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Hideya Watanabe

Publications and source records attributed to Hideya Watanabe.

At least 19 recordsLinked to original sources
arXiv2026

Kashiwara--Nakashima tableaux, Gelfand--Tsetlin patterns, and quantum symmetric pairs

Kashiwara--Nakashima tableaux and Gelfand--Tsetlin patterns of orthogonal type are famous as combinatorial models of the finite-dimensional irreducible representations of the special orthogonal Lie algebras $\mathfrak{so}_N$. The former is constructed based on the representation theory of quantum groups, especially the theory of crystals, while the latter based on the branching rule for $(\mathfrak{so}_N,\mathfrak{so}_{N-1})$. In the present paper, we construct a natural bijection between them by means of the representation theory of quantum symmetric pairs corresponding to $(\mathfrak{so}_N,\mathfrak{so}_{N-1})$.

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arXiv2025

Berele row-insertion and quantum symmetric pairs

The Berele row-insertion is a symplectic analogue of the Schensted row-insertion. In the present paper, we provide it with a representation theoretical interpretation via the quantum symmetric pairs of type $A\mathrm{II}$. As applications, we lift Berele's Robinson--Schensted correspondence and Kobayashi--Matsumura's Robinson--Schensted--Knuth (RSK for short) correspondence to isomorphisms of representations over a quantum symmetric pair coideal subalgebra, and establish the dual RSK correspondence of type $A\mathrm{II}$.

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arXiv2025

Finite-dimensional irreducible representations of twisted loop algebras of the second kind

Twisted loop algebras of the second kind are infinite-dimensional Lie algebras that are constructed from a semisimple Lie algebra and an automorphism on it of order at most $2$. They are examples of equivariant map algebras. The finite-dimensional irreducible representations of an arbitrary equivariant map algebra have been classified by Neher--Savage--Senesi. In this paper, we classify the finite-dimensional irreducible representations of twisted loop algebras of the second kind in a more elementary way.

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arXiv2025

A proof of the Naito--Sagaki conjecture via the branching rule for $\imath$quantum groups

The Naito--Sagaki conjecture asserts that the branching rule for the restriction of finite-dimensional, irreducible polynomial representations of $GL_{2n}(\mathbb{C})$ to $Sp_{2n}(\mathbb{C})$ amounts to the enumeration of certain ``rational paths'' satisfying specific conditions. This conjecture can be thought of as a non-Levi type analog of the Levi type branching rule, stated in terms of the path model due to Littelmann, and was proved combinatorially in 2018 by Schumann--Torres. In this paper, we give a new proof of the Naito--Sagaki conjecture independently of Schumann--Torres, using the branching rule based on the crystal basis theory for $\imath$quantum groups of type $\mathrm{AII}_{2n-1}$. Here, note that $\imath$quantum groups are certain coideal subalgebras of a quantized universal enveloping algebra obtained by $q$-deforming symmetric pairs, and also regarded as a generalization of quantized universal enveloping algebras; these were defined by Letzter in 1999, and since then their representation theory has become an active area of research. The main ingredients of our approach are certain combinatorial operations, such as promotion operators and Kashiwara operators, which are well-suited to the representation theory of complex semisimple Lie algebras.

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arXiv2024

Integrable modules over quantum symmetric pair coideal subalgebras

We introduce the notion of integrable modules over $\imath$quantum groups (a.k.a. quantum symmetric pair coideal subalgebras). After determining a presentation of such modules, we prove that each integrable module over a quantum group is integrable when restricted to an $\imath$quantum group. As an application, we show that the space of matrix coefficients of all simple integrable modules over an $\imath$quantum group of finite type with specific parameters coincides with Bao-Song's coordinate ring of the $\imath$quantum group.

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arXiv2023

Symplectic tableaux and quantum symmetric pairs

We provide a new branching rule from the general linear group $GL_{2n}(\mathbb{C})$ to the symplectic group $Sp_{2n}(\mathbb{C})$ by establishing a simple algorithm which gives rise to a bijection from the set of semistandard tableaux of a fixed shape to a disjoint union of several copies of sets of symplectic tableaux of various shapes. The algorithm arises from representation theory of a quantum symmetric pair of type $A\mathrm{II}_{2n-1}$, which is a $q$-analogue of the classical symmetric pair $(\mathfrak{gl}_{2n}(\mathbb{C}), \mathfrak{sp}_{2n}(\mathbb{C}))$.

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arXiv2023

Stability of $\imath$canonical bases of locally finite type

We prove the stability conjecture of $\imath$canonical bases, which was raised by Huanchen Bao and Weiqiang Wang in 2016, for all locally finite types. To this end, we characterize the trivial module over the $\imath$quantum groups of such type at $q = \infty$. This result can be seen as a very restrictive version of the $\imath$crystal base theory for locally finite types.

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arXiv2023

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

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 $SO_n(\mathbb{C})$.

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arXiv2023

Crystal bases of modified $\imath$quantum groups of certain quasi-split types

In order to see the behavior of $\imath$canonical bases at $q = \infty$, we introduce the notion of $\imath$crystals associated to an $\imath$quantum group of certain quasi-split type. The theory of $\imath$crystals clarifies why $\imath$canonical basis elements are not always preserved under natural homomorphisms. Also, we construct a projective system of $\imath$crystals whose projective limit can be thought of as the $\imath$canonical basis of the modified $\imath$quantum group at $q = \infty$.

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arXiv2022

Based modules over the $\imath$quantum group of type AI

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.

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arXiv2022

Kirillov-Reshetikhin modules and quantum $K$-matrices

From a quantum $K$-matrix of the fundamental representation, we construct one for the Kirillov-Reshetikhin module by fusion construction. Using the $\imath$crystal theory by the last author, we also obtain combinatorial $K$-matrices corresponding to the symmetric tensor representations of affine type $A$ for all quasi-split Satake diagrams.

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arXiv2022

Stability of $\imath$canonical bases of irreducible finite type of real rank one

It has been known since their birth in Bao and Wang's work that the $\imath$canonical bases of $\imath$quantum groups are not stable in general. In the author's previous work, the stability of $\imath$canonical bases of certain quasi-split types turned out to be closely related to the theory of $\imath$crystals. In this paper, we prove the stability of $\imath$canonical bases of irreducible finite type of real rank $1$, for which the theory of $\imath$crystals has not been developed, by means of global and local crystal bases.

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arXiv2021

Alcove paths and Gelfand-Tsetlin patterns

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 model. 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 $A$.

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arXiv2021

Classical weight modules over $\imath$quantum groups

$\imath$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.

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arXiv2020

Canonical bases for tensor products and super Kazhdan-Lusztig theory

We generalize a construction in [BW18] (arXiv:1610.09271) by showing that the tensor product of a based $\textbf{U}^{\imath}$-module and a based $\textbf{U}$-module is a based $\textbf{U}^{\imath}$-module. This is then used to formulate a Kazhdan-Lusztig theory for an arbitrary parabolic BGG category $\mathcal{O}$ of the ortho-symplectic Lie superalgebras, extending a main result in [BW13] (arXiv:1310.0103).

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arXiv2018

Global crystal bases for integrable modules over a quantum symmetric pair of type AIII

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 nodes. Also, we obtain an intrinsic characterization of the $\jmath$-crystal bases, whose original definition is artificial.

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arXiv2018

A combinatorial formula expressing periodic $R$-polynomials

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 of 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.

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