Generalized Versality, Special Points, and Resolvent Degree for the Sporadic Groups

Abstract

Resolvent degree is an invariant measuring the complexity of algebraic and geometric phenomena, including the complexity of finite groups. To date, the resolvent degree of a finite simple group $G$ has only been investigated when $G$ is a cylic group; an alternating group; a simple factor of a Weyl group of type $E_6$, $E_7$, or $E_8$; or $\operatorname{PSL}\left(2, \mathbb{F}_7\right)$. In this paper, we establish upper bounds on the resolvent degrees of the sporadic groups by using the invariant theory of their projective representations. To do so, we introduce the notion of (weak) $\operatorname{RD}_k^{\leq d}$-versality, which we connect to the existence of "special points" on varieties.

Figures (7)

• Figure 1: Historical Organization of Sporadic Groups
• Figure 2: Dimensions of Projective Representations and Degrees of Invariants
• Figure 3: Sporadic Groups Subquotient Table from the $\mathbb{A}\mathbb{T}\mathbb{L}\mathbb{A}\mathbb{S}$, I
• Figure 4: Sporadic Groups Subquotient Table from the $\mathbb{A}\mathbb{T}\mathbb{L}\mathbb{A}\mathbb{S}$, II
• Figure 5: Minimal Linear Representations, Projective Representations, and $X_G$

Theorems & Definitions (63)

• Corollary 1.2: Appears as Corollary \ref{['cor:Explicit Form of Bounds on the Resolvent Degree of the Sporadic Groups']}: Explicit Form of Theorem \ref{['thm:Bounds on the Resolvent Degree of the Sporadic Groups']}
• Theorem 1.3: Appears as Theorem \ref{['thm:RDKD-versality and Special Points']}: Generalized Versality and Special Points
• Definition 2.1: $G$-Torsors
• Definition 2.2: Twists
• Definition 2.3: Integral Models
• Definition 2.4: Definition 4.1 of FarbKisinWolfson2023
• Definition 2.5: Definition 4.1 of FarbKisinWolfson2023 via Field Extensions
• Lemma 2.6: Equivalence of Definitions
• proof
• Definition 2.7: Saturation and Closure Under Extensions