### Abstract

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 c_{d} (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 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 o_{n}. The proof involves a recent similar result for partitions, proved by Bessenrodt and the authors.

Original language | English |
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Pages (from-to) | 335-350 |

Number of pages | 16 |

Journal | Proceedings of the Edinburgh Mathematical Society |

Volume | 56 |

Issue number | 2 |

Early online date | 21 Mar 2013 |

DOIs | |

Publication status | Published - Jun 2013 |

### Fingerprint

### Keywords

- bar lengths
- bar partitions
- covering groups
- partitions
- symmetric group

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Proceedings of the Edinburgh Mathematical Society*,

*56*(2), 335-350. https://doi.org/10.1017/S0013091512000387

**On bar lengths in partitions.** / Gramain, Jean Baptiste; Olsson, Jørn B.

Research output: Contribution to journal › Article

*Proceedings of the Edinburgh Mathematical Society*, vol. 56, no. 2, pp. 335-350. https://doi.org/10.1017/S0013091512000387

}

TY - JOUR

T1 - On bar lengths in partitions

AU - Gramain, Jean Baptiste

AU - Olsson, Jørn B.

N1 - J.-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.

PY - 2013/6

Y1 - 2013/6

N2 - 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.

AB - 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.

KW - bar lengths

KW - bar partitions

KW - covering groups

KW - partitions

KW - symmetric group

UR - http://www.scopus.com/inward/record.url?scp=84910627765&partnerID=8YFLogxK

U2 - 10.1017/S0013091512000387

DO - 10.1017/S0013091512000387

M3 - Article

VL - 56

SP - 335

EP - 350

JO - Proceedings of the Edinburgh Mathematical Society

JF - Proceedings of the Edinburgh Mathematical Society

SN - 0013-0915

IS - 2

ER -