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

TL;DR

This paper introduces a new tableau model based on quantum group theory to compute characters of irreducible polynomial representations of the special orthogonal group, facilitating tensor product and branching rule descriptions.

Contribution

The paper develops a novel tableau model derived from $ extit{i}$-quantum group theory, providing a combinatorial framework for $SO_n( extbf{C})$ representations.

Findings

New tableau model for $SO_n( extbf{C})$ representations

Combinatorial description of tensor products

Branching rules from $GL_n( extbf{C})$ to $SO_n( extbf{C})$

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 $SO_n(\mathbb{C})$. This model originates from the representation theory of the $\imath$quantum group (also known as the quantum symmetric pair coideal subalgebra) of type $\mathrm{A\!I}$, and is equipped with a combinatorial structure, which we call $\mathrm{A\!I}$-crystal structure. This structure enables us to describe combinatorially the tensor product of an $SO_n(\mathbb{C})$-module and a $GL_n(\mathbb{C})$-module, and the branching from $GL_n(\mathbb{C})$ to $SO_n(\mathbb{C})$.