Number Of Simple Modules Research Articles

Weights in a Benson-Solomon block

AbstractTo each pair consisting of a saturated fusion system over ap-group together with a compatible family of Külshammer-Puig cohomology classes, one can count weights in a hypothetical block algebra arising from these data. When the pair arises from a genuine block of a finite group algebra in characteristicp, the number of conjugacy classes of weights is supposed to be the number of simple modules in the block. We show that there is unique such pair associated with each Benson-Solomon exotic fusion system, and that the number of weights in a hypothetical Benson-Solomon block is$12$, independently of the field of definition. This is carried out in part by listing explicitly up to conjugacy all centric radical subgroups and their outer automorphism groups in these systems.

Classifying torsion classes for algebras with radical square zero via sign decomposition

To study the set of torsion classes of a finite dimensional basic algebra over a field, we use a decomposition, called sign-decomposition, parameterized by elements of {±1}n where n is the number of simple modules. If A is an algebra with radical square zero, then for each ϵ∈{±1}n there is a hereditary algebra Aϵ! with radical square zero and a bijection between the set of torsion classes of A associated to ϵ and the set of faithful torsion classes of Aϵ!. Furthermore, this bijection preserves the property of being functorially finite. From a point of view of tilting theory, it implies that there is a bijection between the set of isomorphism classes of basic two-term silting complexes for A associated to ϵ and the set of isomorphism classes of basic tilting Aϵ!-modules. As an application, we prove that the number of two-term tilting complexes over Brauer line algebras (respectively, Brauer cycle algebras) having n edges is (2nn) (respectively, 22n−1 if n is odd, and ∞ if n is even).

Hochschild cohomology of polynomial representations of GL2

We compute the Hochschild cohomology algebras of Ringel-self-dual blocks of polynomial representations of GL2 over an algebraically closed field of characteristic p>2 , that is, of any block whose number of simple modules is a power of p>2. These algebras are finite-dimensional and we provide an explicit description of their bases and multiplications.

Hochschild Cohomology of Polynomial Representations of $\mathrm{GL}_2$

We compute the Hochschild cohomology algebras of Ringel-self-dual blocks of polynomial representations of \mathrm{GL}_2 over an algebraically closed field of characteristic p>2 , that is, of any block whose number of simple modules is a power of p . These algebras are finite-dimensional and we provide an explicit description of their bases and multiplications.

On the number of simple modules in a block of a finite group

We prove that if B is a p-block with non-trivial defect group D of a finite p-solvable group G, then ℓ(B)<pr, where r is the sectional rank of D. We remark that there are infinitely many p-blocks B with non-Abelian defect groups and ℓ(B)=pr−1.We conjecture that the inequality ℓ(B)≤pr holds for an arbitrary p-block with defect group of sectional rank r. We show this to hold for a large class of p-blocks of various families of quasi-simple and nearly simple groups.

Affine algebras with arbitrary simple modules

ABSTRACTWe construct affine algebras with an arbitrary number of simple modules of each finite dimension.

The Cartan matrix of the Schur algebra S(2, r)

Let k be an infinite field of prime characteristic and let r be a positive integer. Using admissible decompositions, we determine explicitly the entries of the decomposition matrix of the Schur algebra S(2, r) over k and prove that any two blocks with the same number of simple modules have the same decomposition matrix and hence the same Cartan matrix.

On Alperin's Conjecture and Certain Subgroup Complexes

We prove a new formula about local control of the number of p-regular conjugacyclasses of a finite group. We then relate the results to Alperin’s weight conjecture to obtain newresults describing the number of simple modules for a finite group in terms of weights of solvablesubgroups. Finally, we use the results to obtain new formulations of Alperin’s weight conjecture,and to obtain restrictions on the structure of a minimal counterexample.

On a conjecture of Smith's

AbstractIn this paper we prove a conjecture of Smith's on the number of simple modules of algebras which are similar to the enveloping algebra of sl(2).

On the number of simple modules of certain tame blocks and algebras

On the number of simple modules of certain tame blocks and algebras

Almost split sequences for group algebras of finite representation type

Let k be an algebraically closed field of characteristic p and G a finite group such that p divides the order of G. We compute all almost split sequences over kG when kG is of finite representation type, or more generally, for a finite dimensional k-algebra Λ \Lambda given by a Brauer tree. We apply this to show that if Λ \Lambda and Λ ′ \Lambda ’ are stably equivalent k-algebras given by Brauer trees, then they have the same number of simple modules.