The primitive idempotents of the p-permutation ring

Bouc, Serge; Thévenaz, Jacques

doi:10.1016/j.jalgebra.2009.11.036

Cited by 11 publications

(15 citation statements)

References 6 publications

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“…Then the algebra A(1) is isomorphic to the algebra Fpp k (H). This algebra is split semisimple, and its primitive idempotents F H Q,s have been determined in [4]…”

Section: Shifted Functors Of Linear Representationsmentioning

confidence: 99%

The center of a Green biset functor

Bouc

Romero

2019

Pacific J. Math.

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For a Green biset functor A, we define the commutant and the center of A and we study some of their properties and their relationship. This leads in particular to the main application of these constructions: the possibility of splitting the category of A-modules as a direct product of smaller abelian categories. We give explicit examples of such decompositions for some classical shifted representation functors. These constructions are inspired by similar ones for Mackey functors for a fixed finite group. We fix a non-empty class D of finite groups closed under subquotients and cartesian products, and a set D of representatives of isomorphism classes of groups in D. We denote by RD the full subcategory of RC consisting of groups in D, so in particular RD is a replete subcategory of RC, in the sense of [2], Definition 4.1.7. The category of biset functors, i.e. the category of R-linear functors from RC to the category R-Mod of all R-modules, will be denoted by Fun R . The category Fun D,R of D-biset functors is the category of R-linear functors from RD to R-Mod.

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“…Then the algebra A(1) is isomorphic to the algebra Fpp k (H). This algebra is split semisimple, and its primitive idempotents F H Q,s have been determined in [4]…”

Section: Shifted Functors Of Linear Representationsmentioning

confidence: 99%

The center of a Green biset functor

Bouc

Romero

2019

Pacific J. Math.

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“…It was shown in [5] that the primitive idempotents of the (commutative) ring K ⊗ Z pp k (G) are indexed by the orbits of G on Q G,p : the idempotent F G R,s associated to the orbit of the pair (R, s) is equal to (2.13)…”

Section: 6mentioning

confidence: 99%

On the Cartan matrix of Mackey algebras

Bouc

2011

Trans. Amer. Math. Soc.

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Abstract. Let k be a field of characteristic p > 0, and let G be a finite group. The first result of this paper is an explicit formula for the determinant of the Cartan matrix of the Mackey algebra μ k (G) of G over k. The second one is a formula for the rank of the Cartan matrix of the cohomological Mackey algebra coμ k (G) of G over k, and a characterization of the groups G for which this matrix is nonsingular. The third result is a generalization of this rank formula and characterization to blocks of coμ k (G): in particular, if b is a block of kG, the Cartan matrix of the corresponding block coμ k (b) of coμ k (G) is nonsingular if and only if b is nilpotent with cyclic defect groups.

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“…Of course, p is a prime. We shall be considering p-permutation modules for finite groups over an algebraically closed field F of characteristic p. A review of the theory of ppermutation modules can be found in Bouc-Thévenaz [5,Section 2].…”

Section: Introductionmentioning

confidence: 99%

“…To motivate further study of the algebras Z[1/p]T (G) and KT (G), we mention that, notwithstanding the formulas for the primitive idempotents of KT (G) in Boltje [4, 3.6], Bouc-Thévenaz [5,4.12] and [1], the relationship between those idempotents and the basis {[M G P,E ] : (P, E) ∈ G P(E)} remains mysterious. In Section 4, we shall prove that KT (G) is K-semisimple as well as commutative, in other words, the primitive idempotents of KT (G) comprise a basis for KT (G).…”

Section: Introductionmentioning

confidence: 99%

A new canonical induction formula for p-permutation modules

Barker

Mutlu

2019

Comptes Rendus. Mathématique

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Applying Robert Boltje's theory of canonical induction, we give a restrictionpreserving 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. 2010 Mathematics Subject Classification: 20C20.

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The primitive idempotents of the p-permutation ring

Cited by 11 publications

References 6 publications

The center of a Green biset functor

The center of a Green biset functor

On the Cartan matrix of Mackey algebras

A new canonical induction formula for p-permutation modules

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