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.     

A Combinatorial Description of the Syzygies of Certain Weyl Modules

Abstract

A basis for the syzygies of the resolution of 3-rowed Weyl modules with at most one triple overlap presented by Buchsbaum and Rota (Buchsbaum D., Rota, G.-C. (1994). A new construction in homological algebra. Proc. Natl. Acad. Sci. 91:4115–4119.) is constructed using Letter-Place techniques. This basis is explicitly given by computing the image under the differential of a conveniently chosen subset of the canonical Letter-Place basis of each term in the resolution.     

Superdecomposable Pure-Injective Modules and Integral Group Rings

We prove that if G is a non-trivial finite group, then the integralgroup ring ZG possesses a superdecomposable pure-injective module.     

The Extension Group of the Simple Modules Over the First Weyl Algebra

The aim of this paper is to prove that for certain generalizedWeyl algebras A (including the first Weyl algebra A₁ over afield of characteristic zero) and for every simple left (right)A-module M, there are infinitely many non-isomorphic simpleleft (right) A-modules {N_i} such that (respectively ). 1991 Mathematics Subject Classification 18G15, 16G.     

Extended Affine Weyl Groups: Presentation by Conjugation via Integral Collection

We give several necessary and sufficient conditions for the existence of the presentation by conjugation for a non-simply laced extended affine Weyl group. We invent a computational tool by which one can determine simply the existence of the presentation by conjugation for an extended affine Weyl group. As an application, we determine the existence of the presentation by conjugation for a large class of extended affine Weyl groups.     

Essential Dimension of Algebraic Groups and Integral Representations of Weyl Groups

Upper bounds on the essential dimension of algebraic groups can be found by examining related questions about the integral representation theory of lattices for their Weyl groups. We examine these questions in detail for all simple affine algebraic groups, expanding on work of Lorenz and Reichstein for PGL_n. This results in upper bounds on the essential dimensions of these simple affine algebraic groups which match or improve on the previously known upper bounds.     

B型和C型Weyl模的一个新重数公式

A monomial basis and a filtration of subalgebras for the universal enveloping algebra U(gl) of a complex simple Lie algebra gl of type Bl and Cl are given, and the decomposition of the Weyl module V (λ) as a U(gl)-module into a direct sum of Weyl modules V (μ)’s as U(gl-1)modules is described. In particular, a new multiplicity formula for the Weyl module V (λ) is obtained in this note.     

Homotopies for the Generalized Bar Complex Associated to Certain 3-Rowed Weyl Modules

In Buchsbaum and Rota (1994 Buchsbaum , D. , Rota , G.-C. ( 1994 ). A new construction in homological algebra . Proc. Nat. Acad. Sci. 91 : 4115 – 4119 . [CSA] [CROSSREF] [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), the authors presented a generalized bar complex associated to certain 3-rowed Weyl modules and proved that this complex is in fact a resolution via an induction on the number of overlaps between the second and third rows and a fundamental exact sequence (Akin and Buchsbaum, 1985 Akin , K. , Buchsbaum , D. ( 1985 ). Characteristic-free representation theory of the general linear group . Adv. Math 58 ( 2 ): 149 – 200 . [CSA]  [Google Scholar]). In this article we study the structure of this resolution by constructing a splitting contracting homotopy for the complexes corresponding to certain shapes.     

Forcing Linearity Numbers of Semisimple Modules over Integral Domains

In this paper we determine the forcing linearity numbers for semisimple modules over integral domains.     

Box Dimension of Weyl Fractional Integral of Continuous Functions with Bounded Variation

We know that the Box dimension of f(x) ∈ C~1[0,1] is 1. In this paper, we prove that the Box dimension of continuous functions with bounded variation is still1. Furthermore, Box dimension of Weyl fractional integral of above function is also 1.     

Some Weyl Modules for Simple Algebraic Groups of Type B4 and D4

Let G be a simply connected, semisimple algebraic group of type B₄ or D₄ over an algebraically closed field of characteristic p > 0. We determine the characters of certain simple modules for these groups by calculating the composition factors of the Weyl modules.