Transition matrices and Pieri-type rules for polysymmetric functions

Abstract

Asvin G and Andrew O'Desky recently introduced the graded algebra P$\hat{I}›$ of polysymmetric functions as a generalization of the algebra $\hat{I}›$ of symmetric functions. This article develops combinatorial formulas for some multiplication rules and transition matrix entries for P$\hat{I}›$ that are analogous to well-known classical formulas for $\hat{I}›$. In more detail, we consider pure tensor bases $\{s_{\ddot{I}}^{\otimes }„\}$, $\{p_{\ddot{I}}^{\otimes }„\}$, and $\{m_{\ddot{I}}^{\otimes }„\}$ for P$\hat{I}›$ that arise as tensor products of the classical Schur basis, power-sum basis, and monomial basis for $\hat{I}›$. We find expansions in these bases of the non-pure bases $\{P_{\hat{I}}´\}$, $\{H_{\hat{I}}´\}$, $\{E_{\hat{I}}^{+}´\}$, and $\{E_{\hat{I}}´\}$ 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 $s_{\ddot{I}}^{\otimes }ƒH_{\hat{I}}´$, $p_{\ddot{I}}^{\otimes }ƒE_{\hat{I}}´$, etc.

30 pages, multiple in-line figures