Petrie symmetric functions

Grinberg, Darij

doi:10.5802/alco.232

Grinberg, Darij ¹

¹ Drexel University Korman Center 15 S 33rd Street Office #263 Philadelphia, PA 19104 (USA)

Algebraic Combinatorics, Volume 5 (2022) no. 5, pp. 947-1013.

Abstract

For any positive integer $k$ and nonnegative integer $m$ , we consider the symmetric function $G (k, m)$ defined as the sum of all monomials of degree $m$ that involve only exponents smaller than $k$ . We call $G (k, m)$ a Petrie symmetric function in honor of Flinders Petrie, as the coefficients in its expansion in the Schur basis are determinants of Petrie matrices (and thus belong to $\{0, 1, - 1\}$ by a classical result of Gordon and Wilkinson). More generally, we prove a Pieri-like rule for expanding a product of the form $G (k, m) \cdot s_{μ}$ in the Schur basis whenever $μ$ is a partition; all coefficients in this expansion belong to $\{0, 1, - 1\}$ . We also show that $G (k, 1), G (k, 2), G (k, 3), ...$ form an algebraically independent generating set for the symmetric functions when $1 - k$ is invertible in the base ring, and we prove a conjecture of Liu and Polo about the expansion of $G (k, 2 k - 1)$ in the Schur basis.

Received: 2020-05-08
Revised: 2021-12-31
Accepted: 2022-02-14
Published online: 2022-11-09

DOI: 10.5802/alco.232

Classification: 05E05
Keywords: symmetric functions, Schur functions, Schur polynomials, combinatorial Hopf algebras, Petrie matrices, Pieri rules, Murnaghan–Nakayama rule

Author's affiliations:

Grinberg, Darij ¹

¹ Drexel University Korman Center 15 S 33rd Street Office #263 Philadelphia, PA 19104 (USA)

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {Petrie symmetric functions},
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Grinberg, Darij. Petrie symmetric functions. Algebraic Combinatorics, Volume 5 (2022) no. 5, pp. 947-1013. doi : 10.5802/alco.232. https://alco.centre-mersenne.org/articles/10.5802/alco.232/

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