Related papers: Conjugacy in Permutation Representations of the Sy…

Let $G$ be a finite simple group of Lie type, and let $\pi_G$ be the permutation representation of $G$ associated with the action of $G$ on itself by conjugation. We prove that every irreducible representation of $G$ is a constituent of…

Let SL(2,q) be the group of 2X2 matrices with determinant one over a finite field F of size q. We prove that if q is even, then the product of any two noncentral conjugacy classes of SL(2,q) is the union of at least q-1 distinct conjugacy…

Given a subgroup H of a finite group G, we begin a systematic study of the partial representations of G that restrict to global representations of H. After adapting several results from [DEP00] (which correspond to the case where H is…

The birational $R$-matrix is a transformation that appears in the theory of geometric crystals, the study of total positivity in loop groups, and discrete dynamical systems. This $R$-matrix gives rise to an action of the symmetric group…

Let $B$ be some invertible Hermitian or skew-Hermitian matrix. A matrix $A$ is called $B$-normal if $AA^\star = A^\star A$ holds for $A$ and its adjoint matrix $A^\star := B^{-1}A^HB$. In addition, a matrix $Q$ is called $B$-unitary, if…

We consider the asymptotic behavior as $n\to\infty$ of the spectra of random matrices of the form \[\frac{1}{\sqrt{n-1}}\sum_{k=1}^{n-1}Z_{nk}\rho_n ((k,k+1)),\] where for each $n$ the random variables $Z_{nk}$ are i.i.d. standard Gaussian…

We characterise those classes of permutations having the property that for every tableau shape either every permutation of that shape or no permutation of that shape belongs to the class. The characterisation is in terms of the dominance…

The adjoint representations of the Lie algebras of the classical groups SU(n), SO(n), and Sp(n) are, respectively, tensor, antisymmetric, and symmetric products of two vector spaces, and hence are matrix representations. We consider the…

Let $P(n)$ be the probability that two independent, uniformly random permutations of $[n]$ have the same order, and let $K(n)$ be the probability that they are in the same conjugacy class. Answering a question of Thibault Godin, we prove…

Extending earlier work of Guralnick and of Cai and Zhang, we classify the almost simple groups which have transitive permutation representations of prime power degree $p^k$, and those which have $p$-complements (stabilisers of order coprime…

A set of permutations $I \subset S_n$ is said to be {\em k-intersecting} if any two permutations in $I$ agree on at least $k$ points. We show that for any $k \in \mathbb{N}$, if $n$ is sufficiently large depending on $k$, then the largest…

We study linear and hermitian representations of finite $C_2$-graded groups. We prove that the category of linear representations is equivalent to a category of antilinear representations as an $\infty$-category. We also prove that the…

We classify regular generically free actions of finite groups on the projective plane, up to conjugation in the Cremona group.

We find the range of the major index on the various conjugacy classes of involutions in the symmetric group $S_n$. In addition to indicating the minimum and the maximum values, we show that except for the case of involutions without fixed…

Let G be a simple algebraic group over an algebraically closed field k of bad characteristic. We classify the spherical unipotent conjugacy classes of G. We also show that if the characteristic of k is 2, then the fixed point subgroup of…

We use representation theory to construct spaces of matrices of constant rank. These spaces are parametrized by the natural representation of the general linear group or the symplectic group. We present variants of this idea, with more…

The class of finitely presented algebras over a field $K$ with a set of generators $a_{1},..., a_{n}$ and defined by homogeneous relations of the form $a_{1}a_{2}... a_{n} =a_{\sigma (1)} a_{\sigma (2)} ... a_{\sigma (n)}$, where $\sigma$…

In this paper, we derive an asymptotic formula for the number of conjugacy classes of elements in a class of statistically convex-cocompact actions with contracting elements. Denote by $\mathcal C(o, n)$ (resp. $\mathcal C'(o, n)$) the set…

Consider the Deligne-Simpson problem: {\em give necessary and sufficient conditions for the choice of the conjugacy classes $C_j\subset GL(n,{\bf C})$ (resp. $c_j\subset gl(n,{\bf C})$) so that there exist irreducible $(p+1)$-tuples of…

For a fixed permutation $\sigma \in S_k$, let $N_{\sigma}$ denote the function which counts occurrences of $\sigma$ as a pattern in permutations from $S_n$. We study the expected value (and $d$-th moments) of $N_{\sigma}$ on conjugacy…