_{0}, …, a_{r}) of vertices such that (a_{i−1}, a_{i}) ∈ E for i = 1, …, r, we then say that Γ goes from a_{0} to a_{r}. The length of such a directed path Γ is l(Γ) =^{def} r. If A ⊆ V then the directed graph induced on A by D is (A, E ∩ A^{2}).

We follow [17], Chap. 3, for notation and terminology concerning partially ordered sets. In particular, given a partially ordered set (or, poset, for short) P we let Int(P) =^{def} {(u, v) ∈ P^{2} : u ≤ v}, and given u, v ∈ P we let [u, v] =^{def} {x ∈ P : u ≤ x ≤ v}, and call this an interval of P. We consider [u, v] as a poset with the partial ordering induced by P. We say that a poset P is locally finite if |[u, v]| < +∞ for all (u, v) ∈ Int(P), and we denote by δ_{P} the delta function of P. We will usually omit the index P if there is no danger of confusion.

Recall (see, e.g., [17], §3.6) that given a locally finite poset P and a commutative ring R the incidence algebra of P with coefficients in R, denoted I(P; R), is the set of all functions f : Int(P) → R with sum and product defined by

(f + g)(u, v) =^{def} f(u, v) + g(u, v)

and

(fg)(u, v) =^{def} ∑_{u≤z≤v} f(u, z) g(z, v),

for all f, g ∈ I(P; R) and (u, v) ∈ Int(P). It is well known (see, e.g., [17], §3.6, and Proposition 3.6.2) that I(P; R) is an associative algebra having δ as identity element, and that an element f ∈ I(P; R) is invertible if and only if f(u, u) is invertible in R for all u ∈ P. If f is invertible then we denote by f^{−1} its (two-sided) inverse. We adopt the convention that f(u, v) =^{def} 0 if f ∈ I(P; R) and u, v ∈ P, u ≰ v.

Let P be a locally finite poset, and r : P → Z be such that if u < v then r(u) < r(v). Then ρ(u, v) =^{def} r(v) − r(u) is a weak rank function for P in the sense of [6]. Let I(P) =^{def} I(P; R[q]). Following Stanley (see [18],p. 830, and Proposition 6.11, p. 835) we let

Ĩ(P) =^{def} {f ∈ I(P) : deg(f(u, v)) ≤ ρ(u, v), for all (u, v) ∈ Int(P)},

and

I_{\frac{1}{2}} (P) =^{def} \{f ∈ Ĩ(P) : deg(f(u, v)) < \frac{1}{2}ρ(u, v) for u < v, and f(u, u) = 1\}.

Note that Ĩ(P) is a subalgebra of I(P) and that, if f ∈ I(P) is invertible, then f ∈ Ĩ(P) if and only if f^{−1} ∈ Ĩ(P). Given f ∈ Ĩ(P) we let

\overline{f}(u, v) =^{def} q^{ρ(u,v)}f(u, v) (\frac{1}{q}) ,

for all (u, v) ∈ Int(P). Notice that Ĩ(P), I_{\frac{1}{2}} (P), and the involution ^{−} all depend also on r. However, throughout this work r will always be fixed, so no confusion should arise.

Following [18], Definition 6.2, p. 830, we say that an element K ∈ I(P) is a P -kernel (or, more simply, a kernel) if K is unitary (i.e., K(u, u) = 1 for all u ∈ P) and there exists an element f ∈ I(P) such that: i) f is invertible in I(P);

ii) Kf = \overline{f}.

An element f ∈ I(P) satisfying ii) above is called K-totally acceptable (see [18], Definition 6.2, p.830)^{2}. The next result was first proved by Stanley in the locally graded case (see [18], Corollary 6.7), and by the author in the locally finite one (see [6, Theorem 6.2]).

Theorem 2.1 Let P be a locally finite poset and K ∈ I(P) a P -kernel. Then there exists a unique K-totally acceptable element γ ∈ I_{\frac{1}{2}} (P).

We call the element γ whose existence and uniqueness is guaranteed by the preceding theorem the Kazhdan-Lusztig-Stanley function (or KLS-function, for short) of K. As noted in [18], §§6 and 7, the function γ specializes to many interesting objects depending on the particular choice of the poset P and kernel K.

There is a simple way to decide if a given element K ∈ I(P) is a P -kernel or not. The following result was first proved by Stanley in [18] (see Theorem 6.5, p. 831) in the case that P is locally graded. However, his proof carries over unchanged to the present more general setting.

Theorem 2.2 Let P be a locally finite poset and K ∈ I(P) be unitary. Then K is a P -kernel if and only if K\overline{K} = δ.

