Plethysm and Lattice Point Counting

Abstract

We apply lattice point counting methods to compute the multiplicities in the plethysm of GL(n). Our approach gives insight into the asymptotic growth of the plethysm and makes the problem amenable to computer algebra. We prove an old conjecture of Howe on the leading term of plethysm. For any partition μ of 3, 4, or 5, we obtain an explicit formula in λ and k for the multiplicity of S λ in S μ (S k ).

Plethysm and fast matrix multiplication

“…As symmetric powers (together with exterior powers) are the simplest Schur functors, one could expect that respective formulas for S d (sl n ) are harder. In principle, one could use the methods of [6,7,16] to decompose this plethysm, but this requires a lot of nontrivial character manipulations. Instead, we present a very easy proof of explicit decomposition based on Cauchy formula and Littlewood-Richardson rule in Theorem 1.…”

Non-Normal Very Ample Polytopes – Constructions and Examples

“…The main object of our study are convex lattice polytopes. These combinatorial objects appear in many contexts including: toric geometry, algebraic combinatorics, integer programming, enumerative geometry and many others [3,6,10,15,16,22]. Thus, it is not surprising that there is a whole hierarchy of their properties and invariants.…”

Obstructions to Combinatorial Formulas for Plethysm

“…In [Sta00, Problem 9] Richard Stanley asked for a positive combinatorial method to compute plethysm coefficients. A connection between plethysm and lattice point counting was shown at least in [KM16,Col17,CDW12]. These connections are not direct in the sense that plethysm coefficients are not seen to equal counts, but always involve some opaque arithmetics.…”

“…In previous work the authors gave a formula for plethysm coefficients, which is a sum of Ehrhart functions of various polytopes with (positive and negative) coefficients [KM16]. This allows to gather experimental data on the questions for many rays.…”

“…Remark 9. While this is not visible from the representation in [KM16], a general theorem of Meinrenken and Sjamaar [MS99] implies that the chambers of plethysm quasi-polynomials are polyhedral cones (see [KM16,Remark 3.12], [PV15]). One may thus ask the stronger question if they are Ehrhart quasi-polynomials in general.…”