Identities from representation theory

TL;DR

This paper introduces new combinatorial formulas for characters and multiplicities in type C and B Lie algebras, connecting representation theory with various q-analogues of classical combinatorial numbers.

Contribution

It provides novel determinant formulas for characters and multiplicities, and links these to q-analogues of Catalan, Motzkin, and Riordan numbers using combinatorial and crystal base methods.

Findings

New Jacobi--Trudi-type formula for type C characters

Determinant formulas for tensor product multiplicities in types B and C

Identities relating representation theory to q-analogues of classical numbers

Abstract

We give a new Jacobi--Trudi-type formula for characters of finite-dimensional irreducible representations in type $C_n$ using characters of the fundamental representations and non-intersecting lattice paths. We give equivalent determinant formulas for the decomposition multiplicities for tensor powers of the spin representation in type $B_n$ and the exterior representation in type $C_n$. This gives a combinatorial proof of an identity of Katz and equates such a multiplicity with the dimension of an irreducible representation in type $C_n$. By taking certain specializations, we obtain identities for $q$-Catalan triangle numbers, the $q,t$-Catalan number of Stump, $q$-triangle versions of Motzkin and Riordan numbers, and generalizations of Touchard's identity. We use (spin) rigid tableaux and crystal base theory to show some formulas relating Catalan, Motzkin, and Riordan triangle numbers.