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

For any positive integer k and nonnegative integer m, we consider the symmetric function Gk,m defined as the sum of all monomials of degree m that involve only exponents smaller than k. We call Gk,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 Gk,m·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 Gk,1,Gk,2,Gk,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 Gk,2k-1 in the Schur basis.

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DOI: 10.5802/alco.232
Classification: 05E05
Keywords: symmetric functions, Schur functions, Schur polynomials, combinatorial Hopf algebras, Petrie matrices, Pieri rules, Murnaghan–Nakayama rule
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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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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