Tensor species and symmetric functions.
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.
Méndez
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Departmento de Mathematicas, Universidad de Carabobo, Valencia, Estado Carabobo, Venezuela.