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Tensor species and symmetric functions.

M Méndez1

  • 1Departmento de Mathematicas, Universidad de Carabobo, Valencia, Estado Carabobo, Venezuela.

Proceedings of the National Academy of Sciences of the United States of America
|November 1, 1991
PubMed
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This study defines equivariant representations of the symmetric group Sn using tensor species. Their characteristic generating functions generalize Frobenius characters and correspond to symmetric functions, with combinatorial operations mapping to plethysm.

Area of Science:

  • Algebraic Combinatorics
  • Representation Theory
  • Symmetric Functions

Background:

  • Equivariant representations are a type of tensor species.
  • Characteristic generating functions generalize Frobenius characters.
  • These functions are homogeneous symmetric functions for equivariant representations.

Purpose of the Study:

  • To define and explore equivariant representations of the symmetric group Sn.
  • To establish connections between combinatorial operations on representations and operations on symmetric functions.
  • To construct specific equivariant representations linked to elementary, complete, and Schur functions.

Main Methods:

  • Definition of equivariant representations as tensor species.
  • Generalization of Frobenius characters to characteristic generating functions.

Related Experiment Videos

  • Exploration of combinatorial operations and their correspondence to symmetric function operations like plethysm.

    • Main Results:

    • Characteristic generating functions for equivariant representations are homogeneous symmetric functions.
    • Combinatorial operations on representations correspond to formal operations on characteristic functions.
    • Equivariant representations with elementary, complete, and Schur functions as characteristics were constructed.

    Conclusions:

    • The framework provides a unified approach to studying representations and symmetric functions.
    • Bijective proofs confirm the relationship between constructed representations and monomial symmetric functions.
    • This work deepens the understanding of the interplay between representation theory and combinatorics.