Compression of Nakajima monomials in type A and C

TL;DR

This paper develops an explicit crystal morphism connecting Nakajima monomials with crystal bases for type A and C Lie algebras, linking various combinatorial models and enabling new insertion algorithms.

Contribution

It introduces a novel explicit crystal morphism that bridges Nakajima monomials, Kashiwara tableaux, and the Littelmann path model for types A and C.

Findings

Established a crystal morphism connecting Nakajima monomials and crystal bases.

Connected Nakajima monomials with Kashiwara tableaux and the Littelmann path model.

Developed an insertion scheme for Nakajima monomials compatible with tableau insertions.

Abstract

We describe an explicit crystal morphism between Nakajima monomials and monomials which give a realization of crystal bases for finite dimensional irreducible modules over the quantized enveloping algebra for Lie algebras of type A and C. This morphism provides a connection between arbitrary Nakajima monomials and Nakashima Kashiwara tableaux. This yields a translation of Nakajima monomials to the Littelmann path model. Furthermore, as an application of our results we describe an insertion scheme for Nakajima monomials compatible to the insertion scheme for tableaux.