Séminaire Lotharingien de Combinatoire, 84B.61 (2020), 12 pp.

Darij Grinberg

The Petrie Symmetric Functions

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 contain no exponents larger than k-1. 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 {-1,0,1} 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) . sμ in the Schur basis whenever μ is a partition; all coefficients in this expansion belong to {-1,0,1}. We show a further formula for G(k,m) and prove 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. We prove a conjecture of Liu and Polo about the expansion of G(k,2<k-1) in the Schur basis. We then take our Pieri-like rule as an impetus to pose a different question: What other symmetric functions f have the property that each product fsμ expands in the Schur basis with all coefficients belonging to {-1,0,1}? We call this property MNability due to its most classical instance (besides the Pieri rules, which do not use -1 coefficients) being the Murnaghan-Nakayama rule. Surprisingly, we find a number of infinite families of MNable symmetric functions besides the classical ones.


Received: November 20, 2019. Accepted: February 20, 2020. Final version: April 30, 2020.

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