The ring of $D_4$ triality invariants

TL;DR

This work establishes a precise bridge between D$_4$ triality invariants—holomorphic objects with modular and symmetry properties under a fiber-product symmetry—and classical invariant theory. By introducing a cusp-simplifying injection and expressing the invariants as an intersection of two Seiberg--Witten coefficient rings, it reduces the ring $R^{\mathcal{G}}_{*,*}$ to the joint covariant/semiinvariant ring of a binary quadratic and a binary cubic via the Roberts isomorphism. The authors construct an explicit isomorphism and provide a minimal generating set of 15 invariants, organized through a refined trigrading that aligns with covariant theory, and they present an algorithm to build invariants of given weight and degree. This work connects gauge-theory motivated automorphic structures to classical invariant theory, enabling efficient computation and suggesting directions for higher-rank generalizations and weak invariant structures.

Abstract

$D_4$ triality invariants are modular forms as well as polynomial invariants for a fiber product of the modular group and the Weyl group of type $F_4$. We show that the ring of $D_4$ triality invariants satisfying a certain cusp condition is isomorphic to the ring of joint covariants of a binary cubic and a binary quadratic form.

Theorems & Definitions (53)

• Theorem 1.1: Theorem \ref{['thm:RtriandCov']}
• Definition 2.1
• Remark 2.2
• Proposition 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Lemma 2.6
• proof