Note that Theorem 2.1 defines a map from the set of P -kernels to I_{\frac{1}{2}} (P) and that, by Theorem 2.2, the map f ↦ \overline{f}f^{−1} is its inverse. Thus the correspondence K ↦ γ of Theorem 2.1 is a bijection. We call this bijection the KLS-correspondence of P and the elements of I_{\frac{1}{2}} (P) the KLS-functions of P.

Let P be a countable set, and M be the free Z[q^{\frac{1}{2}} , q^{− \frac{1}{2}} ]-module with basis {m_{i}}_{i∈P}. Let j : M → M be a Z-linear involution such that j(q) = q^{−1} and j(am) = j(a)j̇(m) for all m ∈ M, a ∈ Z[q^{\frac{1}{2}} , q^{− \frac{1}{2}} ], and let r : P → Z. Let L be the free Z[q^{− \frac{1}{2}} ]-module with basis m′_{k} =^{def} q^{− \frac{r(k)}{2}} m_{k} (k ∈ P). So

L = \{∑_{i∈P} a_{i}(q^{− \frac{1}{2}} ) q^{− \frac{r(i)}{2}} m_{i} : a_{i}(t) ∈ Z[t]\} .

2These definitions are slightly different from those in [18] in that they are “left-handed” (Stanley uses “fK” instead of “Kf”). This choice is more convenient for our purposes, and does not affect the validity of any of the results quoted in this section and used in the sequel.

Clearly, L is a Z[q^{− \frac{1}{2}} ]-submodule of M (seen as a Z[q^{− \frac{1}{2}} ]-module).

Following [10] we say that a basis {c_{i}}_{i∈P} of L is the IC basis of M with respect to ({m_{i}}_{i∈P}, j, L) if, for all i ∈ P, we have that:

i) j(c_{i}) = c_{i};

ii) π(c_{i}) = π(m′_{i}), where π : L → L/q^{− \frac{1}{2}} L is the canonical projection;

iii) {c_{i}}_{i∈P} is the only basis of L satisfying i) and ii).

Examples of IC bases include Kazhdan-Lusztig bases of Hecke algebras of Coxeter groups and of q-Schur algebras, as well as canonical bases of quantized enveloping algebras, of quantum linear groups, and of generalized Temperley-Lieb algebras (see [10] and [14] for details, and further references).

We follow [15] for general Coxeter groups notation and terminology. Given a Coxeter system (W, S) and v ∈ W we denote by l(v) the length of v in W, with respect to S, and we let

D(v) =^{def} {s ∈ S : l(sv) < l(v)},

and ε_{v} =^{def} (−1)^{l(v)}. We denote by e the identity of W, and we let T =^{def} {vsv^{−1} : v ∈ W, s ∈ S} be the set of reflections of W. For u, v ∈ W we also write l(u, v) =^{def} l(v) − l(u). We denote by B(W) the Bruhat graph of W. Recall (see, e.g., [15, §8.6], or [12]) that this is the directed graph having W as vertex set and having a directed edge from u to v if and only if u^{−1}v ∈ T and l(u) < l(v). We then write u → v. If u^{−1}v = r ∈ T we also write u →^{r} v. The transitive closure of B(W) is a partial order on W that is usually called the Bruhat order (see, e.g., [15, §5.9]) and that we denote by ≤. Throughout this work, we always assume that W is partially ordered by ≤.

Let A ⊆ T and W′ be the subgroup of W generated by A. Following [15], §8.2, we call W′ a reflection subgroup of W. It is then known (see, e.g., [15], Theorem 8.2) that (W′, S′) is again a Coxeter system where S′ =^{def} {t ∈ T : N(t) ∩ W′ = {t}},

and N(w) =^{def} {t ∈ T : l(wt) < l(w)}, and that T ∩ W′ = {vtv^{−1} : t ∈ S′, v ∈ W′}

([11, Thm. 3.3(i)]). We call the elements of S′ the canonical generators of W′. We say that W′ is a dihedral reflection subgroup if |S′| = 2 (i.e., if (W′, S′) is a dihedral Coxeter system). Following [13] we say that a total ordering ≺ of T is a reflection ordering if, for any dihedral reflection subgroup W′ of W, we have that either a ≺ aba ≺ ababa ≺ … ≺ babab ≺ bab ≺ b or b ≺ bab ≺ babab ≺ … ≺ ababa ≺ aba ≺ a where {a, b} =^{def} S′. The existence of reflection orderings (and many of their properties) is proved in [13], §2, and [1], §5.2.

