Related papers: Supercharacters and pattern subgroups in the upper…

In this paper we give a new formula for characters of finite dimensional irreducible $\frak{gl}(m,n)$ modules. We use two main ingredients: Su-Zhang formula and Brion's theorem.

Let $\mathbb F_q$ be the finite field with $q$ elements, where $q$ is a prime power and, for each integer $n\ge 1$, let $\mathbb F_{q^n}$ be the unique $n$-degree extension of $\mathbb F_q$. The $\mathbb F_q$-orders of an element in…

We classify the irreducible complex characters of the symplectic groups $Sp_{2n}(q)$ and the orthogonal groups $Spin_{2n}^\pm(q)$, $Spin_{2n+1}(q)$ of degrees up to the bound D, where $D=(q^n-1)q^{4n-10}/2$ for symplectic groups,…

The characters of the unitarizable highest weight modules over the N=2 superconformal algebras are presented. This is a slightly extended version of an Encyclopedia entry.

Let G be a connected unipotent group over a finite field F_q with q elements. In this article we propose a definition of L-packets of complex irreducible representations of the finite group G(F_q) and give an explicit description of…

Let F be a finite field with q elements, let A be a finite dimensional F-algebra and let J=J(A) be the Jacobson radical of A. Then G=1+J is a p-group, where p is the characteristic of F. We refer to G as an F-algebra group. A subgroup H of…

The category of finite dimensional (type 1) representations of a quantum affine algebra $U_q(\hat{{\mathfrak g}})$ is not semisimple. However, as any abelian category with finite-length objects, it admits a unique decomposition into a…

Let $S_n$ denote a symmetric group, $\chi$ an irreducible character of $S_n$, and $g\in S_n$ an element which decomposes into $k$ disjoint cycles, where $1$-cycles are included. Then $|\chi(g)|\le k!$, and this upper bound is sharp for…

A general setting to study a certain type of formulas, expressing characters of the symmetric group $\mathfrak{S}_n$ explicitly in terms of descent sets of combinatorial objects, has been developed by two of the authors. This theory is…

In this paper we obtain the orthogonality relations for the supergroup U(m|n), which are remarkably different from the ones for the U(N) case. We extend our results for ordinary representations, obtained some time ago, to the case of…

In order to find a tractable theory to substitute for the wild character theory of the group of $n\times n$ unipotent upper-triangular matrices over a finite field $\mathbb{F}_q$, Andr\'e and Yan introduced the notion of supercharacter…

Given a finite group G, let cd(G) denote the set of degrees of the irreducible complex characters of G. The character degree graph of G is defined as the simple undirected graph whose vertices are the prime divisors of the numbers in cd(G),…

In this work, we classify all finite groups such that for every field extension F of \mathbb{Q}, F is the field of values of at most 3 irreducible characters.

We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in…

Let $G$ be a finite group and $\mathrm{Irr}(G)$ be the set of irreducible characters of $G$. The codegree of an irreducible character $\chi$ of the group $G$ is defined as $\mathrm{cod}(\chi)=|G:\mathrm{ker}(\chi)|/\chi(1)$. In this paper,…

We develop explicit formulae for the eigenvalues of various invariants for highest weight irreducible representations of the quantum supergroup $U_q[gl(m|n)]$. The techniques employed make use of modified characteristic identity methods and…

Following the ideas in~\cite{yM88} and some inspiration from~\cite{KO24}, we construct a bialgebra $T_q(n)$ and a pointed Hopf algebra $UT_q(n)$ which quantize the coordinate rings of the algebra of upper triangular matrices and of the…

Let $G$ be a finite group and $p$ be a prime number dividing the order of $G$. An irreducible character $\chi$ of $G$ is called a quasi $p$-Steinberg character if $\chi(g)$ is nonzero for every $p$-regular element $g$ in $G$. In this paper,…

We describe an easy way how to find supercharacter theories for a finite group, if its character table is known. Namely, we show how an arbitrary partition of the conjugacy classes or of the irreducible characters can be refined to the…

A method is developed to construct irreducible representations(irreps) of the quantum supergroup $U_q(C(n+1))$ in a systematic fashion. It is shown that every finite dimensional irrep of this quantum supergroup at generic $q$ is a…