Related papers: $U_n(q)$ acting on flags and supercharacters

Let $U_n(q)$ denote the upper triangular group of degree $n$ over the finite field $\F_q$ with $q$ elements. It is known that irreducible constituents of supercharacters partition the set of all irreducible characters $\Irr(U_n(q)).$ In…

Representations of Quantum Groups U_q (g_n), g_n any semi simple Lie algebra of rank n, are constructed from arbitrary representations of rank n-1 quantum groups for q a root of unity. Representations which have the maximal dimension and…

Let J be a finite-dimensional nilpotent algebra over a finite field F_q. We formulate a procedure for analysing characters of the group 1+J. In particular, we study characters of the group $U_n (q)$ of unipotent triangular $n\times n$…

Let $U_n$ denote the group of upper $n \times n$ unitriangular matrices over a fixed finite field $\mathbb{F}$ of order $q$. That is, $U_n$ consists of upper triangular $n \times n$ matrices having every diagonal entry equal to $1$. It is…

We construct representations of the quantum algebras ~$U_{q{\bf q}}(gl(n))$ and ~$U_{q{\bf q}}(sl(n))$~ which are in duality with the multiparameter quantum groups ~$GL_{q{\bf q}}(n)$, ~$SL_{q{\bf q}}(n)$,~ respectively. These objects…

A known result for the finite general linear group $\GL(n,\FF_q)$ and for the finite unitary group $\U(n,\FF_{q^2})$ posits that the sum of the irreducible character degrees is equal to the number of symmetric matrices in the group. Fulman…

Let F* be the field of q elements and let P(n,q) denote the projective space of dimension n-1 over F*. We construct a family H^{n}_{k,i} of combinatorial homology modules associated to P(n,q) for a coefficient field F of positive…

Let ${\bf G}$ be a connected reductive group over $\bar{\mathbb{F}}_q$, the algebraically closure of $\mathbb{F}_q$ (the finite field with $q=p^e$ elements), with the standard Frobenius map $F$. Let ${\bf B}$ be an $F$-stable Borel…

Let $G_2(q)$ be the Chevalley group of type $G_2$ defined over a finite field with q=p^n elements, where p is a prime number and $n$ is a positive integer. In this paper, we determine when the restriction of an absolutely irreducible…

The construction approach proposed in the previous paper Ref. 1 allows us there and in the present paper to construct at generic deformation parameter $q$ all finite--dimensional representations of the quantum Lie superalgebra…

Let $q$ be a prime power, $G=GL_n(q)$ and let $U\leqslant G$ be the subgroup of (lower) unitriangular matrices in $G$. For a partition $\lambda$ of $n$ denote the corresponding unipotent Specht module over the complex field $\C$ for $G$ by…

We consider $U_{q}(\mathfrak{gl}_{n})$, the quantum group of type $A$ for $|q| = 1$, $q$ generic. We provide formulas for signature characters of irreducible finite-dimensional highest weight modules and Verma modules. In both cases, the…

We construct a new family of irreducible representations of $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$ and its modular double by quantizing the classical parabolic induction corresponding to arbitrary parabolic subgroups, such that the…

The two-parametric quantum superalgebra $U_{pq}[gl(2/2)]$ and its representations are considered. All finite-dimensional irreducible representations of this quantum superalgebra can be constructed and classified into typical and nontypical…

In this paper we study the minimal faithful permutation representations of $SL_n(\mathbb F_q)$ and $GL_n(\mathbb F_q)$.

Let $q$ be a prime power and $U$ the group of lower unitriangular matrices of order $n$ for some natural number $n$. We give a lower bound for the degrees of irreducible constituents of Andr\'{e}-Yan supercharacters and classify the…

Let $q$ be an odd prime power, $n > 1$, and let $P$ denote a maximal parabolic subgroup of $GL_n(q)$ with Levi subgroup $GL_{n-1}(q) \times GL_1(q)$. We restrict the odd-degree irreducible characters of $GL_n(q)$ to $P$ to discover a…

In this paper, we consider the restriction of finite dimensional $GL_{mn} (\C)$-modules to the subgroup ${GL_m (\C)\times GL_n (\C)}$. In particular, for a Weyl module $V_{\lambda} (\C^{mn})$ of $U_q(gl_{mn})$ we construct a representation…

For all prime powers q we restrict the unipotent characters of the special orthogonal groups SO_5(q) and SO_7(q) to a maximal parabolic subgroup. We determine all irreducible constituents of these restrictions for SO_5(q) and a large part…

The standard supercharacter theory of the finite unipotent upper-triangular matrices $U_n(q)$ gives rise to a beautiful combinatorics based on set partitions. As with the representation theory of the symmetric group, embeddings of…