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

Sebastian Debus, Tobias Metzlaff

TL;DR

This work develops a parallel theory of coinvariant spaces for Weyl groups under additive and multiplicative actions. It proves that multiplicative coinvariants $\\mathbb{C}[\\Omega]_{\\mathcal{W}}$ also carry the regular representation and identifies them with multiplicative harmonics $\\mathcal{L}_{\mathrm{m}}$, mirroring the additive case. An explicit graded transfer algorithm translates symmetry-adapted bases from additive coinvariants $\\mathrm{Sym}(\\Omega)_{\\mathcal{W}}$ to the multiplicative setting, preserving isotypic and graded structures. The paper then constructs type-$(A)$ and type-$(C)$ explicit $\\mathcal{W}$-equivariant homomorphisms and provides concrete bases in small ranks, along with open questions about broader types, Gröbner bases, and diagonal/generalizations. These results enable efficient symmetry-aware computation of multiplicative coinvariants and harmonics with potential applications in representation theory and computational algebra.

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.

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

TL;DR

This work develops a parallel theory of coinvariant spaces for Weyl groups under additive and multiplicative actions. It proves that multiplicative coinvariants also carry the regular representation and identifies them with multiplicative harmonics , mirroring the additive case. An explicit graded transfer algorithm translates symmetry-adapted bases from additive coinvariants to the multiplicative setting, preserving isotypic and graded structures. The paper then constructs type- and type- explicit -equivariant homomorphisms and provides concrete bases in small ranks, along with open questions about broader types, Gröbner bases, and diagonal/generalizations. These results enable efficient symmetry-aware computation of multiplicative coinvariants and harmonics with potential applications in representation theory and computational algebra.

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.

Paper Structure

This paper contains 9 sections, 20 theorems, 67 equations, 1 figure, 3 tables, 2 algorithms.

Table of Contents

  1. Additive and Multiplicative Actions
  2. Multiplicative Coinvariant Spaces
  3. Multiplicative $\mathcal{W}$-Harmonics
  4. Associated Graded Equivariants
  5. Graded Transfer of Coinvariants
  6. Homomorphisms for types $\mathrm{A}_{n-1}$ and $\mathrm{C}_{n}$
  7. Type $\mathrm{A}_{n-1}$
  8. Type $\mathrm{C}_{n}$
  9. Conclusion and Open Questions

Key Result

Theorem 1.1

Let $\mathcal{W}$ be a complex reflection group. Then the following statements hold.

Figures (1)

  • Figure 1: The subsets $\Omega_0 \subset \Omega_1 \subset \Omega_2 \subset \Omega_3 \subset \Omega_4$ of the hexagonal lattice in the plane, the weight lattice $\Omega$ of $\mathrm{A}_{2}$.

Theorems & Definitions (44)

  • Theorem 1.1: Shephard-Todd shephardtodd54, Chevalley chevalley1955
  • Theorem 1.2: Steinberg SteinbergHarmonicPolynomials
  • Theorem 1.3: Bourbaki bourbaki456, Steinberg Steinberg1975, Farkas farkas84
  • Example 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 34 more