The Mathieu group ${\mathsf{M}}_{12}$ vs $\mathbf{S}{\mathbf{L}}_{\mathbf{3}}(\mathbf{3})$ in characteristic 3

We prove that the principal $3$-blocks of the Mathieu group ${\mathsf{M}}_{\mathsf{12}}$ and the special linear group ${\text{SL}}_3(3)$ are splendidly Rickard equivalent, and hence derived equivalent, by applying Rickard’s theorem whose origin goes back to the second author. As a byproduct we answer to a question on the first Hochschild cohomology of the principal $3$-blocks above posed by W. Murphy.