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.

pdf (360 kb)   tex (15 kb)

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.
5. Macdonald I.G., On the degrees of the irreducible representations of symmetric groups, Bull. London Math. Soc. 3 (1971), 189-192.
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.