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 $G\left(k,1\right),G\left(k,2\right),G\left(k,3\right),\dots$ 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.

106 pages. The version at https://www.cip.ifi.lmu.de/~grinberg/algebra/petriesym.pdf will be updated more frequently. See the ancillary file (or http://www.cip.ifi.lmu.de/~grinberg/algebra/petriesym-long.pdf ) for a more detailed version. See https://www.cip.ifi.lmu.de/~grinberg/algebra/fps20pet.pdf for a quick introduction without proofs. v3 adds Section 5.7, which generalizes many results

References in corpus (3)

• Petrie symmetric functions
• On the cohomology of line bundles over certain flag schemes II
• Symmetric Functions and Rectangular Catalan Combinatorics

Cited by in corpus (4)

• Petrie symmetric functions
• On the cohomology of line bundles over certain flag schemes II
• A generalization of complete and elementary symmetric functions
• A parking function interpretation for $(-1{)}^k\nabla m_{2^k{1}^l}$