Transition matrices and Pieri-type rules for polysymmetric functions

Abstract: Asvin G and Andrew O'Desky recently introduced the graded algebra P$\Lambda$ of polysymmetric functions as a generalization of the algebra $\Lambda$ of symmetric functions. This article develops combinatorial formulas for some multiplication rules and transition matrix entries for P$\Lambda$ that are analogous to well-known classical formulas for $\Lambda$. In more detail, we consider pure tensor bases ${s^{\otimes}_{\tau}}$, ${p^{\otimes}_{\tau}}$, and ${m^{\otimes}_{\tau}}$ for P$\Lambda$ that arise as tensor products of the classical Schur basis, power-sum basis, and monomial basis for $\Lambda$. We find expansions in these bases of the non-pure bases ${P_{\delta}}$, ${H_{\delta}}$, ${E^+_{\delta}}$, and ${E_{\delta}}$ studied by Asvin G and O'Desky. The answers involve tableau-like structures generalizing semistandard tableaux, rim-hook tableaux, and the brick tabloids of E\u{g}ecio\u{g}lu and Remmel. These objects arise by iteration of new Pieri-type rules that give expansions of products such as $s^{\otimes}{\sigma}H{\delta}$, $p^{\otimes}{\sigma}E{\delta}$, etc.