Open AccessTensor species and symmetric functions.
Author(s) -
Miguel Méndez
Publication year - 1991
Publication title -
proceedings of the national academy of sciences of the united states of america
Language(s) - English
Resource type - Journals
eISSN - 1091-6490
pISSN - 0027-8424
DOI - 10.1073/pnas.88.21.9892
Subject(s) - equivariant map , stanley symmetric function , mathematics , symmetric function , symmetric group , ring of symmetric functions , complete homogeneous symmetric polynomial , pure mathematics , monomial , bijection , tensor (intrinsic definition) , symmetric tensor , representation theory of the symmetric group , representation (politics) , representation theory , schur polynomial , combinatorics , macdonald polynomials , mathematical analysis , exact solutions in general relativity , politics , political science , law , orthogonal polynomials , difference polynomials
An equivariant representation of the symmetric group Sn (equivariant representation from here on) is defined as a particular type of tensor species. For any tensor species R the characteristic generating function of R is defined in a way that generalizes the Frobenius characters of representations of the symmetric groups. If R is an equivariant representation, then the characteristic is a homogeneous symmetric function. The combinatorial operations on equivariant representations correspond to formal operations on the respective characteristic functions. In particular, substitution of equivariant representations corresponds to plethysm of symmetric functions. Equivariant representations are constructed that have as characteristic the elementary, complete, and Schur functions. Bijective proofs are given for the formulas that connect them with the monomial symmetric functions.