Kazhdan and Lusztig have shown that the partition of the symmetric group of degree $n$ into left cells is given by the Robinson-Schensted correspondence. The aim of this paper is to provide a similar description of the left cells in type $B_n$ for a special class of choices of unequal parameters. This is based on a generalization of the Robinson-Schensted correspondence for type $B_n$. We also give an explicit description of the left cell representations and show that they are irreducible and constructible.
In 1979, Vogan introduced a generalised $\τ$ -invariant for characterising primitive ideals in enveloping algebras. Via a known dictionary this translates to an invariant of left cells in the sense of Kazhdan and Lusztig. Although it is not a complete invariant, it is extremely useful in describing left cells. Here, we propose a general framework for defining such invariants which also applies to Hecke algebras with unequal parameters.
A conjecture of C. Bonnafé, M. Geck, L. Iancu, and T. Lam parameterizes Kazhdan-Lusztig left cells for unequal parameter Hecke algebras in type $B_n$ by families of standard domino tableaux of arbitrary rank. Relying on a family of properties outlined by G. Lusztig and the recent work of C. Bonnafé, we verify the conjecture and describe the structure of each cell as a module for the underlying Weyl group.
Following Lusztig, we consider a Coxeter group $W$ together with a weight function $L$. This gives rise to the pre-order relation $\leq_{L}$ and the corresponding partition of $W$ into left cells. We introduce an equivalence relation on weight functions such that, in particular, $\leq_{L}$ is constant on equivalent classes. We shall work this out explicitly for $W$ of type $F_4$ and check that several of Lusztig's conjectures concerning left cells with unequal parameters hold in this case, even for those parameters which do not admit a geometric interpretation. The proofs involve some explicit…
In 1979, Vogan proposed a generalised $τ$-invariant for characterising primitive ideals in enveloping algebras. Via a known dictionary this translates to an invariant of left cells of finite Weyl groups. Although it is not a complete invariant, it is extremely useful in describing left cells. Here, we propose a general framework for defining such invariants which also applies to Hecke algebras with unequal parameters.
These are notes for lectures given at MIT in 1999 (Fall). They contain a discussion of affine Hecke algebras with possibly unequal parameters including a theory of cells and a partly conjectural theory of J-rings.
Let W be a Coxeter group and L be a weight function on W. Following Lusztig, we have a corresponding decomposition of W into left cells, which have important applications in representation theory. We study the case where $W$ is an affine Weyl group of type $\tilde{G_{2}}$. Using explicit computation with \textsf{CHEVIE}, we show that (1) there are only finitely many possible decompositions into left cells and (2) the number of left cells is finite in each case, thus confirming some of Lusztig's conjectures in this case. For the proof, we show some equalities on the Kazhdan-Lusztig polynomials …
Let $(W,S)$ be a Coxeter system, let $G$ be a finite solvable group of automorphisms of $(W,S)$ and let $φ$ be a weight function which is invariant under $G$. Let $φ_G$ denote the weight function on $W^G$ obtained by restriction from $φ$. The aim of this paper is to compare the ${\mathbf{a}}$-function, the set of Duflo involutions and the Kazhdan-Lusztig cells associated to $(W,φ)$ and to $(W^G,φ_G)$, provided that Lusztig's Conjectures hold.
Computations in small Coxeter groups and infinite dihedral groups suggest that Kazhdan-Lusztig cells for unequal parameters obey to some "semicontinuity" phenomenon (as the parameter vary). The aim of this paper is to provide a rigorous theoretical background that allows to state some precise conjectures.
Computations in small Coxeter groups or dihedral groups suggest that the partition into Kazhdan-Lusztig cells with unequal parameters should obey to some semicontinuity phenomenon (as the parameters vary). The aim of this paper is to provide a rigorous theoretical background for supporting this intuition that will allow to state several precise conjectures.
