The Two-Step Nilpotent Representations of the Extended Affine Hecke Algebra of Type A

We define a set of cell modules for the extended affine Hecke algebra of type A which are parametrised by SL_n()-conjugacy classes of pairs (s, N), where s SL_n() is semisimple and N is a nilpotent element of the Lie algebra which has at most two Jordan blocks and satisfies Ad(s)·N=q ² N. When q ²–1, each of these has irreducible head, and the irreducible representations of the affine Hecke algebra so obtained are precisely those which factor through its Temperley–Lieb quotient. When q ²=–1, the above remarks apply to a subset of the cell modules. Using our work on the cellular nature of those quotients, we are able to obtain complete information on the decomposition of the cell modules in all cases, even when q is a root of unity. They turn out to be multiplicity free, and the composition factors may be precisely described in terms of a partial order on the pairs (s, N). These results give explicit formulae for the dimensions of the irreducibles. Assuming our modules are identified with the standard modules earlier defined by Bernstein–Zelevinski, Kazhdan–Lusztig and others, our results may be interpreted as the determination of certain Kazhdan–Lusztig polynomials. [This has now been proved and will appear in a subsequent work of the authors.]The second author thanks the Australian Research Council and the Alexander von Humboldt Stiftung for support and the Universität Bielefeld for hospitality during the preparation of this work.     

Row and Column Removal Theorems for Homomorphisms of Specht Modules and Weyl Modules

We prove a q-analogue of the row and column removal theorems for homomorphisms between Specht modules proved by Fayers and the first author [16]. These results can be considered as complements to James and Donkin’s row and column removal theorems for decomposition numbers of the symmetric and general linear groups. In this paper we consider homomorphisms between the Specht modules of the Hecke algebras of type A and between the Weyl modules of the q-Schur algebra.This research was supported by ARC grant DP0343023. The first author was also supported by a Sesqui Research Fellowship at the University of Sydney.     

Modular Branching Rules and the Mullineux Map for Hecke Algebras of Type A

We prove the quantum version - for Hecke algebras H A_n of typeA at roots of unity - of Kleshchev's modular branching rulefor symmetric groups. This result describes the socle of therestriction of an irreducible H A_n-module to the subalgebraH A_n–1. As a consequence, we describe the quantum versionof the Mullineux involution describing the irreducible moduleobtained on twisting an irreducible module with the sign representation.1991 Mathematics Subject Classification: 20C05, 20G05.     

Canonical bases and affine Hecke algebras of type D

We prove a conjecture of Miemietz and Kashiwara on canonical bases and branching rules of affine Hecke algebras of type D. The proof is similar to the proof of the type B case in Varagnolo and Vasserot (in press) [15].     

On Representations of Affine Hecke Algebras of Type B

Ariki’s and Grojnowski’s approach to the representation theory of affine Hecke algebras of type A is applied to type B with unequal parameters to obtain – under certain restrictions on the eigenvalues of the lattice operators – analogous multiplicity-one results and a classification of irreducibles with partial branching rules as in type A. Research supported by the Studienstiftung des deutschen Volkes.     

一类组合不等式的简证与推广

利用概率方法给出了形如sum from k=1 to n(1/k)>π/4(sum from k=1 to n((-1)k-1Cnk)1/(k～1/2))与sum from k=1 to n(1/k)<2～(1/2)(sum from k=1 to n((-1)k-1Cnk)1/k2)1/2的组合不等式.     

Yang-Baxter bases of 0-Hecke algebras and representation theory of 0-Ariki-Koike-Shoji algebras

After reformulating the representation theory of 0-Hecke algebras in an appropriate family of Yang-Baxter bases, we investigate certain specializations of the Ariki-Koike algebras, obtained by setting q=0 in a suitably normalized version of Shoji's presentation. We classify the simple and projective modules, and describe restrictions, induction products, Cartan invariants and decomposition matrices. This allows us to identify the Grothendieck rings of the towers of algebras in terms of certain graded Hopf algebras known as the Mantaci-Reutenauer descent algebras, and Poirier quasi-symmetric functions. We also describe the Ext-quivers, and conclude with numerical tables.     

