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

A classification of the simple highest weight bounded modules of the queer Lie superalgebras q(n) is obtained. To achieve this classification we introduce a new combinatorial tool - the star action. Our result leads in particular to a…

We provide a classification and explicit bases of tableaux of all irreducible subquotients of generic Gelfand-Tsetlin modules over Uq(gl(n)) where q different 1 and -1.

We present some observations on a restricted variant of unitary Cayley graphs modulo n, and the implications for a decomposition of elements of symplectic operators over the integers modulo n. We define quadratic unitary Cayley graphs G_n,…

We develop a crystal base theory for the general linear Lie superalgebra $gl(m,n)$. We prove that any irreducible $U_q(gl(m,n))$-module in some category has a crystal base, and prove that its associated crystal base is parameterized by…

To construct a quantum group gauge theory one needs an algebra which is invariant under gauge transformations. The existence of this invariant algebra is closely related with the existence of a differential algebra $\delta _{{\cal H}}…

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 $q=4$ and $k$ a positive integer. In this short note, we present a class of permutation polynomials over $\Bbb F_{q^{3k}}$. We also present a generalization.

We examine super symmetric representations of the B-type Hecke algebra. We exploit such representations to obtain new non-diagonal solutions of the reflection equation associated to the super algebra U_q(gl(m|n)). The boundary super algebra…

Level-one representations of the quantum affine superalgebra $U_q[\hat{gl(N|N)}]$ associated to the appropriate non-standard system of simple roots and $q$-vertex operators (intertwining operators) associated with the level-one modules are…

Let $\mathfrak{g}$ be a semi-simple Lie algebra with fixed root system, and $U_q(\mathfrak{g})$ the quantization of its universal enveloping algebra. Let $\mathcal{S}$ be a subset of the simple roots of $\mathfrak{g}$. We show that the…

Let $U$ be a maximal unipotent subgroup of a connected semisimple group $G$ and $U'$ the derived group of $U$. We study actions of $U'$ on affine $G$-varieties. First, we consider the algebra of $U'$ invariants on $G/U$. We prove that…

The purpose of this paper is to construct new families of irreducible Gelfand-Tsetlin modules for U_q(gl_n). These modules have arbitrary singularity and Gelfand-Tsetlin multiplicities bounded by 2. Most previously known irreducible modules…

The intrinsic symmetries of physical systems have been employed to reduce the number of degrees of freedom of systems, thereby simplifying computations. In this work, we investigate the properties of $\mathcal{M}SU(2^N)$,…

Let $q$ be an odd prime and $k$ be a natural number. We show that a finite subset of integers $S$ that does not contain any perfect $q^{th}$ power, contains a $q^{th}$ power residue modulo almost every natural numbers $N$ with at most $k$…

Let $G_{\mathbb{R}}$ be a real reductive Lie group and $G'_{\mathbb{R}}$ a reductive subgroup of $G_{\mathbb{R}}$ such that $\mathfrak{g'}$ is algebraic in $\mathfrak{g}$. In this paper, we consider restrictions of irreducible…

We show that the decomposition matrix of a given group $G$ is unitriangular, whenever $G$ has a normal subgroup $N$ such that the decomposition matrix of $N$ is unitriangular, $G/N$ is abelian and certain characters of $N$ extend to their…

Let F_q be the finite field with q elements. Consider the standard embedding GL(n,F_q) -> GL(n+1,F_q). In this paper we prove that for every irreducible representation pi of GL(n+1,F_q) over algebraically closed fields of characteristic…

Let $U_q(\hat{sl}_2)^{\geq 0}$ denote the Borel subalgebra of the quantum affine algebra $U_q(\hat{sl}_2)$. We show that the following hold for any choice of scalars $\epsilon_0, \epsilon_1$ from the set ${1,-1}$. (i) Let $V$ be a…

Let F be a non-archimedean local field. The construction of Lubin-Tate $(\phi_q, \Gamma)$-modules attached to p-adic representations of $G_F$ depends on the choice of a uniformizer of F. In this paper, we give a description of a functor…

The singular value decomposition of a complex matrix is a fundamental concept in linear algebra and has proved extremely useful in many subjects. It is less clear what the situation is over a finite field. In this paper, we classify the…