We consider two families of equivalence classes in the Weyl groups of type $B_n$ which are suggested by the study of left cells in unequal parameter Iwahori-Hecke algebras. Both families are indexed by a non-negative integer $r$. For $r=0$, the classes in the first family coincide with left cells corresponding to the equal parameter Iwahori-Hecke algebra. When $r$ is sufficiently large, the equivalence classes in the second family agree with left cells corresponding to a special class of choices of unequal parameters. Our main result shows that the two families of equivalence classes coincide,…
Bremke and Xi determined the lowest two-sided cell for affine Weyl groups with unequal parameters and showed that it consists of at most |W_{0}| left cells where W_{0} is the associated finite Weyl group. We prove that this bound is exact. Previously, this was known in the equal parameter case and when the parameters were coming from a graph automorphism. Our argument uniformly works for any choice of parameters.
We conjecture that the "nilpotent points" of Calogero-Moser space for reflection groups are parametrised naturally by the two-sided cells of the group with unequal parameters. The nilpotent points correspond to blocks of restricted Cherednik algebras and we describe these blocks in the case G(m,1,n) and show that in type B our description produces an existing conjectural description of two-sided cells.
To a Coxeter system $(W,S)$ (with $S$ finite) and a weight function $L : W \to \NM$ is associated a partition of $W$ into Kazhdan-Lusztig (left, right or two-sided) $L$-cells. Let $S^\circ = \{s \in S | L(s)=0\}$, $S^+=\{s \in S | L(s) > 0\}$ and let $C$ be a Kazhdan-Lusztig (left, right or two-sided) $L$-cell. According to the semicontinuity conjecture of the first author, there should exist a positive natural number $m$ such that, for any weight function $L' : W \to \NM$ such that $L(s^+)=L'(s^+) > m L'(s^\circ)$ for all $s^+ \in S^+$ and $s^\circ \in S^\circ$, $C$ is a union of Kazhdan-Lusz…
We consider the partition of a finite Coxeter group $W$ into left cells with respect to a weight function $L$. In the equal parameter case, Lusztig has shown that the representations carried by the left cells are precisely the so-called constructible ones. We show that this holds for general $L$, if the conjectural properties (P1)--(P15) in Lusztig's book on Hecke algebras with unequal parameters hold for $W,L$. Our proofs use the idea (Gyoja, Rouquier) that left cell representations are projective in the sense of modular representation theory. This also gives partly new proofs for Lusztig's r…
Based on empirical evidence obtained using the {\sf CHEVIE} computer algebra system, we present a series of conjectures concerning the combinatorial description of the Kazhdan--Lusztig cells for type $B_n$ with unequal parameters. These conjectures form a far-reaching extension of the results of Bonnafé and Iancu obtained earlier in the so-called ``asymptotic case''. We give some partial results in support of our conjectures.
In this paper we study simple transitive $2$-representations of certain $2$-subcategories of the $2$-category of projective functors over a star algebra. We show that in the simplest case, which is associated to the Dynkin type $A_2$, simple transitive $2$-representations are classified by cell $2$-representations. In the general case we conjecture that there exist many simple transitive $2$-representations which are not cell $2$-representations and provide some evidence for our conjecture.
In 1979, Kazhdan and Lusztig developed a combinatorial theory associated with Coxeter groups. They defined in particular partitions of the group in left and two-sided cells. In 1983, Lusztig generalized this theory to Hecke algebras of Coxeter groups with unequal parameters. We propose a definition of left cells and two-sided cells for complex reflection groups, based on ramification theory for Calogero-Moser spaces. These spaces have been defined via rational Cherednik algebras by Etingof and Ginzburg. We conjecture that these coincide with Kazhdan-Lusztig cells, for real reflection groups. C…
In 1979, Kazhdan and Lusztig introduced the notion of "cells" (left, right and two-sided) for a Coxeter group $W$, a concept with numerous applications in Lie theory and around. Here, we address algorithmic aspects of this theory for finite $W$ which are important in applications, e.g., run explicitly through all left cells, determine the values of Lusztig's $\ba$-function, identify the characters of left cell representations. The aim is to show how type $E_8$ (the largest group of exceptional type) can be handled systematically and efficiently, too. This allows us, for the first time, to solv…
Using a characterization of a generalized $τ$-invariant for intermediate parameter Hecke algebras in type $B_n$, we verify a conjectural description of Kazhdan-Lusztig cells in this setting due to C. Bonnafé, L. Iancu, M. Geck, and T. Lam.
