arXiv subjects
Publications and source records attributed to John Revere McHugh.
We determine which quasi-simple groups have a non-principal $2$-block that is stable under complex conjugation. As a corollary, we determine that the Mathieu group $M_{22}$ is the only simple group not possessing a nontrivial irreducible Brauer character of quadratic type, answering a recent question of Gow and Murray.
For a $p$-permutation equivalence between two block algebras of finite groups, we introduce new square diagrams that link the $p$-permutation equivalence via the Brauer construction to local equivalences between stabilizers of corresponding Brauer pairs. These diagrams can be viewed as lifts of the square diagrams in the definition of isotypies. The proof of the commutativity requires new technical tools, namely a formula for how taking fixed points commutes with extended tensor products of finite sets with group actions and how the Brauer construction commutes with taking extended tensor products of $p$-permutation modules. These fundamental formulas, generalizing earlier results by Boltje-Danz and by Boltje-Perepelitsky, should be of independent interest.
We show that Kessar's isotypy between Galois conjugate blocks of finite group algebras does not always lift to a $p$-permutation equivalence. We also provide examples of Galois conjugate blocks which are isotypic but not $p$-permutation equivalent. These results help to clarify the distinction between a $p$-permutation equivalence and an isotypy, and may be useful in determining necessary and sufficient conditions for when an isotypy lifts to a $p$-permutation equivalence.
We introduce a new type of equivalence between blocks of finite group algebras called a strong isotypy. A strong isotypy is equivalent to a $p$-permutation equivalence and restricts to an isotypy in the sense of Broué. To prove these results we first establish that the group $T_{\mathcal{O}}(B)$ of trivial source $B$-modules, where $B$ is a block of a finite group algebra, is isomorphic to groups of ``coherent character tuples.'' This provides a refinement of work by Boltje and Carman which characterizes the ring $T_{\mathcal{O}}(G)$ of trivial source $\mathcal{O}G$-modules, where $G$ is a finite group, in terms of coherent character tuples.
We examine the cokernel of the canonical homomorphism from the trivial source ring of a finite group to the ring of $p$-rational complex characters. We use Boltje and Coşkun's theory of fibered biset functors to determine the structure of the cokernel. An essential tool in the determination of this structure is Bouc's theory of rational $p$-biset functors.