On bar lengths in partitions

In this paper, we present, given a odd integer $d$, a decomposition of the multiset of bar lengths of a bar partition $λ$ as the union of two multisets, one consisting of the bar lengths in its $\bar{d}$-core partition $\bar{c}_d(λ)$ 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 $\bar{c}_d(λ)$ is a sub-multiset of the multiset of bar lengths in $λ$. 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].

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