A family {R_{τ}}_{τ∈T} of elements of a monoid is called an (extensible) solution to the Yang-Baxter equations for W if for any dihedral reflection subgroup W′ of W with canonical generators a and b, we have

R_{a}R_{aba}R_{ababa} ⋯ R_{bab}R_{b} = R_{b}R_{bab} ⋯ R_{ababa}R_{aba}R_{a} .  (1)

The collection {R_{τ}}_{τ∈T} satisfying the Yang-Baxter equations (1) is sometimes called an (extensible) R-matrix (of the corresponding type). In the case of a Weyl group, equations (1), stated case by case in terms of the root system, were given by I. V. Cherednik (implicit in [8] and explicit in [9, Def. 2.1a]).

Let W′ be a reflection subgroup of W and S′ be its set of canonical generators. Let T′ =^{def} T ∩ W′, and l′ be the length function of W′ with respect to S′. Given u ∈ W it is known (see [11, Corollary 3.4(ii)]) that there is a unique element w_{0} of uW′ having minimal length. We then let u_{0} be the unique element of W′ such that u = w_{0}u_{0}. The following result is an immediate consequence of Theorem 3.3(i) and Corollary 3.4 of [11]. Proposition 2.3 Let W′ be a reflection subgroup of W, and u, v ∈ W be such that v^{−1}u ∈ W′. Then, for each r ∈ T ∩ W′,

u →^{r} v in B(W) ⇔ u_{0} →^{r} v_{0} in B(W) ⇔ u_{0} →^{r} v_{0} in B(W′).

We denote by H(W) the Hecke algebra associated to W. Recall (see, e.g., [15], Chap. 7) that this is the free Z[q, q^{−1}]-module having the set {T_{w} : w ∈ W } as a basis and multiplication such that

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}  (2)

for all w ∈ W and s ∈ S. It is well known that this is an associative algebra having T_{e} as identity and that each basis element is invertible in H(W). More precisely, we have the following result (see [15], Proposition 7.4). Proposition 2.4 Let v ∈ W. Then

(T_{v^{−1}})^{−1} = q^{−l(v)} ∑_{u≤v}(−1)^{l(v)−l(u)} R_{u,v}(q) T_{u} ,

where R_{u,v}(q) ∈ Z[q].

The polynomials R_{u,v} defined by the previous proposition are called the R-polynomials of W. It is easy to see that deg(R_{u,v}) = l(v) − l(u), and that R_{u,u} = 1, for all u, v ∈ W, u ≤ v. It is customary to let R_{u,v} =^{def} 0 if u ≰ v. We then have the following fundamental result that follows from (2) and Proposition 2.4 (see [15], §7.5).

Theorem 2.5 Let u, v ∈ W and s ∈ D(v). Then

R_{u,v}(q) = \{ \begin{matrix}R_{su,sv}(q), & if s ∈ D(u), \\ qR_{su,sv}(q) + (q − 1)R_{u,sv}(q), & if s ∉ D(u). \end{matrix}

Define an element R ∈ I(W) by letting

R(u, v) =^{def} R_{u,v}(q)  (3)

for all (u, v) ∈ Int(W). The next result is well known (although it’s not usually stated in this form) and a proof of it can be found, e.g., in [15, Prop. 7.8] (see also [18, Ex. 6.9]).

Proposition 2.6 R is a W -kernel.

We call R the Kazhdan-Lusztig kernel of W. Let P be the KLS-function of R. It then follows immediately from the definitions and well known results (see, e.g., [15, §7.10]) that

P(u, v) = P_{u,v}(q)

for all (u, v) ∈ Int(W), where {P_{u,v}(q)}_{u,v∈W} are the Kazhdan-Lusztig polynomials of W.

3 P -kernels and IC bases

In this section we show the equivalence, given a locally finite poset P, of P -kernels and IC bases.

Let (P, ≤) be a locally finite poset, r : P → Z, and M be the free Z[q^{\frac{1}{2}} , q^{− \frac{1}{2}} ]- module with basis {m_{i}}_{i∈P}.

Proposition 3.1 Let K ∈ I(P) be unitary. Then the following are equivalent:

i) K is a P -kernel; ii) the map ι : M → M defined by

ι(m_{j}) =^{def} q^{−r(j)} ∑_{i∈P} K(i, j) m_{i},

for j ∈ P, ι(q) = q^{−1}, and Z-linear extension, is an involution of M.

Proof. We have that

ι^{2}(m_{j}) = ∑_{i∈P} q^{r(j)} K(i, j) (\frac{1}{q}) ι(m_{i})
= ∑_{i≤j} q^{r(i)}\overline{K}(i, j) ∑_{k∈P} q^{−r(i)} K(k, i)m_{k}
= ∑_{k∈P} ( ∑_{k≤i≤j} K(k, i) \overline{K}(i, j)) m_{k}.

