Combinatorial description of Lusztig $q$-weight multiplicity

TL;DR

This paper explores a conjectured relationship between Lusztig q-weight multiplicities and Kirillov-Reshetikhin crystals in type C, introducing a new gl_n-version and proposing positivity conjectures.

Contribution

It establishes a conjectural link between Lusztig q-weight multiplicities and crystal bases, and defines a new version for type C with positivity conjectures.

Findings

Conjectured a precise relationship between Lusztig q-weight multiplicities and Kirillov-Reshetikhin crystals.

Defined a gl_n-version of q-weight multiplicity for type C.

Proposed positivity conjectures for the new q-weight multiplicity.

Abstract

We conjecture a precise relationship between Lusztig $q$-weight multiplicities for type $C$ and Kirillov-Reshetikhin crystals. We also define $\mathfrak{gl}_n$-version of $q$-weight multiplicity for type $C$ and conjecture the positivity.