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Transition matrices and Pieri-type rules for polysymmetric functions

Aditya Khanna, Nicholas A. Loehr

TL;DR

This work generalizes symmetric-function combinatorics to the polysymmetric setting by defining the algebra \\textsf{P}\boldsymbol{\Lambda} as a tensor product of symmetric-function algebras and introducing pure-tensor bases along with four non-pure bases $H$, $E^+$, $E$, and $P$. It develops Pieri-type and Murnaghan–Nakayama–style rules for expanding products with these bases in the $s^{\otimes}$, $p^{\otimes}$, and $m^{\otimes}$ frameworks, introducing tableau-like combinatorial devices such as TRHT, ICRPT, ICRHT, TPRT, HTBT, ETBT, and their duals to encode coefficients. The paper provides explicit transition-matrix formulas and demonstrates how pure-tensor transitions factor across blocks, while giving detailed examples and an appendix of sample matrices to illustrate the constructions. The results extend classical symmetric-function theory to polysymmetric functions, enabling systematic computation of multiplication rules and transition matrices in this richer algebraic setting, with potential applications to representation theory of polysymmetric structures and related combinatorics.

Abstract

Asvin G and Andrew O'Desky recently introduced the graded algebra P$Λ$ 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 P$Λ$ that are analogous to well-known classical formulas for $Λ$. In more detail, we consider pure tensor bases $\{s^{\otimes}_τ\}$, $\{p^{\otimes}_τ\}$, and $\{m^{\otimes}_τ\}$ for P$Λ$ 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 $s^{\otimes}_σH_δ$, $p^{\otimes}_σE_δ$, etc.

Transition matrices and Pieri-type rules for polysymmetric functions

TL;DR

This work generalizes symmetric-function combinatorics to the polysymmetric setting by defining the algebra \\textsf{P}\boldsymbol{\Lambda} as a tensor product of symmetric-function algebras and introducing pure-tensor bases along with four non-pure bases , , , and . It develops Pieri-type and Murnaghan–Nakayama–style rules for expanding products with these bases in the , , and frameworks, introducing tableau-like combinatorial devices such as TRHT, ICRPT, ICRHT, TPRT, HTBT, ETBT, and their duals to encode coefficients. The paper provides explicit transition-matrix formulas and demonstrates how pure-tensor transitions factor across blocks, while giving detailed examples and an appendix of sample matrices to illustrate the constructions. The results extend classical symmetric-function theory to polysymmetric functions, enabling systematic computation of multiplication rules and transition matrices in this richer algebraic setting, with potential applications to representation theory of polysymmetric structures and related combinatorics.

Abstract

Asvin G and Andrew O'Desky recently introduced the graded algebra P of polysymmetric functions as a generalization of the algebra of symmetric functions. This article develops combinatorial formulas for some multiplication rules and transition matrix entries for P that are analogous to well-known classical formulas for . In more detail, we consider pure tensor bases , , and for P that arise as tensor products of the classical Schur basis, power-sum basis, and monomial basis for . We find expansions in these bases of the non-pure bases , , , and 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 , , etc.
Paper Structure (23 sections, 35 theorems, 96 equations, 1 figure)

This paper contains 23 sections, 35 theorems, 96 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of Symmetric Functions
  3. Polysymmetric Functions
  4. The Bases $H$, $E^+$, $E$, and $P$
  5. Transition Matrices for $\textsf{P}\mathsf{\Lambda}$
  6. Main Results
  7. Expansions in the $s^{\otimes}$ Basis
  8. Rule for $s^{\otimes}_{\sigma}P_{d^m}$.
  9. Rule for $s_{\sigma}^{\otimes}P_{\delta}$ and $\mathcal{M}(P,s^{\otimes})$.
  10. Rule for $s^{\otimes}_{\sigma}H_{d^r}$.
  11. Rule for $s_{\sigma}^{\otimes}H_{\delta}$ and $\mathcal{M}(H,s^{\otimes})$.
  12. Rules for $s^{\otimes}_{\sigma}E^+_{d^m}$ and $s^{\otimes}_{\sigma}E_{d^m}$.
  13. Rules for $s_{\sigma}^{\otimes}E^+_{\delta}$, $s_{\sigma}^{\otimes}E_{\delta}$, $\mathcal{M}(E^+,s^{\otimes})$, and $\mathcal{M}(E,s^{\otimes})$
  14. Expansions in the $p^{\otimes}$ Basis
  15. Algebraic Development of $p^{\otimes}$-Expansions
  16. ...and 8 more sections

Key Result

Proposition 3

Let $\{f_{\lambda}\}$ and $\{g_{\lambda}\}$ be bases of $\Lambda$ such that $f_{\lambda},g_{\lambda}\in\Lambda^n$ whenever $\lambda\vdash n$. Let $F_{\tau}=f_{\tau}^{\otimes}$ and $G_{\tau}=g_{\tau}^{\otimes}$ be the corresponding pure tensor bases. For all types $\sigma,\tau$,

Figures (1)

  • Figure 1: ICRPTs in \ref{['ex:ICRPT']}.

Theorems & Definitions (89)

  • Remark 1
  • Example 2
  • Proposition 3
  • proof
  • Proposition 4
  • Example 5
  • Theorem 6
  • proof
  • Example 7
  • Theorem 8
  • ...and 79 more