## 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, 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 language | English |
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Pages (from-to) | 2493-2541 |

Number of pages | 49 |

Journal | Discrete Mathematics |

Volume | 342 |

Issue number | 9 |

DOIs | |

State | Published - Sep 2019 |

### Bibliographical note

Publisher Copyright:© 2019 Elsevier B.V.

## Keywords

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