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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