Transition matrices and Pieri-type rules for polysymmetric functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915618938159104 |
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| author | Khanna, Aditya Loehr, Nicholas A. |
| author_facet | Khanna, Aditya Loehr, Nicholas A. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_13404 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Transition matrices and Pieri-type rules for polysymmetric functions Khanna, Aditya Loehr, Nicholas A. Combinatorics Algebraic Geometry 05E05, 05A17 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. |
| title | Transition matrices and Pieri-type rules for polysymmetric functions |
| topic | Combinatorics Algebraic Geometry 05E05, 05A17 |
| url | https://arxiv.org/abs/2408.13404 |