1. Transition matrices and Pieri-type rules for polysymmetric functions
- Author
-
Khanna, Aditya and Loehr, Nicholas A.
- Subjects
Mathematics - Combinatorics ,Mathematics - Algebraic Geometry ,05E05, 05A17 - 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., Comment: 30 pages, multiple in-line figures
- Published
- 2024