Two remarks on decomposition numbers of standard modules for quantum affine $\mathfrak{sl}_2$

TL;DR

This paper derives explicit positive formulas for decomposition numbers and q-characters of irreducible representations of quantum affine sl_2 using geometric and combinatorial methods.

Contribution

It introduces new closed-form formulas for decomposition numbers and q-characters, advancing understanding of quantum affine algebra representations.

Findings

Explicit positive formulas for certain decomposition numbers.

A piecewise-linear formula for q-characters.

Enhanced understanding of quantum affine sl_2 representations.

Abstract

We use Nakajima's geometric approach to representations of quantum affine algebras and recent results on explicit descriptions of specific canonical basis elements, to derive closed positive formulas for certain decomposition numbers of representations of quantum affine $\mathfrak{sl}_2$. Moreover, we obtain a piecewise-linear closed formula for the $q$-characters of irreducible representations of quantum affine $\mathfrak{sl}_2$.