Generalized hook lengths in symbols and partitions
Christine Bessenrodt
, Jean-Baptiste Gramain
and Jørn B. Olsson
Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167, Hannover, Germany
DOI: 10.1007/s10801-011-0338-9
Abstract
In this paper we present, for any integer d, a description of the set of hooks in a d-symbol. We then introduce generalized hook length functions for a d-symbol, and prove a general result about them, involving the core and quotient of the symbol. We list some applications, for example to the well-known hook lengths in integer partitions. This leads in particular to a generalization of a relative hook formula for the degree of characters of the symmetric group discovered by G. Malle and G. Navarro in Trans. Am. Math. Soc. 363, 6647-6669, 2011.
Pages: 309–332
Keywords: symbols; hooks; hook lengths; partitions; core; quotient
Full Text: PDF
References
James, G., Kerber, A.: The Representation Theory of the Symmetric Group. Encyclopedia of Mathematics and its Applications, vol.
16. Addison-Wesley, Reading (1981) Malle, G.: Unipotente Grade imprimitiver komplexer Spiegelungsgruppen. J. Algebra 177, 768-826 (1995) CrossRef Malle, G., Navarro, G.: Blocks with equal height zero degrees. Trans. Am. Math. Soc. 363, 6647-6669 (2011) CrossRef Olsson, J.B.: Combinatorics and Representations of Finite Groups. Vorlesungen aus dem FB Mathematik der Univ. Essen, vol. 20 (1993) (This book is freely available at the author's homepage)
16. Addison-Wesley, Reading (1981) Malle, G.: Unipotente Grade imprimitiver komplexer Spiegelungsgruppen. J. Algebra 177, 768-826 (1995) CrossRef Malle, G., Navarro, G.: Blocks with equal height zero degrees. Trans. Am. Math. Soc. 363, 6647-6669 (2011) CrossRef Olsson, J.B.: Combinatorics and Representations of Finite Groups. Vorlesungen aus dem FB Mathematik der Univ. Essen, vol. 20 (1993) (This book is freely available at the author's homepage)