Parities of v-decomposition numbers and an application to symmetric group algebras

TL;DR

This paper establishes a parity property of v-decomposition numbers based on partition signs and applies this to confirm Martin's conjecture for weight 3 blocks of symmetric group algebras, revealing structural module properties.

Contribution

It proves a parity criterion for v-decomposition numbers and verifies Martin's conjecture for specific symmetric group blocks, linking combinatorial and algebraic properties.

Findings

v-decomposition number polynomials are even or odd based on partition parity

Martin's conjecture holds for weight 3 blocks of symmetric group algebras

projective modules in these blocks have radical length 7

Abstract

We prove that the v-decomposition number $d_{\lambda\mu}(v)$ is an even or odd polynomial according to whether the partitions $\lambda$ and $\mu$ have the same relative sign (or parity) or not. We then use this result to verify Martin's conjecture for weight 3 blocks of symmetric group algebras -- that these blocks have the property that their projective (indecomposable) modules have a common radical length 7.