A new tableau model for irreducible polynomial representations of the orthogonal group

Hideya Watanabe

Published 2021-07-01Version 1

We provide a new tableau model from which one can easily deduce the characters of irreducible polynomial representations of the orthogonal group $\mathrm{O}_n(\mathbb{C})$. This model originates from representation theory of the $\imath$quantum group of type AI, and is equipped with a combinatorial structure, which we call AI-crystal structure. This structure enables us to describe combinatorially the tensor product of an $\mathrm{O}_n(\mathbb{C})$-module and a $\mathrm{GL}_n(\mathbb{C})$-module, and the branching from $\mathrm{GL}_n(\mathbb{C})$ to $\mathrm{O}_n(\mathbb{C})$.