Symmetric Group Modules with Specht and Dual Specht Filtrations

The author and Nakano recently proved that multiplicities in a Specht filtration of a symmetric group module are well-defined precisely when the characteristic is at least five. This result suggested the possibility of a symmetric group theory analogous to that of good filtrations and tilting modules for GL _n (k). This article is an initial attempt at such a theory. We obtain two sufficient conditions that ensure a module has a Specht filtration, and a formula for the filtration multiplicities. We then study the categories of modules that satisfy the conditions, in the process obtaining a new result on Specht module cohomology.

Next we consider symmetric group modules that have both Specht and dual Specht filtrations. Unlike tilting modules for GL _n (k), these modules need not be self-dual, and there is no nice tensor product theorem. We prove a correspondence between indecomposable self-dual modules with Specht filtrations and a collection of GL _n (k)-modules which behave like tilting modules under the tilting functor. We give some evidence that indecomposable self-dual symmetric group modules with Specht filtrations may be indecomposable self dual trivial source modules.     

On Homomorphisms Indexed by Semistandard Tableaux

We describe techniques which may be used to compute the homomorphism space between Specht modules for the Hecke algebras of type A. We prove a q-analogue of a result of Fayers and Martin and show how it may be applied to construct homomorphisms between Specht modules. In particular, we show that in certain cases the dimension of the homomorphism space is given by the corank of a matrix whose entries we write down explicitly.     

A Characterization of Continuous Module Homomorphisms on Random Semi—Normed Modules and Its Applications

In this paper,a characterization of continuous module homomorphisms on random seminormed modules is first given;then the characterization is further used to show that the Hahn-Banach type of extension theorem is still true for continuous module homomorphisms on random semi-normed modules.     

Verma-Type Modules for Quantum Affine Lie Algebras

Let be an untwisted affine Kac–Moody algebra and M_J() a Verma-type module for with J-highest weight P. We construct quantum Verma-type modules M_J^q() over the quantum group , investigate their properties and show that M_J^q() is a true quantum deformation of M_J() in the sense that the weight structure is preserved under the deformation. We also analyze the submodule structure of quantum Verma-type modules. Presented by A. VerschorenMathematics Subject Classifications (2000) 17B37, 17B67, 81R50.The first author is a Regular Associate of the ICTP. The third author was supported in part by a Faculty Research Grant from St. Lawrence University.     

Dirac Cohomology of some Harish-Chandra Modules

In this paper we determine the Dirac cohomology of certain irreducible Harish-Chandra modules of a semisimple connected Lie group G with finite center: irreducible finite-dimensional modules and unitary A (λ) modules. We also comment on the relationship to (, K)-cohomology. The first named author is partially supported by research grants from RGC of Hong Kong SAR and NSF of China. The second named author is partially supported by the National Natural Science Foundation of China (10501025), Liu Hui Center for Applied Mathematics and Youth Teachers Foundation of Tianjin University. The third named author is partially supported by a grant from the Ministry of Science, Education and Sports of the Republic of Croatia.     

Co-Point Modules Over Frobenius Koszul Algebras

Let A be a Frobenius Koszul algebra such that its Koszul dual A ^! is a quantum polynomial algebra. Co-point modules over A were defined as dual notion of point modules over A ^! with respect to the Koszul duality. In this article, we will see that various important functors between module categories over A used in representation theory of finite dimensional algebras send co-point modules to co-point modules. As a consequence, we will show that if (E, σ) is a geometric pair associated to A ^!, then the map σ:E → E is an automorphism of the point scheme E of A ^!, so that there is a bijection between isomorphism classes of left point modules over A ^! and those of right point modules over A ^!.     

When Products of Self-Small Modules are Self-Small

A module M is called “self-small” if the functor Hom(M, ?) commutes with direct sums of copies of M. The main goal of the present article is to construct a non-self-small product of self-small modules without nonzero homomorphisms between distinct ones and to correct an error in a claim about products of self-small modules published by Arnold and Murley in a fundamental article on this topic. The second part of the article is devoted to the study of endomorphism rings of self-small modules.     

Homomorphisms between Specht modules

In positive characteristic, the Specht modules S corresponding to partitions are not necessarily irreducible, and understanding their structure is vital to understanding the representation theory of the symmetric group. In this paper, we address the related problem of finding the spaces of homomorphisms between Specht modules. Indeed in [2], Carter and Payne showed that the space of homomorphisms from S to S is non-zero for certain pairs of partitions and where the Young diagram for is obtained from that for by moving several nodes from one row to another. We also consider partitions of this type, and, by explicitly examining certain combinations of semi-standard homomorphisms, we are able to give a constructive proof of the Carter–Payne theorem and to generalise it.Mathematics Subject Classification (2000): 20C30     

The Theory of Finite Models without Equal Sign

In this paper, it is the first time ever to suggest that we study the model theory of all finite structures and to put the equal sign in the same situtation as the other relations. Using formulas of infinite lengths we obtain new theorems for the preservation of model extensions, submodels, model homomorphisms and inverse homomorphisms. These kinds of theorems were discussed in Chang and Keisler＇s Model Theory, systematically for general models, but Gurevich obtained some different theorems in this direction for finite models. In our paper the old theorems manage to survive in the finite model theory. There are some differences between into homomorphisms and onto homomorphisms in preservation theorems too. We also study reduced models and minimum models. The characterization sentence of a model is given, which derives a general result for any theory T to be equivalent to a set of existential-universal sentences. Some results about completeness and model completeness are also given.     