Therefore ι^{2}(m_{j}) = m_{j} for all j ∈ P if and only if

∑_{k≤i≤j} K(k, i) \overline{K}(i, j) = δ(k, j)

for all k, j ∈ P, and the result follows from Theorem 2.2. □

Keeping the notation as in the previous proposition, we can now prove the main result of this section. Note that by Theorem 2.2 K ∈ I(P) is a P -kernel if and only if \overline{K} is.

Theorem 3.2 Let K ∈ I(P) be a P -kernel, and γ ∈ I(P) be unitary. Then the following are equivalent:

i) γ is the KLS-function of \overline{K};

ii) \{q^{− \frac{r(i)}{2}} ∑_{k∈P}γ(k, i) m_{k}\}_{i∈P} is the IC basis of M with respect to (\{m_{i}\}_{i∈P} , ι, L).

Proof. Let, for brevity, c_{i} =^{def} q^{− \frac{r(i)}{2}} ∑_{k∈P} γ(k, i)m_{k} for i ∈ P. We then have that

ι(c_{i}) = ∑_{k∈P} q^{\frac{r(i)}{2}} γ(k, i) (\frac{1}{q}) ι(m_{k})
= ∑_{k∈P} q^{− \frac{r(i)}{2}} \overline{γ}(k, i) ∑_{j∈P} K(j, k) m_{j}
= ∑_{j∈P} ( ∑_{j≤k≤i} K(j, k)\overline{γ}(k, i)) q^{− \frac{r(i)}{2}} m_{j}

for all i ∈ P. Therefore ι(c_{i}) = c_{i} for all i ∈ P if and only if

∑_{j≤k≤i} K(j, k)γ̄(k, i) = γ(j, i)

for all i, j ∈ P, namely if and only if \overline{K}γ = \overline{γ} in I(P).

Furthermore, c_{i} ∈ L for all i ∈ P if and only if γ(k, i)q^{\frac{r(k)−r(i)}{2}} ∈ Z[q^{− \frac{1}{2}} ] for all k, i ∈ P, namely if and only if deg(γ(k, i)) ≤ \frac{r(i)−r(k)}{2} for all (k, i) ∈ Int(P). Finally, π(c_{i}) = π(m′_{i}) if and only if γ(k, i)q^{− \frac{r(i)}{2}} m_{k} ∈ q^{− \frac{1}{2}} L for all k ≠ i, which happens if and only if deg(γ(k, i)) < \frac{r(i)−r(k)}{2} for all k < i.

The fact that γ ∈ I(P) and γ is unitary make it clear that {c_{i}}_{i∈P} is a basis of L. Suppose now that {d_{i}}_{i∈P} is another basis of L such that ι(d_{i}) = d_{i} and π(d_{i}) = π(m′_{i}) for all i ∈ P. Then ι(d_{i} − c_{i}) = d_{i} − c_{i} and d_{i} − c_{i} ∈ q^{− \frac{1}{2}} L for all i ∈ P, and hence, by the lemma on p. 138 of [10], d_{i} − c_{i} = 0 for all i ∈ P. □

4 The Kazhdan-Lusztig kernel

In this section we prove the main result of this paper. Namely, given a Coxeter group W partially ordered by Bruhat order we characterize, among all W -kernels, the one that corresponds under Theorems 2.1 and 3.2 to the Kazhdan-Lusztig basis of the Hecke algebra of W.

Let W be a Coxeter group, partially ordered by Bruhat order, and H(W) be the Hecke algebra of W. Given K ∈ I(W) we define a map j : H(W) → H(W) by letting

j(T_{w}) =^{def} q^{−l(w)} ∑_{x≤w} ε_{x}ε_{w}K(x, w) T_{x},  (4)

for all w ∈ W, j(q) =^{def} q^{−1}, and Z-linear extension. It follows easily from Proposition 3.1 and Theorem 2.2 that K is a W -kernel if and only if j is an involution. We wish to characterize, among all possible W -kernels, the Kazhdan-Lusztig kernel R defined by (3). The answer turns out to be extremenly simple and elegant.

Theorem 4.1 Let K ∈ I(W) be a W -kernel. Then the following are equivalent:

i) K = R;

ii) j is a ring homomorphism.

Proof. It is well known that if K = R then the map j defined by (4) is a ring homomorphism (see [15, §7.7] and Proposition 2.4).

So let K be a W -kernel, and suppose that j is a ring homomorphism. It is clear that j(T_{e}) = T_{e}. Furthermore we have that

j(T_{s}^{2}) = j((q − 1)T_{s} + qT_{e})
= (q^{−1} − 1)j(T_{s}) + q^{−1}T_{e}
= (q^{−1} − 1) q^{−1}(T_{s} − K(e, s)T_{e}) + q^{−1}T_{e}
= (q^{−2} − q^{−1})T_{s} + ((q^{−1} − q^{−2})K(e, s) + q^{−1})T_{e}.

