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On Stanley’s inequalities for character multiplicities

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Abstract

Let G be a group of automorphisms of a ranked poset \({{\mathcal Q}}\) and let N k denote the number of orbits on the elements of rank k in \({{\mathcal Q}}\). What can be said about the N k for standard posets, such as finite projective spaces or the Boolean lattice? We discuss the connection of this question to the representation theory of the group, and in particular to the inequalities of Livingstone-Wagner and Stanley. We show that these are special cases of more general inequalities which depend on the prime divisors of the group order. The new inequalities often yield stronger bounds depending on the order of the group.

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Authors and Affiliations

  1. Department of Mathematics, Southern Federal University, Technological Institute, Taganrog, Russia

    V. B. Mnukhin

  2. School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK

    I. J. Siemons

Authors
  1. V. B. Mnukhin
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  2. I. J. Siemons
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Corresponding author

Correspondence to V. B. Mnukhin.

Additional information

V. B. Mnukhin was supported by the London Mathematical Society and the Leverhulme Trust. In addition RFBR grants 10-07-00135, 10-07-00478, 11-07-00591 are acknowledged.

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Mnukhin, V.B., Siemons, I.J. On Stanley’s inequalities for character multiplicities. Arch. Math. 97, 513–521 (2011). https://doi.org/10.1007/s00013-011-0321-7

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  • DOI: https://doi.org/10.1007/s00013-011-0321-7

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