On the resolvent degree of PSU(3,q)

TL;DR

This paper bounds the resolvent degree (RD) for certain finite simple groups by leveraging the GGSW framework of RD^{≤d}-versality and invariant theory. Focusing on PSU(3,q) and PSU(2,q), it identifies small-dimensional irreducible representations and analyzes their invariant forms, showing Sym^2 V^* and Sym^3 V^* lack invariants while quartic invariants exist and suffice to build versal G-varieties. It derives explicit RD bounds for PSU(3,q), delivering small-q bounds and a general asymptotic bound RD(PSU(3,q)) ≤ q^2 - q - log_4(q^2 - q + 6) for large q, and performs extensive computational bounds for PSU(2,q) up to q ≤ 125 with results corroborated by Appendix data. Together, these results illustrate a scalable approach to bounding RD across families of Lie-type simple groups and contribute to the broader program of uniform RD bounds for finite simple groups.

Abstract

Resolvent degree ($\operatorname{RD}$) is an invariant of finite groups in terms of the complexity of their algebraic actions. We address the problem of bounding $\operatorname{RD}(G)$ for all finite simple groups using the methods established by Gómez-Gonzáles-Sutherland-Wolfson in terms of $\operatorname{RD}^{\leq d}_{\mathbb{C}}$-versality and special points. We give upper bounds on $\operatorname{RD}(\operatorname{PSU}(3,q))$ and $\operatorname{RD}(\operatorname{PSU}(2, q))$ in terms of classical invariant theory. In the $\operatorname{PSU}(3,q)$ case, stability of low-degree invariants permit an asymptotic bound on $\operatorname{RD}$ growing in $q$.

Theorems & Definitions (10)

• Theorem 1.1: Bounds on the resolvent degree of $\mathop{\mathrm{PSU}}\nolimits(3,q)$
• Definition 2.1
• Theorem 2.2: Bounds on resolvent degree via invariant theory
• proof : Sketch of Theorem \ref{['thm:the_game']}
• Theorem 3.1
• proof
• proof : Proof of Theorem \ref{['thm:bounds_of_rd_psu3']}
• Example A.1: Klein1878
• Example A.2
• Example A.3