Department of Mathematics - Master Calendar

Speaker: William Erickson (Monmouth College)

Title: Stanley decompositions of modules of covariants

Abstract: We will present an unexpectedly clean combinatorial description of classical modules of covariants, objects once prominent in 19th-century invariant theory and now resurfacing in equivariant machine learning.  Given a complex reductive group G, and G-modules W and U, the module of covariants consists of all G-equivariant polynomial maps from W to U; in this sense it generalizes the ring of invariants (i.e., the case where U is trivial).  In this talk, we will focus on Weyl's classical setting in which G is a classical group acting on any number of vectors and covectors.  Our main result gives an explicit and uniform Stanley decomposition for the modules of covariants for every irreducible G-module U.  The components in these decompositions are indexed by certain combinatorial objects we call jellyfish: families of disjoint lattice paths whose shape is dictated by the highest weight of U.  These jellyfish yield transparent combinatorial interpretations for the Hilbert series, the multiplicity, and related invariants typically approachable only with heavier algebraic machinery.  This is joint work with Markus Hunziker.