On the other hand

j(T_{s})j(T_{s}) = q^{−1}(T_{s} − K(e, s)T_{e})q^{−1}(T_{s} − K(e, s)T_{e})
= q^{−2}(T_{s}^{2} − 2K(e, s)T_{s} + K(e, s)^{2}T_{e})
= q^{−2}((q − 1)T_{s} + qT_{e} − 2K(e, s)T_{s} + K(e, s)^{2}T_{e})
= q^{−2}((q − 1 − 2K(e, s))T_{s} + (q + K(e, s)^{2})T_{e}).

Hence q^{−2}(1−q) = q^{−2}(q−1−2K(e, s)) and q^{−2}((q−1)K(e, s)+q) = q^{−2}(q+K(e, s)^{2}) so

K(e, s) = q − 1.  (5)

Let now w ∈ W and s ∈ S \ D(w). Then we have that

j(T_{s}T_{w}) = j(T_{sw}) = −q^{−l(sw)} ∑_{x∈W} ε_{x}ε_{w}K(x, sw)T_{x}.  (6)

On the other hand

j(T_{s})j(T_{w}) = q^{−1}(T_{s} − K(e, s)T_{e})q^{−l(w)} ∑_{x≤w} ε_{x}ε_{w}K(x, w)T_{x}
= q^{−l(w)−1}(T_{s} + (1 − q)T_{e}) ∑_{x≤w} ε_{x}ε_{w}K(x, w)T_{x}
= q^{−l(sw)}ε_{w} [ ∑_{x≤w} ε_{x}K(x, w)(T_{s}T_{x} + (1 − q)T_{x})]
= q^{−l(sw)}ε_{w} [  ∑_{{x≤w:s∈D(x)}} ε_{x}K(x, w)qT_{sx}

+ ∑_{{x≤w:s∉D(x)}} ε_{x}(K(x, w)T_{sx} + (1 − q)K(x, w)T_{x})]

= q^{−l(sw)}ε_{w} [  ∑_{{x∈W:s∈D(x)}} ε_{x}qK(x, w)T_{sx} + ∑_{{x∈W:s∉D(x)}} ε_{x}K(x, w)T_{sx}
+ ∑_{{x∈W:s∉D(x)}} ε_{x}(1 − q)K(x, w)T_{x}] .  (7)

Extracting the coefficient of T_{y} in (6) and (7) yields that

K(y, sw) = \{ \begin{matrix}K(sy, w), & if s ∈ D(y), \\ q K(sy, w) + (q − 1) K(y, w), & if s ∉ D(y). \end{matrix}

for all y ∈ W. This, by (5) and Theorem 2.5 implies, by induction on l(v), that

K(y, v) = R_{y,v}(q),

for all y, v ∈ W, as desired. □

The KLS-function of the W -kernel R, namely the Kazhdan-Lusztig polynomials of W, have many deep and interesting properties (see, e.g., [15, Chap. 7], or [1, Chap 5]), including (for finite Coxeter groups, and conjecturally for all Coxeter groups) remarkable nonnegativity and monotonicity properties (see, e.g., [16], [2], or [4], for a survey and further references). But the KLS-function of R is, by Theorem 2.1, uniquely determined by (and uniquely determines) R. Therefore, as surprising as it may seem, Theorem 4.1 implies that all these properties are a consequence of the simple fact that the map j defined by (4) is a ring homomorphism.

5 Factorization and the Yang-Baxter equations

In this section we define, for each reflection t ∈ T, a W -kernel K_{t}, we compute explicitly the IC bases corresponding to these kernels, and show that they give a solution to the Yang-Baxter equations for W. These kernels are closely related to the Kazhdan-Lusztig kernel since they provide a factorization of it.

Let W be a Coxeter group, and T be its set of reflections. Given t ∈ T we define an element K_{t} ∈ I(W) by letting

K_{t}(u, v) =^{def} \{ \begin{matrix}q^{\frac{l(u,v)}{2}} (q^{\frac{1}{2}} − q^{− \frac{1}{2}} ), & if u →^{t} v, \\ 1, & if u = v, \\ 0, & otherwise, \end{matrix}

for all (u, v) ∈ Int(W). Given A ⊆ T (A finite or infinite) and a total order ≺ on A we define

K_{A}^{≺} =^{def} ∏_{t∈A} K_{t}  (8)

the product being in I(W), and the factors being taken in the order given by ≺. Note that K_{A}^{≺} is well defined since, if (u, v) ∈ Int(W) then the interval [u, v] is finite, and hence the directed graph induced by B(W) on [u, v] is also finite. So there is a finite subset B ⊆ A such that K_{A}^{≺}(u, v) = K_{B}^{≺}(u, v).

