Petrie symmetric functions

Abstract

For any positive integer $k$ and nonnegative integer $m$, we consider the symmetric function $G\left( k,m\right)$ defined as the sum of all monomials of degree $m$ that involve only exponents smaller than $k$. We call $G\left( k,m\right)$ 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 $\left\{ 0,1,-1\right\} $ by a classical result of Gordon and Wilkinson). More generally, we prove a Pieri-like rule for expanding a product of the form $G\left( k,m\right) \cdot s_{\mu}$ in the Schur basis whenever $\mu$ is a partition; all coefficients in this expansion belong to $\left\{ 0,1,-1\right\} $. We also show that $G\left( k,1\right) ,G\left( k,2\right) ,G\left( k,3\right) ,\ldots$ 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 $G\left( k,2k-1\right)$ in the Schur basis.