Additive and Multiplicative Coinvariant Spaces of Weyl Groups in the Light of Harmonics and Graded Transfer

TL;DR

This paper explores the structure of coinvariant spaces associated with Weyl groups, revealing their harmonic and combinatorial properties, and introduces an algorithm to compute bases linking additive and multiplicative invariants.

Contribution

It establishes the isomorphism between multiplicative coinvariant spaces and harmonic spaces, and presents an algorithm to derive multiplicative bases from additive ones, enhancing computational methods.

Findings

Multiplicative coinvariant space is isomorphic to a space of multiplicative harmonics.

An algorithm is developed to compute multiplicative coinvariant bases from additive bases.

New explicit equivariant maps and combinatorial structures are identified for Weyl groups of type A and C.

Abstract

The action of a Weyl group on the associated weight lattice induces an additive action on the symmetric algebra and a multiplicative action on the group algebra of the lattice. We show that the coinvariant space of the multiplicative action affords the regular representation and is isomorphic to a space of multiplicative harmonics, which corresponds to existing results for additive coinvariants of reflection groups. We then design an algorithm to compute a multiplicative coinvariant basis from an additive one. The algorithm preserves isotypic decomposition and graded structure and enables the study of multiplicative coinvariants by integrating combinatorial knowledge from the additive setting. We investigate the Weyl groups of type A and C to find new explicit equivariant maps and combinatorial structure.