# Matthew Gelvin's research while affiliated with Bilkent University and other places

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## Publications (10)

We show that, given an almost-source algebra 𝐴 of a 𝑝-block of a finite group 𝐺, then the unit group of 𝐴 contains a basis stabilized by the left and right multiplicative action of the defect group if and only if, in a sense to be made precise, certain relative multiplicities of local pointed groups are invariant with respect to the fusion system....

We show that, given an almost-source algebra $A$ of a $p$-block of a finite group $G$, then the unit group of $A$ contains a basis stabilized by the left and right multiplicative action of the defect group if and only if, in a sense to be made precise, certain relative multiplicities of local pointed groups are invariant with respect to the fusion...

Let G be a finite group and k an algebraically closed field of characteristic p>0. We define the notion of a Dade kG-module as a generalization of endo-permutation modules for p-groups. We show that under a suitable equivalence relation, the set of equivalence classes of Dade kG-modules forms a group under tensor product, and the group obtained thi...

Let $G$ be a finite group and $k$ an algebraically closed field of characteristic $p>0$. We define the notion of a Dade $kG$-module as a generalization of endo-permutation modules for $p$-groups. We show that under a suitable equivalence relation, the set of equivalence classes of Dade $kG$-modules form a group under tensor product, and the group o...

Given a block of a finite group, any source algebra has a basis invariant under the multiplicative actions of the defect group. Is such a basis a characteristic biset of the block fusion system? If the basis can be chosen to consist entirely of units, the question is answered in the affirmative. We prove the equivalence of several reformulations of...

Let B be a p-block of the finite group G. We observe that the p-fusion of G constrains the module structure of B: Any basis of B that is invariant under the left and right multiplications of a chosen Sylow p-subgroup S of G must in fact form a semicharacteristic biset for the fusion system on S induced by G. The parameterization of such semicharact...

We show that every saturated fusion system $\mathcal{F}$ has a unique minimal
$\mathcal{F}$-characteristic biset $\Lambda_\mathcal{F}$. We examine the
relationship of $\Lambda_\mathcal{F}$ with other concepts in $p$-local finite
group theory: In the case of a constrained fusion system, the model for the
fusion system is the minimal $\mathcal{F}$-ch...

We consider the Burnside ring $A(\mathcal{F})$ of $\mathcal{F}$-stable
$S$-sets for a saturated fusion system $\mathcal{F}$ defined on a $p$-group
$S$. It is shown by S. P. Reeh that the monoid of $\mathcal{F}$-stable sets is
a free commutative monoid with canonical basis $\{\alpha_P\}$. We give an
explicit formula that describes $\alpha_P$ as an $...

Let p be a prime number and G a finite group of order divisible by p. Quillen
showed that the Brown poset of nonidentity p-subgroups of G is homotopy
equivalent to its subposet of nonidentity elementary abelian subgroups. We show
here that a similar statement holds for the fusion category of nonidentity
p-subgroups of G. Other categories of p-subgr...

This paper introduces the notion of fusion action system, an abstraction of the $p$-local data of a finite group acting on a finite set. Fusion action systems are closely connected with the theory of fusion systems; we detail the relationship of this new definition to the existing literature. We also develop a theory of classifying spaces of fusion...

## Citations

... So Ω ∆(φ) = Ω Q , and similarly for Ω P . Theorem 6.4, together with Proposition 6.3, recovers the main result of Gelvin [2], which asserts that, in the notation of the proposition, any S×S-stable basis of OGb is F S (G)semicharacteristic. ...

... ω F is (F, F)-stable: This is essentially [5, Proposition 9.10], using that F is the centralizer fusion system of a. The difference is that [5] deals with actual bisets, whileω F here is a virtual biset. We adapt the proof used in [5]. ...

... Let p be a prime number and fix an S ∈ Syl p (G). Let F S (G) denote the Frobenius fusion system on S induced by G. Following [6], we define the Burnside ring of F S (G) to be the subring B(F S (G)) of B(S) consisting of the elements a ∈ B(S) such that |a P | = |a Q | whenever P, Q are subgroups of S that are conjugate in G. More generally, for any saturated fusion system F on S, we define the Burnside ring of F to be the subring B(F) of B(S) consisting of the elements a ∈ B(S) such that |a P | = |a Q | whenever P and Q are isomorphic in F. The first main result about B(F) is due to Sune Reeh. ...

... Proof: The homotopy Bousfield-Kan spectral sequence [4, XI. §7] implies that N Γ K is naturally isomorphic to lim 1 π 1 (map(BΓ, P ) 0 ) where P is the homotopy pullback diagram (20) for BK. In particular, N Γ K is naturally isomorphic to V Γ K since π 1 (map(BΓ, BK ∧ p ), 0) vanishes by Theorem A. In the more restricted contexts of [2,25], all V Γ K are trivial. ...

... Proof. The equivalence of (ii) and (iii) is just the characterization of stability in Burnside rings (see page 9). ...