Obstructions to Combinatorial Formulas for Plethysm

  • Thomas Kahle
  • Mateusz Michałek
Keywords: Representation theory, Plethysm, Ehrhart quasi-polynomial, Kronecker coefficients, Lattice point counting


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.

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