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Stability properties of Plethysm: new approach with combinatorial proofs (Extended abstract)

Abstract : Plethysm coefficients are important structural constants in the theory of symmetric functions and in the representations theory of symmetric groups and general linear groups. In 1950, Foulkes observed stability properties: some sequences of plethysm coefficients are eventually constants. Such stability properties were proven by Brion with geometric techniques and by Thibon and Carré by means of vertex operators. In this paper we present a new approach to prove such stability properties. This new proofs are purely combinatorial and follow the same scheme. We decompose plethysm coefficients in terms of other plethysm coefficients (related to the complete homogeneous basis of symmetric functions). We show that these other plethysm coefficients count integer points in polytopes and we prove stability for them by exhibiting bijections between the corresponding sets of integer points of each polytope.
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  • HAL Id : hal-01337765, version 1
  • ARXIV : 1505.03842

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Laura Colmenarejo. Stability properties of Plethysm: new approach with combinatorial proofs (Extended abstract). 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), Jul 2015, Daejeon, South Korea. pp.877-888. ⟨hal-01337765⟩

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