Related papers: On signed $p$-Kostka matrices

We prove the existence and main properties of signed Young modules for the symmetric group, using only basic facts about symmetric group representations and the Brou{\'e} correspondence. We then prove new reduction theorems for the signed…

We study the multiplicities of Young modules as direct summands of permutation modules on cosets of Young subgroups. Such multiplicities have become known as the p-Kostka numbers. We classify the indecomposable Young permutation modules,…

Recently Donkin defined signed Young modules as a simultaneous generalization of Young and twisted Young modules for the symmetric group. We show that in odd characteristic, if a Specht module $S^\lambda$ is irreducible, then $S^\lambda$ is…

By a result of Hemmer, every simple Specht module of a finite symmetric group over a field of odd characteristic is a signed Young module. While Specht modules are parametrized by partitions, indecomposable signed Young modules are…

Let $K$ be a field of characteristic two, and let $\lambda$ be a two-part partition of some natural number $r$. Denote the permutation module corresponding to the (maximal) Young subgroup $\Sigma_\lambda$ in $\Sigma_r$ by $M^\lambda$. We…

Let p be a prime number. We study the dimensions of Brauer constructions of Young and Young permutation modules with respect to p-subgroups of the symmetric groups. They depend only on partitions labelling the modules and the orbits of the…

The decomposition matrix of a finite group in prime characteristic p records the multiplicities of its p-modular irreducible representations as composition factors of the reductions modulo p of its irreducible representations in…

We consider a certain quotient of a polynomial ring categorified by both the isomorphic Green rings of the symmetric groups and Schur algebras generated by the signed Young permutation modules and mixed powers respectively. They have bases…

We study the indecomposable summands of the permutation module obtained by inducing the trivial $\mathbb{F}(S_a\wr S_n)$-module to the full symmetric group $S_{an}$ for any field $\mathbb{F}$ of odd prime characteristic $p$ such that…

Let $n$ be a positive integer and $\lambda$ be a partition of $n$. Let $M^\lambda$ be the Young permutation module labelled by $\lambda$. In this paper, we study symmetric and exterior powers of $M^\lambda$ in positive characteristic case.…

In this article, we study the permutation modules and Young modules of the group algebras of the direct product of symmetric groups $K\mathfrak{S}_{a,b}$, and the walled Brauer algebras $\B_{r,t}(\delta)$. In the category of dual…

We construct a class $\Theta_{\mathscr{R}}$ of homomorphisms from a Specht module $S_{\mathbb{Z}}^{\lambda}$ to a signed permutation module $M_{\mathbb{Z}}(\alpha|\beta)$ which generalises James's construction of homomorphisms whose…

Let $p$ be an odd prime and let $n$ be a natural number. In this article we determine the irreducible constituents of the permutation module induced by the action of the symmetric group $\mathfrak{S}_n$ on the cosets of a Sylow $p$-subgroup…

Let $G$ be a cyclic $p$-group for some prime number $p>0$ and let $R$ be a complete discrete valuation ring in mixed characteristic. In this paper, we present a generalization of two results that characterize $RG$-permutation modules,…

If $[\lambda(j)]$ is a multipartition of the positive integer $n$ (a sequence of partitions with total size $n$), and $\mu$ is a partition of $n$, we study the number $K_{[\lambda(j)]\mu}$ of sequences of semistandard Young tableaux of…

For prime levels $5 \le p \le 19$, sets of $\Gamma_{0}(p)$-permuted theta quotients are constructed that generate the graded rings of modular forms of positive integer weight for $\Gamma_{1}(p)$. An explicit formulation of the permutation…

The signed permutation modules are a simultaneous generalization of the ordinary permutation modules and the twisted permutation modules of the symmetric group. In a recent paper Dave Benson and Peter Symonds defined a new invariant…

Perhaps the most important problem in representation theory in the 1970s and early 1980s was the determination of the multiplicity of composition factors in a Verma module. This problem was settled by the proof of the Kazhdan-Lusztig…

Kostka functions $K^{\pm}_{\lambda, \mu}(t)$ associated to complex reflection groups are a generalization of Kostka polynomials, which are indexed by $r$-partitions $\lambda, \mu$ and a sign $+, -$. It is known that Kostka polynomials have…

Applying Robert Boltje's theory of canonical induction, we give a restriction-preserving formula expressing any $p$-permutation module as a $\mathbb{Z}[1/p]$-linear combination of modules induced and inflated from projective modules…