Decomposition Theorem for Product of Fundamental Crystals in Monomial Realization of type $C_n$

TL;DR

This paper establishes a decomposition theorem for the product of fundamental crystals in monomial realization of type C_n, revealing a new structure different from tensor product decompositions.

Contribution

It introduces a novel decomposition rule for the product of fundamental crystals in monomial realization, distinct from tensor product rules.

Findings

The product of fundamental crystals retains a crystal structure.

The decomposition into irreducible crystals differs from tensor product decompositions.

Provides explicit description of the decomposition process.

Abstract

We consider a product of fundamental crystals of type $C_n$ in monomial realization, where the product means a natural product of Laurent monomials, not a tensor product. Then we shall show that the product still holds a crystal structure and describe how it is decomposed into irreducible crystals, which is, in general, different from the decomposition rule for the tensor product of the fundamental crystals.