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.     

Minimal polynomials of Young permutation modules and idempotents

Applying an epimorphism of the Solomon descent algebra onto the subring of the Green ring spanned by the isomorphism classes of Young permutation modules, we determine a basis of primitive orthogonal idempotents which diagonalise the multiplication maps of Young permutation modules. We determine direct sum decompositions of tensor products of hook Young permutation modules, the minimal polynomials of all Young permutation modules, and of the Young module Y^(r?1,1).     

The complexity of the Specht modules corresponding to hook partitions

We show that the complexity of the Specht module corresponding to any hook partition is the p-weight of the partition. We calculate the variety and the complexity of the signed permutation modules. Let E _s be a representative of the conjugacy class containing an elementary abelian p-subgroup of a symmetric group generated by s disjoint p-cycles. We give formulae for the generic Jordan types of signed permutation modules restricted to E _s and of Specht modules corresponding to hook partitions μ restricted to E _s where s is the p-weight of μ.      

The Benson - Symonds Invariant for Permutation Modules

In a recent paper, Dave Benson and Peter Symonds defined a new invariant γ_G(M) for a finite dimensional module M of a finite group G which attempts to quantify how close a module is to being projective. In this paper, we determine this invariant for permutation modules of the symmetric group corresponding to two-part partitions using tools from representation theory and combinatorics.

     

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.     

Uniformly generated submodules of permutation modules: Over fields of characteristic 0

This paper is motivated by a link between algebraic proof complexity and the representation theory of the finite symmetric groups. Our perspective leads to a new avenue of investigation in the representation theory of S_n. Most of our technical results concern the structure of “uniformly” generated submodules of permutation modules. For example, we consider sequences of submodules of the permutation modules M^(n−k,1k) and prove that if the sequence W_n is given in a uniform (in n) way – which we make precise – the dimension p(n) of W_n (as a vector space) is a single polynomial with rational coefficients, for all but finitely many “singular” values of n. Furthermore, we show that dim(W_n)<p(n) for each singular value of n≥4k. The results have a non-traditional flavor arising from the study of the irreducible structure of the submodules W_n beyond isomorphism types. We sketch the link between our structure theorems and proof complexity questions, which are motivated by the famous NP vs. co-NP problem in complexity theory. In particular, we focus on the complexity of showing membership in polynomial ideals, in various proof systems, for example, based on Hilbert's Nullstellensatz.     

Lie powers and Witt vectors

In the study of Lie powers of a module V in prime characteristic p, a basic role is played by certain modules B _n introduced by Bryant and Schocker. The isomorphism types of the B _n are not fully understood, but these modules fall into infinite families , one family B(k) for each positive integer k not divisible by p, and there is a recursive formula for the modules within B(k). Here we use combinatorial methods and Witt vectors to show that each module in B(k) is isomorphic to a direct sum of tensor products of direct summands of the kth tensor power V ^{⊗ k} . To the memory of Manfred Schocker.     

Modular Lie powers and the Solomon descent algebra

Let V be an r-dimensional vector space over an infinite field F of prime characteristic p, and let L_n(V) denote the nth homogeneous component of the free Lie algebra on V. We study the structure of L_n(V) as a module for the general linear group GL_r(F) when n=pk and k is not divisible by p and where r≥n. Our main result is an explicit 1-1 correspondence, multiplicity-preserving, between the indecomposable direct summands of L_k(V) and the indecomposable direct summands of L_n(V) which are not isomorphic to direct summands of V^⊗n. Our approach uses idempotents of the Solomon descent algebras, and in addition a correspondence theorem for permutation modules of symmetric groups. Second author supported by Deutsche Forschungsgemeinschaft (DFG-Scho 799).     

The complexity of certain Specht modules for the symmetric group

During the 2004–2005 academic year the VIGRE Algebra Research Group at the University of Georgia (UGA VIGRE) computed the complexities of certain Specht modules S ^λ for the symmetric group Σ _d , using the computer algebra program Magma. The complexity of an indecomposable module does not exceed the p-rank of the defect group of its block. The UGA VIGRE Algebra Group conjectured that, generically, the complexity of a Specht module attains this maximal value; that it is smaller precisely when the Young diagram of λ is built out of p×p blocks. We prove one direction of this conjecture by showing these Specht modules do indeed have less than maximal complexity. It turns out that this class of partitions, which has not previously appeared in the literature, arises naturally as the solution to a question about the p-weight of partitions and branching.     

On the Iwasawa Theory of p-Adic Lie Extensions

In this paper, the new techniques and results concerning the structure theory of modules over noncommutative Iwasawa algebras are applied to arithmetic: we study Iwasawa modules over p-adic Lie extensions k of number fields k 'up to pseudo-isomorphism'. In particular, a close relationship is revealed between the Selmer group of Abelian varieties, the Galois group of the maximal Abelian unramified p-extension of k as well as the Galois group of the maximal Abelian p-extension unramified outside S where S is a certain finite setof places of k. Moreover, we determine the Galois module structure of local units and other modules arising from Galois cohomology.     

