Related papers: Representations and binomial coefficients
We present a combinatorial proof of the $q$-Pfaff--Saalsch\"utz identity by a composition of explicit bijections, in which $q$-binomial coefficients are interpreted as counting subspaces of $\mathbb{F}_q$-vector spaces. As a corollary, we…
We present new proofs and generalizations of unimodality of the q-binomial coefficients \binom{n}{k}_q as polynomials in q. We use an algebraic approach by interpreting the differences between numbers of certain partitions as Kronecker…
We give a description of simple functors taking finitely generated values, from a small additive category to the category of vector spaces over a field. This result is analogous to Steinberg's tensor product theorems in group representation…
We study a reproducing kernel Hilbert space of functions defined on the positive integers and associated to the binomial coefficients. We introduce two transforms, which allow us to develop a related harmonic analysis in this Hilbert space.…
For a split quasireductive supergroup $G$ defined over a field, we study structure and representation of Frobenius kernels $G_r$ of $G$ and we give a necessary and sufficient condition for $G_r$ to be unimodular in terms of the root system…
In this short note we show that representation and character varieties of discrete groups can be viewed as tensor products of suitable functors over the PROP of cocommutative Hopf algebras. Such view point has several interesting…
In this paper, we study the tensor product of two unitary irreducible representations, as well as the tensor product of a unitary irreducible representation with a finite-dimensional one, and determine the corresponding Clebsch-Gordan…
We consider quantum group representations Rep(G_q) for a semisimple algebraic group G at a complex root of unity q. Here we allow q to be of any order. We first show that the Tannakian center in Rep(G_q) is calculated via a twisting of…
We introduce the rigid tensor category of tubular partitions, and use it to provide a combinatorial model for the representation category of the quantum automorphism group of a homogeneous rooted tree.
In this paper, we consider the necessary and sufficient conditions for the tensor product of the fundamental representations for the restricted quantum loop algebras of type A at roots of unity to be irreducible.
An operator system modulo the kernel of a completely positive linear map of the operator system gives rise to an operator system quotient. In this paper, operator system quotients and quotient maps of certain matrix algebras are considered.…
Motivated by the Saxl conjecture and the tensor square conjecture, which states that the tensor squares of certain irreducible representations of the symmetric group contain all irreducible representations, we study the tensor squares of…
The reduced Kronecker coefficients are particular instances of Kronecker coefficients that contain enough information to recover them. In this notes we compute the generating function of a family of reduced Kronecker coefficients. We also…
For each compact, simple, simply-connected Lie group and each integer level we construct a modular tensor category from a quotient of a certain subcategory of the category of representations of the corresponding quantum group. We determine…
Generators and relations are given for the subalgebra of cocommutative elements in the quantized coordinate rings of the classical groups, where the deformation parameter q is transcendental. This is a ring theoretic formulation of the well…
The purpose of this paper is to compute the Drinfel'd polynomials for two types of evaluation representations of quantum affine algebras at roots of unity and construct those representations as the submodules of evaluation Schnizer modules.…
The authors continue a series of articles studying certain unitary representations of the Richard Thompson groups $F,T,V$ called Pythagorean. They all extend to the Cuntz algebra $\mathcal{O}$ and conversely all representations of…
The operation of tensor product of Cohomological Field Theories (or algebras over genus zero moduli operad) introduced in an earlier paper by the authors is described in full detail, and the proof of a theorem on additive relations between…
An algebraic interpretation of the one-variable quantum $q$-Krawtchouk polynomials is provided in the framework of the Schwinger realization of $\mathcal{U}_{q}(sl_{2})$ involving two independent $q$-oscillators. The polynomials are shown…
Loeb showed that a natural extension of the usual binomial coefficient to negative (integer) entries continues to satisfy many of the fundamental properties. In particular, he gave a uniform binomial theorem as well as a combinatorial…