Two-sided Cells Research Articles

Kawanaka Invariants for Representations of Weyl Groups

Let W be a Weyl group and let V be the natural CW-module, i.e., the reflection representation. For a complex irreducible character χ of W, we consider the invariantIχ;q≔|W|−1∑w∈Wχw2det1+qw|V/det1−qw|Vintroduced by N. Kawanaka. We determine I(χ;q) explicitly. Looking over these results, we observe a relation between Kawanaka's invariants I(χ;q) and the two-sided cells. For example, if a two-sided cell consists of a single element χ, then the Kawanaka invariant I(χ;q) can be expressed as ∏li=1(1+qhi)/(1−qhi) with some integers hi. This expression can be regarded as a quantization of the usual hook formula for the dimension of irreducible representations of symmetric groups.

The distinguished involutions of the weyl group of type E 6

In [1] Kazhdan and Lusztig defined the concepts of left. right. and two-sided cells for the Coxeter group (W,S) which play an important role in the representation theory of its associated Hecke algebra. The concept of left cell has its origin in the work of Robinson and Shensted, who described a 1-1 correspondence between the synlmetric group S n and the set of pairs of standard Yong tableallx of the sanle shape, of size n. In this picture, a left cell of S n. appears as the subset of S n corresponding to the set of pairs with fixed. In 1982, Barbasch and Vogan clescribe explicitly the left cells of the Weyl group of the type B n, D n by using prin1itive ideals in the envelopillg algebra of a con1plex sen1isin1ple Lie algebra. In [8] Tong (jIlangqing describes explicitly the left cells of the Weyl group of the type E6 by using the method introduced by Shiin [6]. Lusztig shows that each 1eft cell of W contains a unique distinguished involution which play an imporatant role in the ring J given in G. Lllsztig [3]. In this paper vve find all distinguished involutions for the Weyl group of type E6. The paper is organized as follows. In Section 1 we recall some general results on the relation the a-function, and distinguished involutions for Weyl groups. In Section:2 we describe canonical expressions for elernents of the Weyl group of type E6. In Section 3 we describe explicitly a11 distinguished involutions for the Weyl group W of type E6.

Invariants for representations of Weyl groups and two-sided cells

The notion of two-sided cell, which was originally introduced by A. Joseph and reformulated by D. Kazhdan and G. Lusztig, has played an important role in the representation theory. Results concerning them have been obtained by very deep and sometimes ad hoc arguments. Here we introduce certain polynomial invariants for irreducible representations of Weyl groups. Our invariants are easily calculated, and the calculational results show that they almost determine the two-sided cells. Moreover, the factorization pattern of our polynomial invariants seems to be controlled by the natural parameter set M(G) of each two-sided cell.

Support varieties for quantum groups

For any module $M$ over small quantum group one defines the support variety using construction from the theory of restricted Lie algebras. It is a closed conical subset of nilpotent cone of the corresponding Lie algebra. If module $M$ is a module over the quantum group $U_{\xi}$ with divided powers then its support variety is invariant under the action of the corresponding algebraic group. In this case we relate codimension $2a$ of the support variety of $M$ in nilpotent cone and dimension of $M$. Namely, we prove that $\dim M$ is `almost' divisible by $l^a$. Further, we give an a priori estimate for support variety of a module in a given linkage class. We compute the support varieties for Weyl modules. Also we compute the support varieties for tilting modules over quantum $SL_n$ and verify in this case Humphreys' Conjecture which relates support varieties of tilting modules with Lusztig's bijection between nilpotent orbits and two-sided cells in the affine Weyl group.

Left Cells in the Affine Weyl Group of TypeF̃4

By applying an algorithm designed before, we complete the description for all the left cells of the affine Weyl groupWaof typeF̃4by finding a representative set of its left cells together with all its left cell graphs (or with all the associated essential graphs) in each of its two-sided cells. The generalized τ-invariants of left cells ofWaare exhibited graphically. A group-theoretical interpretation is given on the numbers of left cells ofWain some two-sided cells. Thus so far the left cells of all the affine Weyl groups of ranks less than or equal to 4 have been known explicitly. Some techniques are developed in applying the algorithm. As a consequence, we complete the verification of a conjecture concerning the characterization of left cells of Weyl groups and affine Weyl groups.

Cells in Affine Weyl Groups and Tensor Categories

In this paper we construct a tensor (or monoidal) category for any two-sided cell in a finite or affine Weyl group. The isomorphism classes of simple objects of this category correspond to the elements of the two-sided cell; the tensor product arises by a certain truncation procedure from the standard convolution of perverse sheaves on a flag manifold, which imitates geometrically the definition of the ringJgiven in G. Lusztig (J. Algebra109, 1987, 536–548). Similarly, we construct, for any left cell, a tensor category in which the isomorphism classes of simple objects correspond to the elements in the left cell intersected with its inverse (a right cell). This contains as a special case the category of equivariant perverse sheaves on an affine grassmannian, which, as a consequence of G. Lusztig (Astérisque101–102(1983), 208–209) is a tensor category for convolution (without truncation), closely related to the tensor category of representations of the group dual, in the sense of Langlands, to the group which gives rise to the affine grassmannian. But our construction can be specialized in other ways, and it gives a (conjectural) realization of the tensor category of representations of the reductive quotient of the centralizer of any unipotent element of the Langlands dual group.

On Generalized Cells in Affine Weyl Groups

We determine the lowest generalized two-sided cell for affine Weyl groups. We show that it consists of at most |W0| generalized left cells, whereW0denotes the corresponding finite Weyl group. For parameters coming from graph automorphisms, we prove that this bound is exact. For such parameters, we also characterize all generalized left cells for finite and affine Weyl groups.

