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The Mathieu group ${\sf M}_{12}$ vs ${\mathbf{SL_3(3)}}$ in characteristic 3

This paper proves that the principal 3-blocks of the Mathieu group ${\sf M}_{12}$ and the special linear group $\SL_3(3)$ are splendidly Rickard equivalent, thereby establishing their derived equivalence and resolving a question posed by W. Murphy regarding their first Hochschild cohomology.

Original authors: Shigeo Koshitani, Tetsuro Okuyama

Published 2026-07-20

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CC0 1.0

Original authors: Shigeo Koshitani, Tetsuro Okuyama

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a universe made entirely of patterns and symmetries, where the laws of physics are replaced by the rigid rules of mathematics. This is the world of modular representation theory, a branch of math that studies how groups of symmetries behave when we count them using a special kind of clock. Instead of counting 1, 2, 3, 4, 5, and then starting over at 1, this clock has a prime number of hours—like 3. When you do math on a clock with only 3 hours, strange and beautiful things happen. The "blocks" of these groups are like different neighborhoods in a city; some neighborhoods are very similar, while others look completely different. Mathematicians have long wondered: if two different groups have neighborhoods that look identical in terms of their underlying symmetry rules, are the neighborhoods themselves actually the same? This question is at the heart of a famous idea called Broué's Abelian Defect Group Conjecture. It suggests that if the "blueprints" of two neighborhoods match perfectly, then the buildings themselves can be transformed into one another without tearing them apart.

For decades, mathematicians have been testing this idea. Most of the time, they looked at cases where the symmetry rules were simple and orderly (called "abelian"). But what happens when the rules get messy and complicated? That is the territory of this paper. The authors, Shigeo Koshitani and Tetsuro Okuyama, decided to tackle a very specific, tricky pair of mathematical neighborhoods: one belonging to the Mathieu group M12 (a famous, rare symmetry group) and the other to the special linear group SL3(3) (a group built from 3x3 grids of numbers). Both of these groups have a "defect group" (the core of their local symmetry) that is non-abelian and messy, specifically a group of order 27. The big question was: even though these groups are totally different animals, are their principal 3-blocks (their main mathematical neighborhoods) secretly the same?

The authors prove that the answer is a resounding yes. They show that the principal 3-blocks of M12 and SL3(3) are splendidly Rickard equivalent. To understand what this means, imagine you have two different Lego castles. One is built with red and blue bricks, the other with green and yellow. They look different, and if you just look at the bricks, they seem unrelated. However, the authors discovered a magical instruction manual (a mathematical tool called a "complex") that allows you to take apart the red-and-blue castle and reassemble it into the green-and-yellow castle without losing a single piece or changing the fundamental structure. This isn't just a superficial similarity; it is a deep, structural identity. Because of this equivalence, the two groups share the same "Hochschild cohomology," which is like having the exact same DNA sequence for their internal algebraic structure.

The paper doesn't just stop at these two groups. Because the connection is so strong, the authors show that this relationship extends to the "autobiographies" of these groups (their automorphism groups), meaning the equivalence holds even when you add a little extra symmetry to the mix. They also answer a specific question posed by another mathematician, W. Murphy, confirming that the first Hochschild cohomology groups of these two blocks are isomorphic as Lie algebras. In plain English, this means that if you were to measure the "twists and turns" of the algebraic structures in both groups, you would get the exact same result.

The proof is a masterclass in connecting different areas of math. The authors use a powerful theorem developed by Rickard, which acts like a bridge between the two groups. To build this bridge, they had to construct specific mathematical objects called "complexes" and verify that they behave exactly as needed. They didn't just guess; they calculated the "Loewy layers" (the internal layers of the groups, like the rings of an onion) and checked the "Cartan invariants" (how the pieces fit together) with extreme precision. They proved that these two seemingly unrelated groups are, in the specific context of their 3-blocks, mathematically indistinguishable. This result is significant because it provides a concrete example of Broué's conjecture holding true even in the messy, non-abelian world, suggesting that the deep connections between symmetry groups are far more robust and universal than previously thought.

Technical Summary: The Principal 3-Blocks of $M_{12}$ and $SL_3(3)$

Problem Statement
This paper addresses a specific instance of Broué's Abelian Defect Group Conjecture (ADGC) within the modular representation theory of finite groups. While the ADGC posits that principal $p$ -blocks with abelian defect groups and identical fusion systems are derived equivalent, the authors investigate the case where the defect group $P$ is non-abelian. Specifically, they examine the principal 3-blocks of the Mathieu group $M_{12}$ and the special linear group $SL_3(3)$ . Both groups share a common Sylow 3-subgroup $P \cong 3^{1+2}_+$ (the extra-special group of order 27 and exponent 3) and possess isomorphic fusion systems over $P$ .

