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We show a correspondence between tensor representations of the super general linear group GL(m|n) and tensor representations of the general linear superalgebra gl(m|n) constructed by Berele and Regev.

We consider general linear superalgebra (type A) and tensor with Laurent polynomial ring in several variables. We then consider the universal central extension of this Lie superalgebra which we call toroidal superalgebra. We give a faithful…

We develop representation theory of general linear groups in the category $\text{Ver}_4^+$, the simplest tensor category which is not Frobenius exact. Since $\text{Ver}_4^+$ is a reduction of the category of supervector spaces to…

We describe the "generic" part of the character ring of general linear groups over a finite field in terms of quiver representations.

Let G be a special linear group over the real, the complex or the quaternion, or a special unitary group. In this note, we determine all special unipotent representations of G in the sense of Arthur and Barbasch-Vogan, and show in…

We define a generalized Berezin transforms on line bundle over the complex hyperbolic space and we give it as a functions of the G-invariant laplacian on the line bundles.

The general structure of the representation theory of a $Z_2$-graded coalgebra is discussed. The result contains the structure of Fourier analysis on compact supergroups and quantisations thereof as a special case. The general linear…

We study generic representations of general linear groups over a finite ring R with coefficients in a field k in which the cardinality of R is invertible, that is functors from finitely-generated projective R-modules to k-vector spaces. We…

Consider the pseidounitary group $G=U(p,q)$ and its compact subgroup $K=U(p)$. We construct an explicit unitary intertwining operator from the tensor product of a holomorphic representation and a antiholomorphic representation of $G$ to the…

Let $F$ be a non-archimedean local field. The classification of the irreducible representations of $GL_n(F)$, $n\ge0$ in terms of supercuspidal representations is one of the highlights of the Bernstein--Zelevinsky theory. We give an…

We give universal upper bounds on the relative dimensions of isotypic components of a tensor product of the linear group GL(n) representations and universal upper bounds on the relative dimensions of irreducible components of a tensor…

For an arbitrary unimodular Lie group $G$, we construct strongly continuous unitary representations in the Bergman space of a naturally constructed strongly pseudoconvex neighborhood of $G$ in the complexification of its underlying…

We consider a new formula for Berezinian (superdeterminant). The Berezinian of a supermatrix $A$ is expressed as the ratio of polynomial invariants of $A$. This formula follows from recurrence relations existing for supertraces of exterior…

Let V be the representation of the quantised enveloping algebra of a general linear group which is the q-analogue of the vector representation. In this paper we construct a basis of the representations obtained by tensoring copies of V and…

We classify indecomposable summands of mixed tensor powers of the natural representation for the general linear supergroup up to isomorphism. We also give a formula for the characters of these summands in terms of composite supersymmetric…

We construct, for any finite commutative ring $R$, a family of representations of the general linear group $\mathrm{GL}_n(R)$ whose intertwining properties mirror those of the principal series for $\mathrm{GL}_n$ over a finite field.

For semisimple Lie superalgebras over an algebraically closed field of characteristic zero, whose category of finite dimensional super representations is semisismple, we classify all irreducible super representations for which the…

We present in detail the classification of the finite dimensional irreducible representations of the super Yangian associated with the Lie superalgebra $gl(1|1)$.

This work presents an approach towards the representation theory of the braid groups $B_n$. We focus on finite-dimensional representations over the field of Laurent series which can be obtained from representations of infinitesimal braids,…

In this note, we construct all irreducible representations of the quantum general linear super group $GL_q(3|1)$ using the double Koszul complex.