On bar lengths in partitions

TL;DR

This paper explores the structure of bar lengths in partitions, decomposing them into core and quotient parts, and applies this to derive formulas for degrees of spin characters in symmetric group extensions.

Contribution

It introduces a novel decomposition of bar lengths in partitions into core and quotient components and derives a relative bar formula for spin character degrees.

Findings

Bar lengths in a partition can be decomposed into core and quotient parts.

The multiset of bar lengths in the core is a sub-multiset of the original.

A new formula relates bar lengths to degrees of spin characters.

Abstract

In this paper, we present, given a odd integer $d$, a decomposition of the multiset of bar lengths of a bar partition $\lambda$ as the union of two multisets, one consisting of the bar lengths in its $\bar{d}$-core partition $\bar{c}_d(\lambda)$ and the other consisting of modified bar lengths in its $\bar{d}$-quotient partition. In particular, we obtain that the multiset of bar lengths in $\bar{c}_d(\lambda)$ is a sub-multiset of the multiset of bar lengths in $\lambda$. Also we obtain a relative bar formula for the degrees of spin characters of the Schur extensions of the symmetric group. The proof involves a recent similar result for partitions, proved in [1].