Weight Multiplicities and Young Tableaux Through Affine Crystals / Jang-Soo Kim, Kyu-Hwan Lee, and Se-jin Oh.

Format:

Book

Author/Creator:

Kim, Jang-Soo, 1952- author.

Lee, Kyu-Hwan, 1970- author.

Oh, Se-jin, 1981- author.

Series:

Memoirs of the American Mathematical Society ; Volume 283.

Memoirs of the American Mathematical Society Series ; Volume 283

Language:

English

Subjects (All):

Affine algebraic groups.

Combinatorial analysis.

Representations of algebras.

Young tableaux.

Kac-Moody algebras.

Quantum groups.

Physical Description:

1 online resource (100 pages)

Edition:

First edition.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

Summary:

"The weight multiplicities of finite dimensional simple Lie algebras can be computed individually using various methods. Still, it is hard to derive explicit closed formulas. Similarly, explicit closed formulas for the multiplicities of maximal weights of affine Kac-Moody algebras are not known in most cases. In this paper, we study weight multiplicities for both finite and affine cases of classical types for certain infinite families of highest weights modules. We introduce new classes of Young tableaux, called the (spin) rigid tableaux, and prove that they are equinumerous to the weight multiplicities of the highest weight modules under our consideration. These new classes of Young tableaux arise from crystal basis elements for dominant maximal weights of the integrable highest weight modules over affine Kac-Moody algebras. By applying combinatorics of tableaux such as the Robinson-Schensted algorithm and new insertion schemes, and using integrals over orthogonal groups, we reveal hidden structures in the sets of weight multiplicities and obtain explicit closed formulas for the weight multiplicities. In particular we show that some special families of weight multiplicities form the Pascal, Catalan, Motzkin, Riordan and Bessel triangles"-- Provided by publisher.

Contents:

Affine Kac-Moody algebras

Crystals and Young walls

Young tableaux and almost even tableaux

Lattice paths and triangular arrays

Dominant maximal weights

Weight multiplicities and (spin) rigid Young tableaux

Level 2 weight multiplicities : Catalan and Pascal triangles

Level 3 weight multiplicities : Motzkin and Riordan triangles

Some level k weight multiplicities when k

> infinity : Bessel triangle

Standard Young tableaux with a fixed number of rows of odd length.

Notes:

Description based on print version record.

Includes bibliographical references.

Other Format:

Print version: Kim, Jang-Soo Weight Multiplicities and Young Tableaux Through Affine Crystals

ISBN:

9781470474027