Browsing by Author "Gelvin, Matthew"
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Item Open Access Conjectural invariance with respect to the fusion system of an almost-source algebra(Walter de Gruyter GmbH, 2022-03-23) Barker, Laurence; Gelvin, MatthewWe 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 also show that, when 𝐺 is 𝑝-solvable, those two equivalent conditions hold for some almost-source algebra of the given 𝑝-block. One motive lies in the fact that, by a theorem of Linckelmann, if the two equivalent conditions hold for 𝐴, then any stable basis for 𝐴 is semicharacteristic for the fusion system.Item Open Access Dade groups for finite groups and dimension functions(Academic Press, 2021-06-15) Gelvin, Matthew; Yalçın, ErgünLet 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 this way is isomorphic to the Dade group D(G) defined by Lassueur. We also consider the subgroup DΩ(G) of D(G) generated by relative syzygies ΩX , where X is a finite G-set. If C(G, p) denotes the group of superclass functions defined on the p-subgroups of G, there are natural generators ωX of C(G, p), and we prove the existence of a well-defined group homomorphism ΨG : C(G, p) → DΩ(G) that sends ωX to ΩX . The main theorem of the paper is the verification that the subgroup of C(G, p) consisting of the dimension functions of k-orientable real representations of G lies in the kernel of ΨG.