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 $mathrm{O}_n(mathbb{C})$.
This paper studies classical weight modules over the $imath$quantum group $mathbf{U}^{imath}$ of type AI. We introduce the notion of based $mathbf{U}^{imath}$-modules by generalizing the notion of based modules over the quantum groups. We prove that
each finite-dimensional irreducible classical weight $mathbf{U}^{imath}$-module with integer highest weight is a based $mathbf{U}^{imath}$-module. As a byproduct, a new combinatorial formula for the branching rule from $mathfrak{sl}_n$ to $mathfrak{so}_n$ is obtained.
$imath$quantum groups are generalizations of quantum groups which appear as coideal subalgebras of quantum groups in the theory of quantum symmetric pairs. In this paper, we define the notion of classical weight modules over an $imath$quantum group,
and study their properties along the lines of the representation theory of weight modules over a quantum group. In several cases, we classify the finite-dimensional irreducible classical weight modules by a highest weight theory.
In their study of the equivariant K-theory of the generalized flag varieties $G/P$, where $G$ is a complex semisimple Lie group, and $P$ is a parabolic subgroup of $G$, Lenart and Postnikov introduced a combinatorial tool, called the alcove paths mod
el. It provides a model for the highest weight crystals with dominant integral highest weights, generalizing the model by semistandard Young tableaux. In this paper, we prove a simple and explicit formula describing the crystal isomorphism between the alcove paths model and the Gelfand-Tsetlin patterns model for type $A$.
In this paper, we study basic properties of global $jmath$-crystal bases for integrable modules over a quantum symmetric pair coideal subalgebra $mathbf{U}^{jmath}$ associated to the Satake diagram of type AIII with even white nodes and no black node
s. Also, we obtain an intrinsic characterization of the $jmath$-crystal bases, whose original definition is artificial.
We study the representation theory of a quantum symmetric pair $(mathbf{U},mathbf{U}^{jmath})$ with two parameters $p,q$ of type AIII, by using highest weight theory and a variant of Kashiwaras crystal basis theory. Namely, we classify the irreducibl
e $mathbf{U}^{jmath}$-modules in a suitable category and associate with each of them a basis at $p=q=0$, the $jmath$-crystal basis. The $jmath$-crystal bases have nice combinatorial properties as the ordinary crystal bases do.
In 1980, Lusztig introduced the periodic Kazhdan-Lusztig polynomials, which are conjectured to have important information about the characters of irreducible modules of a reductive group over a field of positive characteristic, and also about those o
f an affine Kac-Moody algebra at the critical level. The periodic Kazhdan-Lusztig polynomials can be computed by using another family of polynomials, called the periodic $R$-polynomials. In this paper, we prove a (closed) combinatorial formula expressing periodic $R$-polynomials in terms of the doubled Bruhat graph associated to a finite Weyl group and a finite root system.
