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Restriction of odd degree characters of \(\mathfrak{S}_n\). (English) Zbl 1378.20013
Summary: 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 [A. Ayyer et al., Sémin. Lothar. Comb. 75, B75g, 13 p. (2016; Zbl 1339.05014)] and recently developed in [I. M. Isaacs et al., J. Algebra 478, 271–282 (2017; Zbl 1423.20006)].

MSC:
20C30 Representations of finite symmetric groups
05A17 Combinatorial aspects of partitions of integers
20C15 Ordinary representations and characters
05E10 Combinatorial aspects of representation theory
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[1] Ayyer, Arvind and Prasad, Amritanshu and Spallone, Steven, Odd partitions in Young’s lattice, S\'eminaire Lotharingien de Combinatoire, 75, Art. B75g, 13 pages, (2015) · Zbl 1339.05014
[2] Giannelli, Eugenio and Kleshchev, Alexander and Navarro, Gabriel and Tiep, Pham Huu, Restriction of odd degree characters and natural correspondences, International Mathematics Research Notices, (None) · Zbl 1404.20006
[3] Isaacs, I. M. and Navarro, Gabriel and Olsson, J\orn B. and Tiep, Pham Huu, Character restrictions and multiplicities in symmetric groups, Journal of Algebra, 478, 271-282, (2017) · Zbl 1423.20006
[4] James, Gordon and Kerber, Adalbert, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, 16, xxviii+510, (1981), Addison-Wesley Publishing Co., Reading, Mass. · Zbl 1159.20012
[5] Macdonald, I. G., On the degrees of the irreducible representations of symmetric groups , The Bulletin of the London Mathematical Society, 3, 189-192, (1971) · Zbl 0219.20008
[6] Olsson, J. B., Combinatorics and representations of finite groups, (None) · Zbl 0796.05095
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