Permutation presentations of modules over finite groups

We introduce a notion of permutation presentations of modules over finite groups, and completely determine finite groups over which every module has a permutation presentation. To get this result, we prove that every coflasque module over a cyclic p-group is permutation projective.     

Permutation resolutions for Specht modules

For every composition λ of a positive integer r, we construct a finite chain complex whose terms are direct sums of permutation modules M ^μ for the symmetric group \(\mathfrak{S}_{r}\) with Young subgroup stabilizers \(\mathfrak{S}_{\mu}\). The construction is combinatorial and can be carried out over every commutative base ring k. We conjecture that for every partition λ the chain complex has homology concentrated in one degree (at the end of the complex) and that it is isomorphic to the dual of the Specht module S ^λ . We prove the exactness in special cases.     

On permutation characters of wreath products

It is known that the character rings of symmetric groups S_n and the character rings of hyperoctahedral groups S₂?S_n are generated by (transitive) permutation characters. These results of Young are generalized to wreath products G?H (G a finite group, H a permutation group acting on a finite set). It is shown that the character ring of G?H is generated by permutation characters if this holds for G, H and certain subgroups of H. This result can be sharpened for wreath products G?S_n;if the character ring of G has a basis of transitive permutation characters, then the same holds for the character ring of G?S_n.     

On Finite Unions of Submodules

This paper is concerned with finite unions of ideals and modules. The first main result is that, if N ? N ₁ ∪ N ₂ ∪ … ∪ N _s is a covering of a module N by submodules N _i , such that all but two of the N _i are intersections of strongly irreducible modules, then N ? N _k for some k. The special case when N is a multiplication module is considered. The second main result generalizes earlier results on coverings by primary submodules. In the last section unions of cosets is studied.     

Varieties for cohomology with twisted coefficients

Let G be a finite group and k a field of characteristic p > 0. In this paper we consider the support variety for the cohomology module Ext _kG ^* (M, N) where M and N are kG-modules. It is the subvariety of the maximal ideal spectrum of H*(G, k) of the annihilator of the cohomology module. For modules in the principal block we show that that the variety is contained in the intersections of the varieties of M and N and the difference between the that intersection and the support variety of the cohomology module is contained in the group theoretic nucleus. For other blocks a new nucleus is defined and a similar theorem is proven. However in the case of modules in a nonprincipal block several new difficulties are highlighted by some examples. Partially supported by grants from NSF and EPSRC     

The minimal eigenvalue of the Lyapunov transform †

An (n m) hypergraph is a coupleH=(N E), where the vertex set N is {1,…n} and the edge set E is an m-element multiset of nonempty subsets of N. In this paper, we count nonisomorphic hypergraphs where isomorphism of hypergraphs is the natural extension of that of graphs. A main result is an explicit formula for the cycle index of the permutation representation of any permutation group P with object set N acting on the k-element subsets of N. By making a simple substitution in these cycle indices for P the symmetric group S_N and k=1,…,n, we obtain generating functions which enumerate various types of hypergraphs. Using the technique developed, we extend Snapper's results on characteristic polynomials of permutation representations and group characters from the case where the group has odd order to the general case.     

Symmetric semisimple modules of group algebras over finite fields and self-dual permutation codes

A module of a finite group over a finite field with a symmetric non-degenerate bilinear form which is invariant by the group action is called a symmetric module. In this paper, a characterization of indecomposable orthogonal decompositions of symmetric semisimple modules and a criterion for the hyperbolic symmetric modules are obtained, and some applications to the self-dual permutation codes are shown.     

Multiplicity computation of modules overk[x1,…,xn] and an application to weyl algebras

Let A = k[x₁,…,x_n] be the polynomial algebra over a field kof characteristic 0Ian ideal of A, M = A/Iand ^αHP _I the (affine) Hilbert polynomial of M. By further exploring the algorithmic procedure given in [CLO'] for deriving the existence of ^αHP _I , we compute the leading coefficient of ^αHP _I by looking at the leading monomials of a Grobner basis of Iwithout computing ^αHP _I . Using this result and the filtered-graded transfer of Grobner basis obtained in [LW] for (noncommutative) solvable polynomial algebras (in the sense of [K-RW]), we are able to compute the multiplicity of a cyclic module over the Weyl algebra A _n (k) without computing the Hilbert polynomial of that module, and consequently to give a quite easy algorithmic characterization of the “smallest“ modules over Weyl algebras. Using the same methods as before, we also prove that the tensor product of two cyclic modules over the Weyl algebras has the multiplicity which is equal to the product of the multiplicities of both modules. The last result enables us to construct examples of “smallest“ irreducible modules over Weyl algebras.     

Self-duality of rank 2 reflexive modules

Every finitely generated rank 2 third syzygy module over a regular local ring is known to be self-dual. We show more generally that any finitely generated rank 2 reflexive module is self-dual, and that the isomorphism is skew symmetric. We use this ismorphism to estimate how large k may be if the module is a kth syzygy, and how closely allied rank 3 modules are related to their duals.