Petrie symmetric functions

Grinberg, Darij

Algebraic Combinatorics, Tome 5 (2022) no. 5, pp. 947-1013.

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 $G\left(k,m\right)·s_{\mu }$ 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 $G\left(k,1\right),G\left(k,2\right),G\left(k,3\right),...$ 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.

Classification : 05E05
Mots clés : symmetric functions, Schur functions, Schur polynomials, combinatorial Hopf algebras, Petrie matrices, Pieri rules, Murnaghan–Nakayama rule

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