The Partial Order on Two-Sided Cells of Certain Affine Weyl Groups

The Partial Order on Two-Sided Cells of Certain Affine Weyl Groups

Canonical bases for the Brauer centralizer algebra

In this paper we construct canonical bases for the Birman-Wenzl algebra BWn, the q-analogue of the Brauer centralizer algebra, and so define left, right and two-sided cells. We describe these objects combinatorially (generalizing the Robinson-Schensted algorithm for the symmetric group) and show that each left cell carries an irreducible representation of BWn. In particular, we obtain canonical bases for each representation, defined over Z. The same technique generalizes to an arbitrary tangle algebra and Rmatrix [R]; in particular to centralizers of the quantum group action on V ⊗r, for V a finite dimensional representation of a quantum group. BWn occurs for particular values of the parameters (q, r, x) as the centralizers of the action of Uqsp2k or Uqok on the n-th tensor power of its standard representation V . One may presumably transfer the bases of the BWn modules to give a basis of representations occurring in V ⊗n (as in [GL]), and it is natural to conjecture that the basis so obtained coincides with that of [L,§27]. Of the Weyl groups, only in the symmetric group are the cell representations irreducible. In this respect BWn is similar to Sn. One would expect this because of the relation with quantum groups, which also behave like Hecke algebras of type A [L]. Moreover, our main new insight into the structure of BWn is precisely of this form—we show that every representation is induced from a representation of a symmetric group in a precise way (see §6.5). This paper is essentially self-contained, except for an appeal to the solution of the corresponding problem for Sn in [KL,1.4]. In particular, we make no further mention of quantum groups and use no previous work on the structure of BWn (e.g. [BW,HR,W]) except for its description as a

Left cells in affine Weyl groups

We prove a property of left cells in certain crystallographic groups W , by which we formulate an algorithm to find a representative set of left cells of W in any given two-sided cell. As an illustration, we make some applications of this algorithm to the case where W is the affine Weyl group of type e F4. The cells of affine Weyl groups W , as defined by Kazhdan and Lusztig in [6], have been described explicitly in certain special cases: for W of rank 2, see [11]; for W of type An, see [16], [10]; for W of rank 3, see [1], [4]; for the cells with a-values 1, 2 and |Φ|/2 in a general W , see [2], [8], [9], [18], [19], where Φ is the root system determined by W . It is known that there exists a bijection between the set of two-sided cells in an affine Weyl group W and the set of unipotent classes in a certain complex reductive group G associated with W . It is also known that the value of the a-function on a two-sided cell of W is equal to the dimension of the variety of Borel subgroups of G containing an element of the corresponding unipotent class (see [14]). Thus for an affine Weyl group W , the two-sided cells of W are relatively well understood to certain extent. But the classification of left cells in a given two-sided cell of W is not known in general, even the number of these left cells. In the present paper, we shall introduce an algorithm to find a representative set of left cells of W ′ in a given two-sided cell, where W ′ is a group belonging to certain family of crystallographic

The based ring of the lowest two-sided cell of an affine Weyl group. II

The based ring of the lowest two-sided cell of an affine Weyl group. II

The based ring of the lowest two-sided cell of an affine Weyl group

We show that Lusztig’s homomorphism from an affine Hecke algebra to the direct summand of its asymptotic Hecke algebra corresponding to the lowest two-sided cell is related to the homomorphism constructed by Chriss and Ginzburg using equivariant K-theory by a matrix over the representation ring of the associated algebraic group.

Cells and the Reflection Representation of Weyl Groups and Hecke Algebras

Let $\mathcal {H}$ be the generic algebra of the finite crystallographic Coxeter group $W$, defined over the ring $\mathbb {Q}[{u^{1/2}},{u^{ - 1/2}}]$. First, the two-sided cell corresponding to the reflection representation of $\mathcal {H}$ is shown to consist of the nonidentity elements of $W$ having a unique reduced expression. Next, the matrix entries of this representation are computed in terms of certain Kazhdan-Lusztig polynomials. Finally, the Kazhdan-Lusztig polynomials just mentioned are described in case $W$ is of type ${{\text {A}}_{l - 1}}$ or ${{\text {B}}_l}$.

Cells and the reflection representation of Weyl groups and Hecke algebras

Let H \mathcal {H} be the generic algebra of the finite crystallographic Coxeter group W W , defined over the ring Q [ u 1 / 2 , u − 1 / 2 ] \mathbb {Q}[{u^{1/2}},{u^{ - 1/2}}] . First, the two-sided cell corresponding to the reflection representation of H \mathcal {H} is shown to consist of the nonidentity elements of W W having a unique reduced expression. Next, the matrix entries of this representation are computed in terms of certain Kazhdan-Lusztig polynomials. Finally, the Kazhdan-Lusztig polynomials just mentioned are described in case W W is of type A l − 1 {{\text {A}}_{l - 1}} or B l {{\text {B}}_l} .

Induced cells

We define the concept of induced cells for affine Weyl groups which is compatible with the concept of induced unipotent classes under Lusztig’s bijection between the set of two-sided cells of an affine Weyl group and the set of unipotent classes of a corresponding connected reductive algebraic group over ${\mathbf {C}}$.

Two-sided cells in the affine Weyl group of type [formula omitted

Two-sided cells in the affine Weyl group of type [formula omitted

Two-Sided Cells of the Affine Weyl Group of TypeC˜3

Two-Sided Cells of the Affine Weyl Group of TypeC˜3

A Two-Sided Cell in an Affine Weyl Group, II

A Two-Sided Cell in an Affine Weyl Group, II

The lowest two-sided cell for an affine weyl group

The lowest two-sided cell for an affine weyl group

A Two-Sided Cell in an Affine Weyl Group

A Two-Sided Cell in an Affine Weyl Group