For u, v ∈ W we denote by B_{A}^{≺}(u, v) the set of all the directed paths u = a_{0} →^{t_{0}}

a_{1} →^{t_{1}} … →^{t_{r−1}} a_{r} = v in B(W) such that:

i) t_{i} ∈ A for all i = 0, …, r − 1;

ii) t_{0} ≺ t_{1} ≺ … ≺ t_{r−1}.

In other words, if we label each directed edge u →^{t} v in the Bruhat graph with the reflection t, then B_{A}^{≺}(u, v) is the set of all the directed paths from u to v in B(W) with edges whose labels are in A, and are increasing, along the path, with respect to the total order ≺.

The verification of the next simple result is left to the reader.

Proposition 5.1 Let A ⊆ T and ≺ be a total order on A. Then

K_{A}^{≺}(u, v) = q^{\frac{l(u,v)}{2}} ∑_{Γ∈B_{A}^{≺}(u,v)}(q^{\frac{1}{2}} − q^{− \frac{1}{2}} )^{l(Γ)}

for all (u, v) ∈ Int(W). □

In particular, using a well known result on the Kazhdan-Lusztig R-polynomials (see, e.g., [1, Theorem 5.3.4]), we obtain the following factorization result.

Corollary 5.2 Let ≺ be a reflection ordering on T. Then

R = K_{T}^{≺}

in I(W).

Our aim is to show that the elements K_{t} are W -kernels, to compute their corresponding IC-bases, and to show that they give a solution to the Yang-Baxter equations for W. All these results will follow from the next one, which is the main result of this section.

Let W′ be a reflection subgroup of W, and T′ =^{def} T ∩ W′. Denote by ≤′ the Bruhat order of W′. We define an element P_{T′} ∈ I(W) by letting

P_{T′}(u, v) = \{ \begin{matrix}q^{\frac{l(u,v)−l′(u_{0},v_{0})}{2}} P′_{u_{0},v_{0}}(q), & if v^{−1}u ∈ W′, \\ 0, & otherwise, \end{matrix}  (9)

for all (u, v) ∈ Int(W), where l′, u_{0}, and v_{0} have the same meaning as in Proposition 2.3, and P′_{u_{0},v_{0}}(q) denotes the Kazhdan-Lusztig polynomial of u_{0}, v_{0} as elements of W′. Note that it is possible that u_{0} ≰′ v_{0} even if u ≤ v. In this case P′_{u_{0},v_{0}}(q) = 0 and so also P_{T′}(u, v) = 0.

Theorem 5.3 Let W′ be a reflection subgroup of W and ≺′ be a reflection ordering of W′. Then K_{T′}^{≺′} is a W -kernel and P_{T′} is its KLS-function.

Proof. It is clear that P_{T′} is unitary and that deg(P_{T′}(u, v)) < \frac{l(u,v)}{2} if u < v (u, v ∈ W). Therefore, by Theorem 2.1 and the definition of kernel we only have to show that

K_{T′}^{≺′} P_{T′} = \overline{P_{T′}}  (10)

in I(W).

Note that (K_{T′}^{≺′} P_{T′})(u, v) = P_{T′}(u, v) = 0 if v^{−1}u ∉ W′ (for if there is a directed path in B_{T′}^{≺′}(u, v) then necessarily v^{−1}u ∈ W′) so (10) certainly holds in this case. So suppose that uW′ = vW′ and let w_{0} be the element of minimal length in this coset. Then we have that

(K_{T′}^{≺′} P_{T′})(u, v) = ∑_{u≤a≤v} K_{T′}^{≺′}(u, a) P_{T′}(a, v)
= ∑_{a∈[u,v]∩uW′} K_{T′}^{≺′}(u, a) P_{T′}(a, v)

Now, if a ∈ [u, v] ∩ uW′ is such that P_{T′}(a, v) ≠ 0 and K_{T′}^{≺′}(u, a) ≠ 0 then a_{0} ≤′ v_{0} and there is at least a directed path in B_{T′}^{≺′}(u, a). Therefore all the vertices of this path are in uW′. Hence, by Proposition 2.3, there is a corresponding path in B(W′) and so u_{0} ≤′ a_{0}. Therefore u_{0} ≤′ a_{0} ≤′ v_{0}. Conversely, if b_{0} ∈ W′ is such that u_{0} ≤′ b_{0} ≤′ v_{0} then by Proposition 2.3 u ≤ w_{0}b_{0} ≤ v. Therefore

(K_{T′}^{≺′}P_{T′})(u, v) = q^{\frac{l(u,v)}{2}} ∑_{u_{0}≤′a_{0}≤′v_{0}} q^{− \frac{l′(a_{0},v_{0})}{2}} ∑_{Γ∈B_{T′}^{≺′}(w_{0}u_{0},w_{0}a_{0})}(q^{\frac{1}{2}} − q^{− \frac{1}{2}} )^{l(Γ)} P′_{a_{0},v_{0}}(q).

