Asvin G and Andrew O’Desky recently introduced the graded algebra $\textsf {P}\mathsf {\Lambda }$ 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 $\textsf {P}\mathsf {\Lambda }$ that are analogous to well-known classical formulas for $\Lambda $. In more detail, we consider pure tensor bases $\lbrace s^{\otimes }_{\tau }\rbrace $, $\lbrace p^{\otimes }_{\tau }\rbrace $, and $\lbrace m^{\otimes }_{\tau }\rbrace $ for $\textsf {P}\mathsf {\Lambda }$ 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 $\lbrace P_{\delta }\rbrace $, $\lbrace H_{\delta }\rbrace $, $\lbrace E^+_{\delta }\rbrace $, and $\lbrace E_{\delta }\rbrace $ 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 $s^{\otimes }_{\sigma }H_{\delta }$, $p^{\otimes }_{\sigma }E_{\delta }$, etc.
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Keywords: symmetric functions, polysymmetric functions, transition matrices, plethysm, Pieri rules, Murnaghan–Nakayama rule, rim-hook tableaux, brick tabloids, types
Khanna, Aditya 1; Loehr, Nicholas A. 1
CC-BY 4.0
@article{ALCO_2025__8_4_1085_0,
author = {Khanna, Aditya and Loehr, Nicholas A.},
title = {Transition matrices and {Pieri-type} rules for polysymmetric functions},
journal = {Algebraic Combinatorics},
pages = {1085--1117},
publisher = {The Combinatorics Consortium},
volume = {8},
number = {4},
year = {2025},
doi = {10.5802/alco.436},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.436/}
}
TY - JOUR AU - Khanna, Aditya AU - Loehr, Nicholas A. TI - Transition matrices and Pieri-type rules for polysymmetric functions JO - Algebraic Combinatorics PY - 2025 SP - 1085 EP - 1117 VL - 8 IS - 4 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.436/ DO - 10.5802/alco.436 LA - en ID - ALCO_2025__8_4_1085_0 ER -
%0 Journal Article %A Khanna, Aditya %A Loehr, Nicholas A. %T Transition matrices and Pieri-type rules for polysymmetric functions %J Algebraic Combinatorics %D 2025 %P 1085-1117 %V 8 %N 4 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.436/ %R 10.5802/alco.436 %G en %F ALCO_2025__8_4_1085_0
Khanna, Aditya; Loehr, Nicholas A. Transition matrices and Pieri-type rules for polysymmetric functions. Algebraic Combinatorics, Volume 8 (2025) no. 4, pp. 1085-1117. doi : 10.5802/alco.436. https://alco.centre-mersenne.org/articles/10.5802/alco.436/
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