Some reducible Specht modules for Iwahori–Hecke algebras of type A with q=−1

The reducibility of the Specht modules for the Iwahori–Hecke algebras in type A is still open in the case where the defining parameter q equals ?1. We prove the reducibility of a large class of Specht modules for these algebras.     

Two theorems on the vertices of Specht modules

Two theorems about the vertices of indecomposable Specht modules for the symmetric group, defined over a field of prime characteristic p, are proved: 1. The indecomposable Specht module $S^\lambda$ has non-trivial cyclic vertex if and only if $\lambda$ has p-weight 1. 2. If p does not divide n and $S^{(n-r, 1^r)}$ is indecomposable then its vertex is a p-Sylow subgroup of $S_{n-r-1} \times S_r$.Received: 15 August 2002     

Specht modules and semisimplicity criteria for Brauer and Birman–Murakami–Wenzl algebras

A construction of bases for cell modules of the Birman–Murakami–Wenzl (or B–M–W) algebra B _n (q,r) by lifting bases for cell modules of B _n−1(q,r) is given. By iterating this procedure, we produce cellular bases for B–M–W algebras on which a large Abelian subalgebra, generated by elements which generalise the Jucys–Murphy elements from the representation theory of the Iwahori–Hecke algebra of the symmetric group, acts triangularly. The triangular action of this Abelian subalgebra is used to provide explicit criteria, in terms of the defining parameters q and r, for B–M–W algebras to be semisimple. The aforementioned constructions provide generalisations, to the algebras under consideration here, of certain results from the Specht module theory of the Iwahori–Hecke algebra of the symmetric group. Research supported by Japan Society for Promotion of Science.     

Transitioning Between Tableaux and Spider Bases for Specht Modules

Regarding the Specht modules associated to the two-row partition (n, n), we provide a combinatorial path model to study the transitioning matrix from the tableau basis to the A₁-web basis (i.e. cup diagrams), and prove that the entries in this matrix are positive in the upper-triangular portion with respect to a certain partial order.

     

The reducible Specht modules for the Hecke algebra \mathcal{H}_{\mathbb{C},{-}1}(\mathfrak{S}_{n})

The reducible Specht modules for the Hecke algebra $\mathcal {H}_{\mathbb{F},q}(\mathfrak{S}_{n})$ have been classified except when q=?1. We prove one half of a conjecture which we believe classifies the reducible Specht modules when q=?1.     

Rouquier blocks

This paper investigates the Rouquier blocks of the Hecke algebras of the symmetric groups and the Rouquier blocks of the q-Schur algebras. We first give an algorithm for computing the decomposition numbers of these blocks in the ``abelian defect group case' and then use this algorithm to explicitly compute the decomposition numbers in a Rouquier block. For fields of characteristic zero, or when q=1 these results are known; significantly, our results also hold for fields of positive characteristic with q≠1. We also discuss the Rouquier blocks in the ``non–abelian defect group' case. Finally, we apply these results to show that certain Specht modules are irreducible.     

Specht Modules and Simple Modules for Centralizer Algebras of the Symmetric Group

Let Σ_n be the symmetric group on n letters. For l ≤ n identify Σ_l with a subgroup of Σ_n in the natural way. Let k be an algebraically closed field of characteristic p. This article begins to develop a theory for modules over the centralizer algebras kΣ_n^Σ_l that is analogous to James's theory of permutation modules, Specht modules, and simple modules over kΣ_n. We make a conjecture about how to construct all simple kΣ_n^Σ_l-modules, we develop tools to test the conjecture, and we prove that it is correct for all n when l < p.     

On Coprime Modules and Comodules

Many observations about coalgebras were inspired by comparable situations for algebras. Despite the prominent role of prime algebras, the theory of a corresponding notion for coalgebras was not well understood so far. Coalgebras C over fields may be called coprime provided the dual algebra C* is prime. This definition, however, is not intrinsic—it strongly depends on the base ring being a field. The purpose of the article is to provide a better understanding of related notions for coalgebras over commutative rings by employing traditional methods from (co)module theory, in particular (pre)torsion theory.

Dualizing classical primeness condition, coprimeness can be defined for modules and algebras. These notions are developed for modules and then applied to comodules. We consider prime and coprime, fully prime and fully coprime, strongly prime and strongly coprime modules and comodules. In particular, we obtain various characterisations of prime and coprime coalgebras over rings and fields.     

CS Modules Relative to a Torsion Theory

In this article we introduce two new concepts, those of a τ-CS and a s-τ-CS module, which are both torsion-theoretic analogues of CS modules. We investigate their relationship with the familiar concepts of τ-injective, τ-simple and τ-uniform modules and compare them with τ-complemented (τ-injective) modules, which were considered by other authors as torsion-theoretic analogues of CS modules. We are interested in decomposing a relatively CS module into indecomposable submodules, and in determining when a direct sum of relatively CS modules is relatively CS. This paper forms part of the Ph.D. thesis of the first author, written under the supervision of the second author. The first author gratefully acknowledges the support of the Commonwealth Scholarship and Fellowship Committee of New Zealand.