We exhibit a set of operators on pairs of domino tableaux of the same shape sending them to other such pairs with the same right tableau, in such a way that any two pairs with the same right tableau are conjugate by some composition of the operators. Using these operators we give explicit bases for the isotypic components of a classical Kazhdan-Lusztig left cell in terms of Kazhdan-Lusztig basis vectors.
Using the representation theory of Cherednik algebras at $t=0$ and a Galois covering of the Calogero-Moser space, we define the notions of left, right and two-sided Calogero-Moser cells for any finite complex reflection group. To each Caloger-Moser two-sided cell is associated a Calogero-Moser family, while to each Calogero-Moser left cell is associated a Calogero-Moser cellular representation. We study properties of these objects and we conjecture that, whenever the reflection group is real (i.e. is a Coxeter group), these notions coincide with the one of Kazhdan-Lusztig left, right and two-s…
Let "$\leq_L$" be the Kazhdan-Lusztig left cell preorder on the symmetric group $S_n$. Let $w\mapsto (P(w),Q(w))$ be the Robinson-Schensted-Knuth correspondence between $S_n$ and the set of standard tableaux with the same shapes. We prove that for any $x,y\in S_n$, $x\leq_L y$ only if $Q(y)\unrhd Q(x)$, where "$\unrhd$" is the dominance (partial) order between standard tableaux. As a byproduct, we generalize an earlier result of Geck by showing that each Kazhdan-Lusztig basis element $C'_w$ can be expressed as a linear combination of some $m_{uv}$ which satisfies that $u\unrhd P(w)^*$, $v\unrh…
We study Kazhdan-Lusztig cells and the corresponding representations of right-angled Coxeter groups and Hecke algebras associated to them. In case of the infinite groups generated by reflections in the hyperbolic plane about the sides of right-angled polygons we obtain an explicit description of the left and two-sided cells. In particular, we prove that there are infinitely many left cells but they all form only three two-sided cells.
Let W~ be an affine Weyl group, and let C be a left, right, or two-sided Kazhdan--Lusztig cell in W~. Let Reduced (C) be the set of all reduced expressions of elements of C, regarded as a formal language in the sense of the theory of computation. We show that Reduced (C) is a regular language. Hence the reduced expressions of the elements in any Kazhdan--Lusztig cell can be enumerated by a finite state automaton.
Using the representation theory of Cherednik algebras at t=0 and a Galois covering of the Calogero-Moser space, we define the notions of left, right and two-sided Calogero-Moser cells for any finite complex reflection group. To each Caloger-Moser two-sided cell is associated a Calogero-Moser family, while to each Calogero-Moser left cell is associated a Calogero-Moser cellular representation. We study properties of these objects and we conjecture that, whenever the reflection group is real (i.e. is a Coxeter group), these notions coincide with the one of Kazhdan-Lusztig left, right and two-sid…
In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent paper, the authors established a close conn…
We study natural bases for two constructions of the irreducible representation of the symmetric group corresponding to $[n,n,n]$: the {\em reduced web} basis associated to Kuperberg's combinatorial description of the spider category; and the {\em left cell basis} for the left cell construction of Kazhdan and Lusztig. In the case of $[n,n]$, the spider category is the Temperley-Lieb category; reduced webs correspond to planar matchings, which are equivalent to left cell bases. This paper compares the images of these bases under classical maps: the {\em Robinson-Schensted algorithm} between perm…
Consider a weighted Coxeter system $(W,S,\mathscr{L})$. Via its associated Iwahori-Hecke algebra, we may determine the partition of $W$ into Kazhdan-Lusztig cells. In this paper, we use the theory of Vogan classes introduced by Bonnafé--Geck to obtain a combinatorial description of the left cells of type $B_n$ when the ratio of the weights of the first to second generator is $n-1$. We further give information on the left cells when this ratio lies in the interval $(n-2,n-1)$.
