Browsing by Author "Barker, Laurence"

Now showing 1 - 7 of 7

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, Matthew
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 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.
Open Access
An inversion formula for the primitive idempotents of the trivial source algebra
(Elsevier, 2019) Barker, Laurence
Formulas for the primitive idempotents of the trivial source algebra, in characteristic zero, have been given by Boltje and Bouc–Thévenaz. We shall give another formula for those idempotents, expressing them as linear combinations of the elements of a canonical basis for the integral ring. The formula is an inversion formula analogous to the Gluck–Yoshida formula for the primitive idempotents of the Burnside algebra. It involves all the irreducible characters of all the normalizers of p-subgroups. As a corollary, we shall show that the linearization map from the monomial Burnside ring has a matrix whose entries can be expressed in terms of the above Brauer characters and some reduced Euler characteristics of posets
Open Access
A new canonical induction formula for p-permutation modules [Une nouvelle formule d'induction canonique pour modules de p-permutation]
(Elsevier, 2019) Barker, Laurence; Mutlu, Hatice
Applying Robert Boltje's theory of canonical induction, we give a restriction-preserving formula expressing any p-permutation module as a Z[1/p]-linear combination of modules induced and inflated from projective modules associated with subquotient groups. The underlying constructions include, for any given finite group, a ring with a Z-basis indexed by conjugacy classes of triples (U,K,E)where U is a subgroup, K is a p′-residue-free normal subgroup of U, and E is an indecomposable projective module of the group algebra of U/K.
Open Access
On contractibility of the Orbit Space of a G-Poset of Brauer Pairs
(1999) Barker, Laurence
Given ap-blockbof a finite groupG, we show that theG-poset of Brauer pairs strictly containing (1,b) has contractibleG-orbit space. A similar result is proved for certainG-posets ofp-subgroups. Both results generalise P. Symonds' verification of a conjecture of P. Webb.
Open Access
Rhetorical biset functors, rational p-biset functors and their semisimplicity in characteristic zero
(Academic Press, 2008) Barker, Laurence
Rhetorical biset functors can be defined for any family of finite groups that is closed under subquotients up to isomorphism. The rhetorical p-biset functors almost coincide with the rational p-biset functors. We show that, over a field with characteristic zero, the rhetorical biset functors are semisimple and, furthermore, they admit a character theory involving primitive characters of automorphism groups of cyclic groups.
Open Access
Some deformations of the fibred biset category
(TÜBİTAK, 2020) Barker, Laurence; Öğüt, İsmail Alperen
We prove the well-definedness of some deformations of the fibred biset category in characteristic zero. The method is to realize the fibred biset category and the deformations as the invariant parts of some categories whose compositions are given by simpler formulas. Those larger categories are constructed from a partial category of subcharacters by linearizing and introducing a cocycle.
Open Access
Tornehave morphisms, II : the lifted Tornehave morphism and the dual of the Burnside functor
(2010) Barker, Laurence
We introduce the lifted Tornehave morphism tornπ : K → B*, an inflation Mackey morphism for finite groups, π being a set of primes, K the kernel of linearization, and B* the dual of the Burnside functor. For p-groups, tornp is unique up to scalar multiples. It induces two morphisms of biset functors, one with a codomain associated with a subgroup of the Dade group, the other with a codomain associated with a quotient of the Burnside unit group.