Clifford theory on rational Cherednik algebras of imprimitive groups

Ram and Rammage have introduced an automorphism and Clifford theory on affine Hecke algebras. Here we will extend them to cyclotomic Hecke algebras and rational Cherednik algebras.     

Seminormal Representations of Weyl Groups and Iwahori-Hecke Algebras

The purpose of this paper is to describe a general procedurefor computing analogues of Young's seminormal representationsof the symmetric groups. The method is to generalize the Jucys-Murphyelements in the group algebras of the symmetric groups to arbitraryWeyl groups and Iwahori-Hecke algebras. The combinatorics ofthese elements allows one to compute irreducible representationsexplicitly and often very easily. In this paper we do thesecomputations for Weyl groups and Iwahori-Hecke algebras of typesA_n, B_n, D_n, G₂. Although these computations are in reach fortypes F₄, E₆ and E₇, we shall postpone this to another work.1991 Mathematics Subject Classification: primary 20F55, 20C15;secondary 20C30, 20G05.     

Cyclic generators for irreducible representations of affine Hecke algebras

We give a detailed account of a combinatorial construction, due to Cherednik, of cyclic generators for irreducible modules of the affine Hecke algebra of the general linear group with generic parameter q.     

On simple modules of Hecke algebras of type Dn

This is a continuation of our previous work. We classify all the simple ℋ_q(D _n )-modules via an automorphismh defined on the set { λ | D^λ ≠ 0}. Whenf _n(q) ≠ 0, this yields a classification of all the simple ℋ ^q (D _n)- modules for arbitrary n. In general ( i. e., q arbitrary), if λ⁽¹⁾ = λ⁽²⁾,wegivea necessary and sufficient condition ( in terms of some polynomials ) to ensure that the irreducible ℋ_q,1(B _n )- module D^λ remains irreducible on restriction to ℋ_q(D _n ).     

Springer Correspondence for Disconnected Exceptional Groups

The determination of the generalized Springer correspondencefor disconnected reductive groups is completed here by beingachieved in the case of disconnected groups of types D₄ in characteristic3 and E₆ in characteristic 2. It is expected that this correspondencewill provide an important ingredient for the computation ofcharacter values of finite groups of Lie type. Some evidenceis provided in support of the latter statement. 2000 MathematicsSubject Classification 20G05 (primary), 20C33 (secondary).     

Extensions of Modules over Schur Algebras, Symmetric Groups and Hecke Algebras

We study the relation between the cohomology of general linear and symmetric groups and their respective quantizations, using Schur algebras and standard homological techniques to build appropriate spectral sequences. As our methods fit inside a much more general context within the theory of finite-dimensional algebras, we develop our results first in that general setting, and then specialize to the above situations. From this we obtain new proofs of several known results in modular representation theory of symmetric groups. Moreover, we reduce certain questions about computing extensions for symmetric groups and Hecke algebras to questions about extensions for general linear groups and their quantizations.     

On Gouvêa's conjecture in the unobstructed case

In this article, for a residual modular representation defined over an arbitrary finite field, Gouvêa's conjecture which says that the universal deformation ring is isomorphic to a certain Hecke algebra is proven in the unobstructed case.     

A Uniform Generalization of Some Combinatorial Hopf Algebras

We generalize the Hopf algebras of free quasisymmetric functions, quasisymmetric functions, noncommutative symmetric functions, and symmetric functions to certain representations of the category of finite Coxeter systems and its dual category. We investigate their connections with the representation theory of 0-Hecke algebras of finite Coxeter systems. Restricted to type B and D we obtain dual graded modules and comodules over the corresponding Hopf algebras in type A.     

一个组合恒等式的再推广

推广了相关文献中的一个组合恒等式,并得到了与相关文献相同的结果.