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 finite-dimensional 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 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.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2016
- DOI:
- 10.48550/arXiv.1602.00181
- arXiv:
- arXiv:1602.00181
- Bibcode:
- 2016arXiv160200181H
- Keywords:
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- Mathematics - Quantum Algebra
- E-Print:
- 77 pages, proofs of several lemmas corrected