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

Abstract

We provide a new tableau model from which one can easily deduce the characters of finite-dimensional irreducible polynomial representations of the special orthogonal group $S{O}_n(\mathbb{C})$. This model originates from the representation theory of the $ı$quantum group (also known as the quantum symmetric pair coideal subalgebra) of type $\mathrm{A}\mathrm{I}$, and is equipped with a combinatorial structure, which we call $\mathrm{A}\mathrm{I}$-crystal structure. This structure enables us to describe combinatorially the tensor product of an $S{O}_n(\mathbb{C})$-module and a $G{L}_n(\mathbb{C})$-module, and the branching from $G{L}_n(\mathbb{C})$ to $S{O}_n(\mathbb{C})$.

39 pages. Presentation is modified in order to make it clearer

References in corpus (2)

• Flagged Littlewood-Richardson tableaux and branching rule for classical groups
• Based modules over the $ı$quantum group of type AI

Cited by in corpus (1)

• Crystal bases of modified $ı$quantum groups of certain quasi-split types