Obstructions to combinatorial formulas for plethysm

TL;DR

This paper investigates the nature of quasi-polynomials in plethysm formulas, showing they are not always counting functions of lattice points in polytopes, which challenges previous assumptions in the field.

Contribution

It demonstrates that certain plethysm-related quasi-polynomials cannot be represented as lattice point counts in polytopes, revealing obstructions to combinatorial formulas.

Findings

Quasi-polynomials for S^3(S^k) and S^k(S^3) are not counting functions of inhomogeneous polytopes.

These functions are not counting functions of lattice points in scaled convex bodies.

Results also impact understanding of rectangular Kronecker coefficients.

Abstract

Motivated by questions of Mulmuley and Stanley we investigate quasi-polynomials arising in formulas for plethysm. We demonstrate, on the examples of $S^3(S^k)$ and $S^k(S^3)$, that these need not be counting functions of inhomogeneous polytopes of dimension equal to the degree of the quasi-polynomial. It follows that these functions are not, in general, counting functions of lattice points in any scaled convex bodies, even when restricted to single rays. Our results also apply to special rectangular Kronecker coefficients.