But by Proposition 2.3 multiplication on the left by w_{0} gives a bijection between B_{T′}^{≺′}(u, a) and B_{T′}^{≺′}(u_{0}, a_{0}) (where we consider u_{0}, a_{0} as elements of W′, and W′ as a Coxeter system in its own right). Therefore, by Proposition 5.1 and Corollary 5.2,

(K_{T′}^{≺′}P_{T′})(u, v) = q^{\frac{l(u,v)}{2}} ∑_{u_{0}≤′a_{0}≤′v_{0}} q^{− \frac{l′(a_{0},v_{0})}{2}} ∑_{Γ∈B_{T′}^{≺′}(u_{0},a_{0})}(q^{\frac{1}{2}} − q^{− \frac{1}{2}} )^{l(Γ)} P′_{a_{0},v_{0}}(q)
= q^{\frac{l(u,v)}{2}} ∑_{u_{0}≤′a_{0}≤′v_{0}} q^{− \frac{l′(u_{0},v_{0})}{2}} R′_{u_{0},a_{0}}(q)P′_{a_{0},v_{0}}(q)
= q^{\frac{l(u,v)−l′(u_{0},v_{0})}{2}} q^{l′(u_{0},v_{0})} P′_{u_{0},v_{0}} (\frac{1}{q})
= q^{l(u,v)} P_{T′}(u, v) (\frac{1}{q}) ,

(where R′_{u_{0},a_{0}}(q) denotes the R-polynomial of u_{0}, a_{0} as elements of W′) and the result follows. □

As a consequence of Theorem 5.3, we can now derive the results stated after Corollary 5.2. The first one can also be verified directly.

Corollary 5.4 Let t ∈ T. Then K_{t} is a W -kernel and its KLS-function P_{t} is given by

P_{t}(u, v) = \{ \begin{matrix}q^{\frac{l(u,v)−1}{2}} , & if u →^{t} v, \\ 1, & if u = v, \\ 0, & otherwise, \end{matrix}  (11)

for all (u, v) ∈ Int(W).

Proof. Take W′ =^{def} {e, t} in Theorem 5.3. □

Corollary 5.5 Let W′ be a dihedral reflection subgroup of W, and {s, t} be its canonical generators.Then:

i) K_{t}K_{tst} … K_{sts}K_{s} is a W -kernel;

ii) its KLS-function P_{⟨s,t⟩} is given by

P_{⟨s,t⟩}(u, v) = \{ \begin{matrix}q^{\frac{l(u,v)−l′(u_{0},v_{0})}{2}} , & if v^{−1}u ∈ W′, and u_{0} ≤′ v_{0}, \\ 0, & otherwise, \end{matrix}

for all (u, v) ∈ Int(W).

iii) K_{s}K_{sts} … K_{tst}K_{t} = K_{t}K_{tst} … K_{sts}K_{s}; iv) P_{s}P_{sts} … P_{tst}P_{t} = P_{t}P_{tst} … P_{sts}P_{s}.

In particular, the families {P_{t}}_{t∈T}, {K_{t}}_{t∈T} ⊆ I(W) satisfy the Yang-Baxter equations for W.

Proof. i) and ii) follow directly from Theorem 5.3 using the reflection ordering t ≺ tst ≺ … ≺ sts ≺ s and the well known fact (see, e.g., [15, §7.12]) that P_{u,v}(q) = 1 for all u ≤ v in a dihedral Coxeter group. Regarding iii), since the role of s and t is symmetrical we deduce from i) that K_{s}K_{sts} … K_{tst}K_{t} is also a W -kernel and from ii) that P_{⟨s,t⟩} is its KLS-function. But then K_{s}K_{sts} … K_{tst}K_{t} and K_{t}K_{tst} … K_{sts}K_{s} are two W -kernels that have the same KLS-function, hence they are equal.

To prove iv) note that

(P_{t}P_{tst} … P_{sts}P_{s})(u, v) = q^{\frac{l(u,v)}{2}} ∑_{Γ∈B_{T′}^{≺}(u,v)} q^{− \frac{l(Γ)}{2}} .

