arXiv subjects

Patrick Serwene

Publications and source records attributed to Patrick Serwene.

6 recordsLinked to original sources

arXiv2025

A reduction theorem for non-vanishing of Hochschild cohomology of block algebras and Happel's property

In this short research note we obtain a reduction theorem for the non-vanishing of the first Hochschild cohomology of block algebras of finite groups with non-trivial defect groups. Along the way we investigate this problem for the blocks of some simple finite group algebras. Mimicking the case of blocks of finite group algebras we find some examples of category algebras that satisfy Happel's property.

math.RT↗

arXiv2024

Weight conjectures for Parker--Semeraro fusion systems

We prove that the Parker--Semeraro systems satisfy six of the nine Kessar--Linckelmann--Lynd--Semeraro weight conjectures for saturated fusion systems. As a by-product we obtain that Robinson's ordinary weight conjecture holds for the principal $3$-block of Aut$(G_2(3))$, the principal $5$-blocks of $HN$, $BM$, Aut$(HN)$, $Ly$, the principal $7$-block of $M$, and the principal $p$-blocks of $G_2(p)$ for $p\geq 3 $.

math.RT↗

arXiv2024

Exotic Fusion System on a Subgroup of the Monster

We prove that an exotic fusion system described by Grazian on a subgroup of the Monster group is block-exotic, thus proving that exotic and block-exotic fusion systems are the same for all $p$-groups with sectional rank 3, where $p \geq 5$.

math.GR↗

arXiv2023

Proving a conjecture for fusion systems on a class of groups

We prove the conjecture that exotic and block-exotic fusion systems coincide holds for all fusion systems on exceptional $p$-groups of maximal nilpotency class, where $p \geq 5$. This is done by considering a family of exotic fusion systems discovered by Parker and Stroth. Together with a previous result by the author, which we also generalise in this paper, and a result by Grazian and Parker this implies the conjecture for fusion systems on such groups. Considering small primes, there are no exotic fusion systems on $2$-groups of maximal class and for $p = 3$, we prove block-exoticity of two exotic fusion systems described by Diaz--Ruiz--Viruel.

math.GR↗

arXiv2022

Reduction Theorems for Generalised Block Fusion Systems

We generalise the Third Main Theorem by Brauer, the First and Second Fong Reduction to generalised block fusion systems and apply the Second Fong Reduction to extend a result by Cabanes about the non-exoticity of fusion systems of unipotent blocks for finite groups of Lie type.

math.RT↗

arXiv2019

Block-Exoticity of a Family of Exotic Fusion Systems

We prove that each exotic fusion system $\mathcal F$ on a Sylow $p$-subgroup of $G_2(p)$ for an odd prime $p$ with $\mathcal O_p(\mathcal F)=1$ is block-exotic. This gives evidence for the conjecture that each exotic fusion system is block-exotic. We prove two reduction theorems for block-realisable fusion systems.

math.RT↗