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Branching of symplectic groups is not multiplicity-free. We describe a new approach to resolving these multiplicities that is based on studying the associated branching algebra $B$. The algebra $B$ is a graded algebra whose components…

The variety of principal minors of $n\times n$ symmetric matrices, denoted $Z_{n}$, is invariant under the action of a group $G\subset \GL(2^{n})$ isomorphic to $\G$. We describe an irreducible $G$-module of degree $4$ polynomials…

Let $\text{GL}(n) = \text{GL}(n, {\mathbb C})$ denote the complex general linear group and let $G \subset \text{GL}(n)$ be one of the classical complex subgroups $\text{O}(n)$, $\text{SO}(n)$, and $\text{Sp}(2k)$ (in the case $n = 2k$). We…

Let $A$ be a ring with $1\neq 0$, not necessarily finite, endowed with an involution~$*$, that is, an anti-automorphism of order $\leq 2$. Let $H_n(A)$ be the additive group of all $n\times n$ hermitian matrices over $A$ relative to $*$.…

Let $G= SL_{n+1}$ be defined over an algebraically closed field of characteristic $p > 2$. For each $n \geq 1$ there exists a singular block in the category of $G_1$-modules which contains precisely $n+1$ irreducible modules. We are…

Let $\mathbb{F}$ be an algebraically closed field and $G$ be an almost quasi-simple group. An important problem in representation theory is to classify the subgroups $H<G$ and $\mathbb{F} G$-modules $L$ such that the restriction…

We construct families of TQFT's over the finite field Z/pZ starting from an integral TQFT obtained by Frohman and Nicas. These TQFT's are likely to describe the constant order contributions of the cyclotomic integer expansions of the…

In this article, we study the irreducibility of representations of the singular braid group on $n$ strands, namely $SB_n$. Our first finding is the determination of the forms of all irreducible representations $\rho : SB_2 \to…

We prove a result on the asymptotic proportion of randomly chosen pairs of permutations in the symmetric group $S_n$ which "invariably" generate a nonsolvable subgroup, i.e., whose cycle structures cannot possibly both occur in the same…

In this paper, we study the combinatorics of congruence subgroups of the modular group by generalizing results obtained in the non-modular case. For this, we define a notion of irreducible solutions from which we can build all the…

We investigate the homogeneous $2$-local representations of the twin group $T_n$ for all integers $n\geqslant 2$. A complete classification is obtained, yielding three distinct families of representations. We show that each of these…

In this paper we study the determinant of irreducible representations of the generalized symmetric groups $\mathbb{Z}_r \wr S_n$. We give an explicit formula to compute the determinant of an irreducible representation of $\mathbb{Z}_r \wr…

Using the general framework of polynomial representations defined by Doty and generalizing the definition given by Doty, Nakano and Peters for $G = \mathrm{GL}_n$, we consider polynomial representations of $G_r T$ for an arbitrary closed…

The problem of computing the characters of the finite dimensional irreducible representations of the Lie superalgebra $\mathfrak{gl}(m|n)$ over $\C$ was solved a few years ago by V. Serganova. In this article, we present an entirely…

The classification of the finite subgroups of $\mathrm{GL}_n(\mathbb{C})$ and $\mathrm{PGL}_n(\mathbb{C})$ is a classical problem in the field of finite group theory, dating back to the late 19th century with authors like Klein, Jordan,…

This paper provides a complete classification of $\mathrm{GL}_n(\mathbb{R})$-distinguished irreducible representations of $\mathrm{GL}_n(\mathbb{C})$ when the representations are either generic or unitary. Additionally, for each such…

The aim of the present paper is to obtain a classification of all the irreducible modular representations of the symmetric group on $n$ letters of dimension at most $n^3$, including dimension formulae. This is achieved by improving an idea,…

Every finite group $G$ has a normal series each of whose factors is either a solvable group or a direct product of nonabelian simple groups. The minimum number of nonsolvable factors attained on all possible such series is called the…

Let $p$ be a prime and let $\mathbb{C}$ be the complex field. We explicitly classify the finite solvable irreducible monomial subgroups of $\mathrm{GL}(p,\mathbb{C})$ up to conjugacy. That is, we give a complete and irredundant list of…

In this paper, we calculate the space $Ext^1_{GL(n)}(L_n(\lambda),L_n(\mu))$, where GL(n) is the general linear group of degree $n$ over an algebraically closed field of positive characteristic, $L_n(\lambda)$ and $L_n(\mu)$ are rational…