We present, given an 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 d -core partition cd (λ) and the other consisting of modified bar lengths in its d -quotient partition. In particular, we obtain that the multiset of bar lengths in cd (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 on. The proof involves a recent similar result for partitions, proved by Bessenrodt and the authors.
Bibliographical noteJ.-B.G. gratefully acknowledges financial support from a grant of the Agence Nationale de la Recherche (ANR-10-PDOC-021-01). He also expresses his gratitude to J. B. Olsson and J. Grodal for their (not only financial) support during
his stay at the University of Copenhagen, where most of this work was done. Finally, the authors thank C. Bessenrodt for useful discussions and for a careful reading of the manuscript, thereby pointing out a problem in an earlier version of this work.
- bar lengths
- bar partitions
- covering groups
- symmetric group