TY - JOUR

T1 - Identities from representation theory

AU - Oh, Se jin

AU - Scrimshaw, Travis

N1 - Funding Information:
SjO was partially supported by the National Research Foundation of Korea (NRF) Grant funded by the Korea government (MSIP) (NRF-2016R1C1B2013135).TS was partially supported by the National Science Foundation RTG, United States grant NSF/DMS-1148634 and the Australian Research CouncilDP170102648. The authors would like to thank Jang Soo Kim, Christian Stump, and Ole Warnaar for useful discussions. The authors additionally thank Ole Warnaar and Christian Stump for comments on earlier versions of this manuscript. Furthermore, the authors thank Jang Soo Kim for an initial proof of Proposition 5.30 using [8, Lemma 3.3] for dimV(rϖ˜
Publisher Copyright:
© 2019 Elsevier B.V.

PY - 2019/9

Y1 - 2019/9

N2 - 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.

AB - 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.

KW - Catalan number

KW - Jacobi–Trudi formula

KW - q-analog

UR - http://www.scopus.com/inward/record.url?scp=85066748381&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2019.05.020

DO - 10.1016/j.disc.2019.05.020

M3 - Article

AN - SCOPUS:85066748381

SN - 0012-365X

VL - 342

SP - 2493

EP - 2541

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 9

ER -