Crystal interpretation of a formula on the branching rule of types $B_{n}$, $C_{n}$, and $D_{n}$
Abstract
The branching coefficients of the tensor product of finitedimensional irreducible $U_{q}(\mathfrak{g})$modules, where $\mathfrak{g}$ is $\mathfrak{so}(2n+1,\mathbb{C})$ ($B_{n}$type), $\mathfrak{sp}(2n,\mathbb{C})$ ($C_{n}$type), and $\mathfrak{so}(2n,\mathbb{C})$ ($D_{n}$type), are expressed in terms of LittlewoodRichardson (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 $A_{n1}$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.
 Publication:

arXiv eprints
 Pub Date:
 January 2016
 arXiv:
 arXiv:1602.00181
 Bibcode:
 2016arXiv160200181H
 Keywords:

 Mathematics  Quantum Algebra
 EPrint:
 77 pages, proofs of several lemmas corrected