Therefore,

(P_{s}P_{sts} … P_{tst}P_{t})(u, v) = q^{\frac{l(u,v)}{2}} ∑_{Γ∈B_{T′}^{≺^{∗}}(u,v)} q^{− \frac{l(Γ)}{2}}

where ≺^{∗} is the ordering of T′ =^{def} W′ ∩ T which is the opposite of ≺. But by Proposition 5.1 (applied to A = W′ ∩ T) and part iii) we have that

∑_{Γ∈B_{T′}^{≺}(u,v)}(q^{\frac{1}{2}} − q^{− \frac{1}{2}} )^{l(Γ)} = ∑_{Γ∈B_{T′}^{≺^{∗}}(u,v)}(q^{\frac{1}{2}} − q^{− \frac{1}{2}} )^{l(Γ)}

for all (u, v) ∈ Int(W). Since lim_{q→0^{+}}(q^{\frac{1}{2}} − q^{− \frac{1}{2}} ) = −∞ and lim_{q→+∞}(q^{\frac{1}{2}} − q^{− \frac{1}{2}} ) = +∞ this implies that

∑_{Γ∈B_{T′}^{≺}(u,v)} x^{l(Γ)} = ∑_{Γ∈B_{T′}^{≺^{∗}}(u,v)} x^{l(Γ)}  (12)

in N[x] for all (u, v) ∈ Int(W), and iv) also follows. □

Part iii) of Corollary 5.5 generalizes to any Coxeter group a special case of the main result of [7] (Theorem 3.1).

Note that the condition “u_{0} ≤′ v_{0}” cannot be omited from part ii) of Corollary 5.5. For example, if W = S_{4} and W′ = {4231, 1324, 4321, 1234} then s = 1324 and t = 4231 are the canonical generators of W′ and s ≤ t in W. However s_{0} = s ≰′ t = t_{0} so P_{⟨s,t⟩}(s, t) = 0.

We should mention that it is possible to give an explicit combinatorial formula for the polynomial in (12). In fact, by Proposition 2.3, the directed graph induced on W′ by the Bruhat graph of W is isomorphic to the Bruhat graph of W′ (as a Coxeter system). Using this one can show that

∑_{Γ∈B_{T′}^{≺}(u,v)} x^{l(Γ)} = \{ \begin{matrix}\begin{matrix}⌊ & \frac{k}{2} & ⌋ \\ ∑ \\ i=0 \end{matrix}C_{o}(k, k − 2i) x^{k−2i}, & if v^{−1}u ∈ W′, and u ≤′ v, \\ 0, & otherwise, \end{matrix}  (13)

where k =^{def} l′(u_{0}, v_{0}) and C_{o}(k, k − 2i) denotes the number of compositions of k into k − 2i odd parts (so C_{o}(k, k − 2i) = (\begin{matrix}k−i−1 \\ i \end{matrix}) and C_{o}(0, 0) =^{def} 1).

Note that it is not true that any product of the kernels K_{t} (t ∈ T) is again a W -kernel. For example, if W is a noncommutative Coxeter group, and s, t ∈ S are such that st ≠ ts then one can verify that

(K_{s}K_{t}\overline{K}_{s}\overline{K}_{t})(e, st) = −q(q^{\frac{1}{2}} − q^{− \frac{1}{2}} )^{2},

so K_{s}K_{t}\overline{K}_{s}\overline{K}_{t} ≠ δ, and hence, by Theorem 2.2, K_{s}K_{t} is not a W -kernel. On the other hand, if s, t ∈ S are such that st = ts then it follows from Corollary 5.5 that K_{s}K_{t} is a W -kernel. It would be interesting to know for which subsets A ⊆ T and for which orderings ≺ of A the product K_{A}^{≺} defined by (8) is a W -kernel.

Since the Kazhdan-Lusztig kernel R is a product of the W -kernels {K_{t}}_{t∈T} (Corollary 5.2) it is natural to wonder whether the KLS-function of R can be expressed in some way as a product, in I(W), of the elements {P_{t}}_{t∈T}. Unfortunately, this is false. In fact, let Π be such a product. Then it is not hard to see that

Π(u, v) = q^{\frac{l(u,v)}{2}} ∑_{Γ} q^{− \frac{l(Γ)}{2}}  (14)

where Γ runs over some subset of the directed paths from u to v in B(W). But if [u, v] is a lattice (e.g., if u = 2143, v = 4231 in S_{4}) then it follows from (the proof of) Prop. 3.3 of [12] that all the directed paths from u to v in B(W) have length l(u, v). Therefore, by (14), Π(u, v) is a constant. On the other hand, by Theorem 6.3 of [3] the Kazhdan-Lusztig polynomial of such a pair u, v equals the g-polynomial of the poset [u, v]^{∗} (see, e.g., [17, §3.14] for the definition of the g-polynomial of an Eulerian poset) and this is not always a constant (for example, P_{2143,4231}(q) = 1+q).

Acknowledgements. I would like to thank Richard Stanley for some useful comments .

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