ON BAR LENGTHS IN PARTITIONS

We present, given an 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 (d) over bar -core partition (c) over bar (d)(lambda) and the other consisting of modified bar lengths in its (d) over bar -quotient partition. In particular, we obtain that the multiset of bar lengths in (c) over bar (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 G(n). The proof involves a recent similar result for partitions, proved by Bessenrodt and the authors.