Transition matrices and Pieri-type rules for polysymmetric functions

Asvin G and Andrew O’Desky recently introduced the graded algebra P$\mathrm{\Lambda }$ of polysymmetric functions as a generalization of the algebra $\mathrm{\Lambda }$ of symmetric functions. This article develops combinatorial formulas for some multiplication rules and transition matrix entries for P$\mathrm{\Lambda }$ that are analogous to well-known classical formulas for $\mathrm{\Lambda }$. In more detail, we consider pure tensor bases $\{s_{\tau }^{\otimes }\}$, $\{p_{\tau }^{\otimes }\}$, and $\{m_{\tau }^{\otimes }\}$ for P$\mathrm{\Lambda }$ that arise as tensor products of the classical Schur basis, power-sum basis, and monomial basis for $\mathrm{\Lambda }$. 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 Eğecioğlu and Remmel. These objects arise by iteration of new Pieri-type rules that give expansions of products such as $s_{\sigma }^{\otimes }{H}_{\delta }$, $p_{\sigma }^{\otimes }{E}_{\delta }$, etc.