Identities from representation theory

Se jin Oh, Travis Scrimshaw

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

We give a new Jacobi–Trudi-type formula for characters of finite-dimensional irreducible representations in type Cn 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 Bn and the exterior representation in type Cn. This gives a combinatorial proof of an identity of Katz and equates such a multiplicity with the dimension of an irreducible representation in type Cn. By taking certain specializations, we obtain identities for q-Catalan triangle numbers, a slight modification of 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.

Original languageEnglish
Pages (from-to)2493-2541
Number of pages49
JournalDiscrete Mathematics
Volume342
Issue number9
DOIs
StatePublished - Sep 2019

Bibliographical note

Publisher Copyright:
© 2019 Elsevier B.V.

Keywords

  • Catalan number
  • Jacobi–Trudi formula
  • q-analog

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