### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 070, 10 pages      arXiv:1705.08655      https://doi.org/10.3842/SIGMA.2017.070
Contribution to the Special Issue on the Representation Theory of the Symmetric Groups and Related Topics

### Restriction of Odd Degree Characters of $\mathfrak{S}_n$

Christine Bessenrodt a, Eugenio Giannelli b and Jørn B. Olsson c
a) Institute for Algebra, Number Theory and Discrete Mathematics, Leibniz Universität Hannover, Welfengarten 1, D-30167 Hannover, Germany
b) Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WA, United Kingdom
c) Department of Mathematical Sciences, University of Copenhagen, DK-2100 Copenhagen Ø, Denmark

Received May 25, 2017, in final form August 30, 2017; Published online September 05, 2017

Abstract
Let $n$ and $k$ be natural numbers such that $2^k$ < $n$. We study the restriction to $\mathfrak{S}_{n-2^k}$ of odd-degree irreducible characters of the symmetric group $\mathfrak{S}_n$. This analysis completes the study begun in [Ayyer A., Prasad A., Spallone S., Sém. Lothar. Combin. 75 (2015), Art. B75g, 13 pages] and recently developed in [Isaacs I.M., Navarro G., Olsson J.B., Tiep P.H., J. Algebra 478 (2017), 271-282].

Key words: characters of symmetric groups; hooks in partitions.

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References

1. Ayyer A., Prasad A., Spallone S., Odd partitions in Young's lattice, Sém. Lothar. Combin. 75 (2015), Art. B75g, 13 pages, arXiv:1601.01776.
2. Giannelli E., Kleshchev A., Navarro G., Tiep P.H., Restriction of odd degree characters and natural correspondences, Int. Math. Res. Not., to appear, arXiv:1601.04423.
3. Isaacs I.M., Navarro G., Olsson J.B., Tiep P.H., Character restrictions and multiplicities in symmetric groups, J. Algebra 478 (2017), 271-282.
4. James G., Kerber A., The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, Vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981.
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6. Olsson J.B., Combinatorics and representations of finite groups, Vorlesungen aus dem Fachbereich Mathematik der Universität Essen, Heft 20, 1994, available at http://www.math.ku.dk/~olsson/manus/comb_rep_all.pdf.