On bar lengths in partitions

Jean Baptiste Gramain; Jørn B. Olsson

doi:10.1017/S0013091512000387

On bar lengths in partitions

Jean Baptiste Gramain^*, Jørn B. Olsson

^*Corresponding author for this work

Mathematical Science

Research output: Contribution to journal › Article › peer-review

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
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	https://doi.org/10.1017/S0013091512000387
Publication status	Published - Jun 2013

Keywords

bar lengths
bar partitions
covering groups
partitions
symmetric group

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title = "On bar lengths in partitions",

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 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.",

keywords = "bar lengths, bar partitions, covering groups, partitions, symmetric group",

author = "Gramain, {Jean Baptiste} and Olsson, {J{\o}rn B.}",

note = "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.",

year = "2013",

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doi = "10.1017/S0013091512000387",

language = "English",

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publisher = "Cambridge University Press",

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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

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DO - 10.1017/S0013091512000387

M3 - Article

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SP - 335

EP - 350

JO - Proceedings of the Edinburgh Mathematical Society

JF - Proceedings of the Edinburgh Mathematical Society

SN - 0013-0915

IS - 2

ER -

On bar lengths in partitions

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Keywords

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