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.     

Symmetric Group Decomposition Numbers for Some Three-Part Partitions

The decomposition numbers d _λμ for Specht modules S ^λ of partitions λ with three parts and whose third part is at most p ? 1 are obtained by induction and by using “node removal rules” developed in James and Williams (2000 James , G. D. , Williams , A. L. (2000). Decomposition numbers of symmetric groups by induction. J. Algebra 228:119–142. [CSA] [CROSSREF]  [Google Scholar]).     

Normal coverings of finite symmetric and alternating groups

In this paper we investigate the minimum number of maximal subgroups Hi, i=1,…,k of the symmetric group Sn (or the alternating group An) such that each element in the group Sn (respectively An) lies in some conjugate of one of the Hi. We prove that this number lies between a?(n) and bn for certain constants a,b, where ?(n) is the Euler phi-function, and we show that the number depends on the arithmetical complexity of n. Moreover in the case where n is divisible by at most two primes, we obtain an upper bound of 2+?(n)/2, and we determine the exact value for Sn when n is odd and for An when n is even.     

A simplified presentation of Specht modules

Fulton and Kraskiewicz gave a presentation of Specht modules as a quotient of the space of column tabloids by dual Garnir relations. We simplify this presentation by showing that it can be generated by a single relation for each pair of columns of a tableau with ordered columns, thereby significantly reducing the number of generators given in the original construction. Our presentation applies to all Specht modules, and is of a similar nature to a recent result by Friedmann-Hanlon-Stanley-Wachs that applies to staircase partitions. We show that our presentation implies the Friedmann-Hanlon-Stanley-Wachs presentation.     

Local representation theory and möbius inversion

Various representation-theoretic parameters of a finite group are shown to satisfy formulas similar to formulas appearing in reformulations by Kölshammer, Robinson, and Thévenaz of Alperin's Conjecture. The conjecture itself is reformulated again, now as a statement not mentioning characters or conjugacy classes.     

Qualitative uncertainty principles for groups with finite dimensional irreducible representations

Let G be a locally compact group of type I and its dual space. Roughly speaking, qualitative uncertainty principles state that the concentration of a nonzero integrable function on G and of its operator-valued Fourier transform on is limited. Such principles have been established for locally compact abelian groups and for compact groups. In this paper we prove generalizations to the considerably larger class of groups with finite dimensional irreducible representations.     

Restrictions on the structure of subgroup lattices of finite alternating and symmetric groups

Let G be a finite alternating or symmetric group. We describe an infinite class of finite lattices, none of which is isomorphic to any interval [H,G] in the subgroup lattice of G.     

Partial sum of matrix entries of representations of the symmetric group and its asymptotics

Many aspects of the asymptotics of Plancherel distributed partitions have been studied in the past fifty years, in particular the limit shape, the distribution of the longest rows, connections with random matrix theory and characters of the representation matrices of the symmetric group. Regarding the latter, we extend a celebrated result of Kerov on the asymptotic of Plancherel distributed characters by studying partial trace and partial sum of a representation matrix. We decompose each of these objects into a main term and a reminder, and for each such a decomposition we prove a central limit theorem for the main term. We apply these results to prove a law of large numbers for the partial sum. Our main tool is the expansion of symmetric functions evaluated on Jucys–Murphy elements.     

Constructing irreducible representations of discrete groups

The decomposition of unitary representations of a discrete group obtained by induction from a subgroup involves commensurators. In particular Mackey has shown that quasi-regular representations are irreducible if and only if the corresponding subgroups are self-commensurizing. The purpose of this work is to describe general constructions of pairs of groups Γ₀ with Γ its own commensurator in Γ. These constructions are then applied to groups of isometries of hyperbolic spaces and to lattices in algebraic groups.     

Pentavalent symmetric graphs admitting transitive non-abelian characteristically simple groups

Let $\Gamma$ be a graph and let $G$ be a group of automorphisms of $\Gamma$. The graph $\Gamma$ is called $G$-normal if $G$ is normal in the automorphism group of $\Gamma$. Let $T$ be a finite non-abelian simple group and let $G=T^l$ with $l\ge 1$. In this paper we prove that if every connected pentavalent symmetric $T$-vertex-transitive graph is $T$-normal, then every connected pentavalent symmetric $G$-vertex-transitive graph is $G$-normal. This result, among others, implies that every connected pentavalent symmetric $G$-vertex-transitive graph is $G$-normal except $T$ is one of 57 simple groups. Furthermore, every connected pentavalent symmetric $G$-regular graph is $G$-normal except $T$ is one of 20 simple groups, and every connected pentavalent $G$-symmetric graph is $G$-normal except $T$ is one of 17 simple groups.     

On irreducible representations of discrete Heisenberg groups

We prove that the irreducible representations with finite weight of discrete nilpotent groups of class 2, including the discrete Heisenberg groups, are monomial representations and obtain their classification for the group Heis(3, ?).     

Construct irreducible representations of quantum groups U q (ƒ m (K))

In this paper, we construct families of irreducible representations for a class of quantum groups U _q (ƒ _m (K)). First, we give a natural construction of irreducible weight representations for U _q (ƒ _m (K)) using methods in spectral theory developed by Rosenberg. Second, we study the Whittaker model for the center of U _q (ƒ _m (K)). As a result, the structure of Whittaker representations is determined, and all irreducible Whittaker representations are explicitly constructed. Finally, we prove that the annihilator of a Whittaker representation is centrally generated.