Portal de Periódicos da CAPES

Petrie symmetric functions

2022; Volume: 5; Issue: 5 Linguagem: Inglês

10.5802/alco.232

2589-5486

Darij Grinberg,

Finite Group Theory Research

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.