In this paper, we study Lusztig's $a$-function for a Coxeter group with unequal parameters. We determine that function explicitly in the ``asymptotic case'' in type $B_n$, where the left cells have been determined in terms of a generalized Robinson--Schensted correspondence by Bonnafé and the second author. As a consequence, we can also show that all of Lusztig's conjectural properties (P1)--(P15) hold in this case, except possibly (P9), (P10) and (P15). Our methods rely on the ``leading matrix coefficients'' introduced by the first author. We also interprete the ideal structure defined by the…
In this paper, we give precise description for the lowest lowest two-sided cell c_{0} and the left cells in it for a weighted Coxeter group of rank 3. Then we show conjectures P1-P15 and \widetilde{P} hold for c_{0} and do some calculation for the based ring of c_{0}.
Combinatorics of Kazhdan-Lusztig cells in affine type $A$ was originally developed by Lusztig, Shi, and Xi. Building on their work, Chmutov, Pylyavskyy, and Yudovina introduced the affine matrix-ball construction (abbreviated AMBC) which gives an analog of Robinson-Schensted correspondence for affine symmetric groups. An alternative approach to Kazhdan-Lusztig theory in affine type $A$ was developed by Blasiak in his work on catabolism. He introduced sign insertion algorithm and conjectured that if one fixes the two-sided cell, the recording tableau of sign insertion process determines uniquel…
Following Lusztig, we consider a Coxeter group $W$ together with a weight function. Geck showed that the Kazhdan-Lusztig cells of $W$ are compatible with parabolic subgroups. In this paper, we generalize this argument to some subsets of $W$ which may not be parabolic subgroups. We obtain two applications: we show that under specific technical conditions on the parameters, the cells of a certain finite parabolic subgroup of $W$ are cells in the whole group, and we decompose the affine Weyl group $\tilde{G}_{2}$ into left and two-sided cells for a whole class of weight functions.
We identify a subalgebra \pH_n of the extended affine Hecke algebra \eH_n of type A. The subalgebra \pH_n is a \u-analogue of the monoid algebra of §_n \ltimes \ZZ_{\geq 0}^n and inherits a canonical basis from that of \eH_n. We show that its left cells are naturally labeled by tableaux filled with positive integer entries having distinct residues mod n, which we term \emph{positive affine tableaux} (PAT). We then exhibit a cellular subquotient \R_{1^n} of \pH_n that is a \u-analogue of the ring of coinvariants \CC[y_1,...,y_n]/(e_1,...,e_n) with left cells labeled by PAT that are essentially …
According to an old result of Schützenberger, the involutions in a given two-sided cell of the symmetric group $\SG_n$ are all conjugate. In this paper, we study possible generalisations of this property to other types of Coxeter groups. We show that Schützenberger's result is a special case of a general result on "smooth" two-sided cells. Furthermore, we consider Kottwitz' conjecture concerning the intersections of conjugacy classes of involutions with the left cells in a finite Coxeter group. Our methods lead to a proof of this conjecture for classical types; combined with previous work, thi…
From the combinatorial characterizations of the right, left, and two-sided Kazhdan-Lusztig cells of the symmetric group, 'RSK bases' are constructed for certain quotients by two-sided ideals of the group ring and the Hecke algebra. Applications to invariant theory, over various base rings, of the general linear group and representation theory of the symmetric group are discussed.
We consider an involution on the affine Weyl group of type $A$ induced from the nontrivial automorphism on the (finite) Dynkin diagram. We prove that the number of left cells fixed by this involution in each two-sided cell is given by a certain Green polynomial of type $A$ evaluated at -1.
We introduce cell modules for the tabular algebras defined in a previous work (math.QA/0107230); these modules are analogous to the representations arising from left Kazhdan--Lusztig cells. The standard modules of the title are constructed in an elementary way by suitable tensoring of the cell modules. We show how a certain extended affine Hecke algebra of type $A$ equipped with its Kazhdan--Lusztig basis is an example of a tabular algebra, and verify that in this case our standard modules coincide with other standard modules defined in the literature.
