Identities from representation theory

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.

arXiv comment68 pages, 8 figures

References in corpus (12)

• Advanced Determinant Calculus
• Existence of Kirillov-Reshetikhin crystals for nonexceptional types
• Kirillov--Reshetikhin crystals for nonexceptional types
• Virtual crystals and Kleber's algorithm
• Discrete analogues of Macdonald-Mehta integrals
• On bijections between 231-avoiding permutations and Dyck paths
• A simple approach to some Hankel determinants
• FindStat - the combinatorial statistics database
• Crystal structure on rigged configurations and the filling map
• Weight multiplicities and Young tableaux through affine crystals
• Bijections for Dyck paths with all peak heights of the same parity
• Factorial characters of classical Lie groups and their combinatorial realisations

Cited by in corpus (5)

• Promotion and growth diagrams for fans of Dyck paths and vacillating tableaux
• Skew Howe duality and limit shapes of Young diagrams
• $\mathcal{N}=4$ line defect correlators of type BCD
• Fluctuations of Young diagrams for symplectic groups and semiclassical orthogonal polynomials
• Cellular subalgebras of the partition algebra