Sign sequences and decomposition numbers

TL;DR

This paper provides a closed formula for certain $v$-decomposition numbers related to the Fock space representation of quantum affine algebras, connecting them to classical decomposition numbers upon evaluation at $v=1$.

Contribution

It introduces a new closed formula for $v$-decomposition numbers associated with partitions obtained by moving nodes of the same $e$-residue.

Findings

Derived a closed formula for $v$-decomposition numbers.

Connected $v$-decomposition numbers to classical decomposition numbers at $v=1.

Extended understanding of the combinatorial structure of decomposition numbers.

Abstract

We obtain a closed formula for the $v$-decomposition numbers $d_{\lambda\mu}(v)$ arising from the canonical basis of the Fock space representation of $U_v(\hat{\mathfrak{sl}}_e)$, where the partition $\lambda$ is obtained from $\mu$ by moving some nodes in its Young diagram, all of which having the same $e$-residue. We also show that when these $v$-decomposition numbers are evaluated at $v=1$, we obtain the corresponding decomposition numbers for the Schur algebras and symmetric groups.