Transition matrices and Pieri-type rules for polysymmetric functions

Asvin G and Andrew O’Desky recently introduced the graded algebra PΛ of polysymmetric functions as a generalization of the algebra Λ of symmetric functions. This article develops combinatorial formulas for some multiplication rules and transition matrix entries for PΛ that are analogous to well-known classical formulas for Λ. In more detail, we consider pure tensor bases {s τ ⊗ }, {p τ ⊗ }, and {m τ ⊗ } for PΛ that arise as tensor products of the classical Schur basis, power-sum basis, and monomial basis for Λ. We find expansions in these bases of the non-pure bases {P δ }, {H δ }, {E δ + }, and {E δ } studied by Asvin G and O’Desky. The answers involve tableau-like structures generalizing semistandard tableaux, rim-hook tableaux, and the brick tabloids of Eğecioğlu and Remmel. These objects arise by iteration of new Pieri-type rules that give expansions of products such as s σ ⊗ H δ , p σ ⊗ E δ , etc.