Related papers: Identities from representation theory
Assuming standard conjectures, we show that the canonical symmetrizing trace evaluated at powers of a Coxeter element produces rational Catalan numbers for irreducible spetsial complex reflection groups. This extends a technique used by…
We examine complexity and versatility of five modulo 9 Kanade--Russell identities through their finite (aka polynomial) versions and images under the $q\mapsto1/q$ reflection.
We revisit the local metaplectic correspondence previously constructed and studied by Flicker and Kazhdan. After restoring and generalizing some of their results, we get several interesting applications to the representation theory of a…
Recently a remarkable map between 4-dimensional superconformal field theories and vertex algebras has been constructed \cite{BLLPRV15}. This has lead to new insights in the theory of characters of vertex algebras. In particular it was…
We state and prove various new identities involving the Z_K parafermion characters (or level-K string functions) for the cases K=4, K=8, and K=16. These identities fall into three classes: identities in the first class are generalizations…
In this paper, we study congruences on sums of products of binomial coefficients that can be proved by using properties of the Jacobi polynomials. We give special attention to polynomial congruences containing Catalan numbers, second-order…
A generalization of the Catalan numbers is considered. New results include binomial identities, recursive relations and a close formula for the multivariate generating function. A simple expression for the Catalan determinant is derived.
In the first paper of this series, we gave infinite families of coloured partition identities which generalise Primc's and Capparelli's classical identities. In this second paper, we study the representation theoretic consequences of our…
For given representation of finite groups on a finite dimension complex vector space, we can define exterior powers of representations. In 1973, Knutson found one of methods of calculating the character of exterior powers of representations…
We present an explicit formula for the transition matrix $\mathcal{C}$ from the type $C_n$ degeneration of the Koornwinder polynomials $P_{(1^r)}(x\,|\,a,-a,c,-c\,|\,q,t)$ with one column diagrams, to the type $C_n$ monomial symmetric…
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 provide a short proof of an algebraic identity. For integers $n\ge 2$ and variables $x,y,z$, it represents $(x^n+y^n-z^n)$ as a value of the quadratic form $\mathcal A^2+\mathcal B^2-\mathcal C^2$ after multiplication by an explicit…
Recently the second named author discovered a combinatorial identity in the context of vertex representations of quantum Kac-Moody algebras. We give a direct and elementary proof of this identity. Our method is to show a related identity of…
In this paper, bicomplex k-Fibonacci quaternions are defined. Also, some algebraic properties of bicomplex k-Fibonacci quaternions which are connected with bicomplex numbers and k-Fibonacci numbers are investigated. Furthermore, the…
In [APS], the authors characterize the partitions of $n$ whose corresponding representations of $S_n$ have nontrivial determinant. The present paper extends this work to all irreducible finite Coxeter groups $W$. Namely, given a nontrivial…
Aspects of the theory of characteristic modes, based on their variational formulation, are presented and an explicit form of a related functional, involving only currents in a spatial domain, is derived. The new formulation leads to deeper…
In this paper, we show combinatorial identities that represent powers of positive integers using multinomial coefficients, which do not come from the multinomial theorem and the multinomial Vandermonde's convolution.
We exhibit a faithful representation of the stylic monoid of every finite rank as a monoid of upper unitriangular matrices over the tropical semiring. Thus, we show that the stylic monoid of finite rank $n$ generates the pseudovariety…
We show that a relation among minimal non-faces of a fillable complex $K$ yields an identity of iterated (higher) Whitehead products in a polyhedral product over $K$. In particular, for the $(n-1)$-skeleton of a simplicial $n$-sphere, we…
The main purpose of this note is to provide an elementary discussion of some simple triangles of integer numbers in particular through their connections with representation theory of $sl_2$. The triangles under consideration are the Catalan…