Crystal interpretation of a formula on the branching rule of types $B_{n}$, $C_{n}$, and $D_{n}$

zbMATH Open1435.05221arXiv1602.00181MaRDI QIDQ5217127

Publication date: 21 February 2020

Abstract: The branching coefficients of the tensor product of finite-dimensional irreducible

U_{q} (m a t h f r a k g)

-modules, where

m a t h f r a k g

is

m a t h f r a k s o (2 n + 1, m a t h b b C)

(

B_{n}

-type),

m a t h f r a k s p (2 n, m a t h b b C)

(

C_{n}

-type), and

m a t h f r a k s o (2 n, m a t h b b C)

(

D_{n}

-type), are expressed in terms of Littlewood-Richardson (LR) coefficients in the stable region. We give an interpretation of this relation by Kashiwara's crystal theory by providing an explicit surjection from the LR crystal of type

C_{n}

to the disjoint union of Cartesian product of LR crystals of

A_{n - 1}

-type and by proving that LR crystals of types

B_{n}

and

D_{n}

are identical to the corresponding LR crystal of type

C_{n}

in the stable region.

Full work available at URL: https://arxiv.org/abs/1602.00181

zbMATH Keywords

Kashiwara-Nakashima tableaux Kashiwara crystals branchlng rule Littlewood-Richardson crystals

Mathematics Subject Classification ID

Combinatorial aspects of representation theory (05E10) Quantum groups (quantized function algebras) and their representations (20G42)

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