Prior to this work, Y. Usami had established the existence of a perfect isometry between these two blocks (a weaker condition than derived equivalence). The central problem addressed here is to determine whether a stronger equivalence—specifically, a splendid Rickard equivalence (derived equivalence induced by a splendid complex of bimodules)—holds between the principal 3-blocks of $M_{12}$ and $SL_3(3)$ .

Methodology
The authors employ a constructive approach based on Rickard's theorem (specifically [Ric02, Theorem 6.1]), which provides sufficient conditions for two algebras to be splendidly Rickard equivalent. The methodology involves the following steps:

Stable Equivalence Construction: The authors first utilize the Scott module $M = Sc(G \times H, \Delta P)$ to establish a stable equivalence of Morita type between the principal blocks $A = B_0(kM_{12})$ and $B = B_0(kSL_3(3))$ . This relies on the fact that the fusion systems $F_P(G)$ and $F_P(H)$ are identical and saturated.
Functor Definition: They define a functor $F: \text{mod-}A \to \text{mod-}B$ via tensor product with the Scott module. This functor preserves vertices and commutes with outer automorphisms.
Explicit Complex Construction: The core of the proof involves explicitly constructing a complex $X^\bullet(S)$ $X^{∙} (S)$ in the bounded derived category $D^b(\text{mod-}B)$ $D^{b} (mod- B)$ for each simple $A$ $A$ -module $S \in \text{Irr}_k(A)$ $S \in Irr_{k} (A)$ .
- For simple modules with vertex $P$ (e.g., $kG, 15, 34$), the complexes are concentrated in degree 0.
- For simple modules with vertex $Q$ (where $Q \cong C_3 \times C_3$ ), specifically $10, 10', 45, 45'$ , the authors construct non-trivial complexes involving projective covers and specific homomorphisms derived from the Green correspondence and the structure of the Scott module.
- Detailed calculations of Loewy structures, socle series, and Green correspondents for modules over the normalizer $N = N_G(Q)$ are performed to define the differentials in these complexes.
Verification of Rickard's Conditions: The authors verify the three conditions of Rickard's theorem:
- (a) The complexes $X^\bullet(S)$ have homology only in degrees 0 and 1 (or 0 for some), and specific Ext-vanishing conditions hold between the homology groups.
- (b) The set $\{X^\bullet(S) \mid S \in \text{Irr}_k(A)\}$ generates $D^b(\text{mod-}B)$ as a triangulated category.
- (c) The complexes satisfy orthogonality conditions ( $\text{Hom}_{D}(X^\bullet(S), X^\bullet(S')[i]) = 0$ for $i \neq 0$ and $S \neq S'$ ).

Key Contributions and Results
The primary result of the paper is Theorem 1.1, which states:

The principal 3-blocks of the Mathieu group $M_{12}$ and the special linear group $SL_3(3)$ are splendidly Rickard (splendidly derived) equivalent.

This result is established by constructing the specific complexes described above and proving they satisfy the criteria for a splendid equivalence.

Corollaries:

Extension to Automorphism Groups: As a direct consequence (Corollary 1.2), the derived equivalence lifts to the principal 3-blocks of the automorphism groups $\text{Aut}(M_{12}) \cong M_{12} \rtimes C_2$ and $\text{Aut}(SL_3(3)) \cong SL_3(3) \rtimes C_2$ .
Hochschild Cohomology: The derived equivalence implies that the first Hochschild cohomology groups of the two blocks are isomorphic as Lie algebras (Corollary 1.3). This specifically answers a question posed by W. Murphy regarding the structure of $HH^1(B_0(kM_{12}))$ and $HH^1(B_0(kSL_3(3)))$ .

Significance and Claims
The authors emphasize that while the result concerns two specific finite simple groups, the significance lies in the application of Rickard's theorem to a non-abelian defect group case ( $3^{1+2}_+$ ). The paper highlights that the origin of the theorem used ([Ric02]) traces back to the second author, Okuyama, and demonstrates that "very concrete calculations" of module structures and Green correspondents are powerful tools for verifying such equivalences.

The paper modestly frames its contribution as a confirmation of a stronger equivalence than the previously known perfect isometry (Usami, 1997) and as a solution to a specific cohomological question. It does not propose new general conjectures but rather solidifies the understanding of the relationship between these two blocks in the context of Broué's conjecture for non-abelian defect groups. The work relies heavily on prior classifications of $p$ -blocks with defect group $p^{1+2}_+$ by the first author (Koshitani) and subsequent developments by Usami and others.

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