commutator theory without join distributivity we develop commutator theory for congruences of general alge braic systems henceforth called algebras assuming only the existence of a ternary term d such that d a b b a a a a a d b b a whenever a is a congruence and aab our results apply in particular to congruence modular and n permutable varieties to most locally finite varieties and to inverse semigroups we obtain results concerning permutability of congruences abelian and solv able congruences connections between congruence identities and commutator identities we show that many lattices cannot be embedded in the congruence lattice of algebras satisfying our hypothesis for other lattices some intervals are forced to be abelian and others are forced to be nonabelian we give simplified proofs of some results about the commutator in modu lar varieties and generalize some of them to single algebras having a modular congruence lattice

commutator theory for relatively modular quasivarieties we develop a commutator theory for relatively modular quasivarieties that extends the theory for modular varieties we characterize relatively modular quasivarieties prove that they have an almost equational axiomatization and we investigate the lattice of subquasivarieties we derive the result that every finitely generated relatively modular quasivariety of semigroups is finitely based

blocks with frobenius inertial quotients abstract in the paper we prove that a perfect isometry between a block of a finite group and its brauer correspondent can be extended to a perfect isometry between a block of a larger finite group and its brauer correspondent with suitable hypotheses we also obtain a similar result for derived equivalences

a local property of basic rickard equivalences abstract let p be a prime and k be an algebraically closed field of characteristic p let g and g be two finite groups b and b be blocks of g and g and p e and p e be maximal brauer b g and b g pairs if kgb and k g b are basic rickard equivalent then there is a group isomorphism p p such that it induces an equivalence between the brauer categories of kgb and k g b and that for any brauer b g pair q f contained in p e and any subgroup k of the image of n g q f in aut q the block algebras k n g k q f and k n g k q f are basic rickard equivalent where q q f is the block of k c g q such that q f p e and k denotes the image of k in aut q

lifting induction theorems abstract we show that in some suitable sense any induction theorem for the character ring of a finite group can be lifted to an induction theorem for the green ring a precise statement see theorem a herein this provides a uniform proof of important induction theorems as for example the ones of conlon and dress moreover we prove an analogue of brauer s induction theorem for the green ring cf theorem b herein

dichotomies structure and concentration in normed spaces abstract we use probabilistic topological and combinatorial methods to establish the following deviation inequality for any normed space x r n there exists an invertible linear map t r n r n with p t g e t g e e t g c exp c max e 2 e log n e 0 where g is the standard n dimensional gaussian vector and c c 0 are universal constants it follows that for every e 0 1 and for every normed space x r n there exists a k dimensional subspace of x which is 1 e euclidean and k c e log n log 1 e this improves by a logarithmic on e term the best previously known result due to g schechtman

on stable orthogonal decompositions of idempotents in fact the two relative projectivities of a pointed group are equivalentin some sense and it is the same for the two relative freeness see 2 2 and2 29 below it is clear that the relative freeness implies the relative projectivity theconverse is certainly false to some extent our aim is just to look for theconditions which make the converse true if

averages of norms and quasi norms we compute the number of summands in q averages of norms needed to approximate an euclidean norm it turns out that these numbers depend on the norm involved essentially only through the maximal ratio of the norm and the euclidean norm particular attention is given to the case q in which the average is replaced with the maxima this is closely connected with the behavior of certain families of projective caps on the sphere

extensions of rickard equivalences broue conjectures that a block of a finite group g with abelian defect group p is rickard equivalent to its brauer correspondent in the normalizer ng p we find a character theoretic criterion on extensions of rickard equivalences between blocks as an application we prove that the conjecture holds for the principal block of a finite group with abelian defect group and cyclic hyperfocal subgroup

two observations regarding embedding subsets of euclidean spaces in normed spaces abstract this paper contains two results concerning linear embeddings of subsets of euclidean space in low dimensional normed spaces the first is an improvement of the known dependence on in dvoretzky s theorem from order of 2 to order of except for log factors the second is a joint generalization of milman s version of dvoretzky s theorem and a recent generalization by klartag and mendelson of the johnson lindenstrauss lemma

g algebras clifford theory and the green correspondence let g be a finite group with subgroup h loosely speaking clifford theory as developed by e c dade and others is a tool for constructing reduction arguments in the case where h is normal in g the green correspondence is a tool for constructing reduction arguments in the case where h contains the normaliser in g of a given p subgroup or more generally where h contains the normaliser of a given local pointed group here we cover both of these situations by considering the case where no h contains the normaliser in g of a given local pointed group a brief discussion in 1 indicates how puig s theory subsumes the green correspondence and aspects of clifford theory here this theme is given a more systematic treatment we shall see in particular that defect pointed subgroups the green correspondence and brauer s extended first main theorem are all manifestations of a more general mechanism let be a complete local commutative noetherian ring with an algebraically closed residue field k of prime characteristic p throughout we fix a g algebra a all algebras are deemed to possess a unity element all algebras and modules are to be free over or over k and of finite rank for the rest of this section let us fix points 3 a of h g respectively on a and suppose that ht3 g have a common defect pointed subgroup pr such that nc p ng h it is easy to show see section 4 t h a t if p is another defect pointed subgroup of ht3 then p is also a defect pointed subgroup of g and ng p ng h it t ranspires again see section 4 t h a t ht3 g if and only if a c trgh ah 3ah when these equivalent

trivial source bimodule rings for blocks and p permutation equivalences we associate with any p block of a finite group a grothendieck ring of certain p permutation bimodules we extend the notion of p permutation equivalences introduced by boltje and xu 4 to source algebras of p blocks of finite groups we show that a p permutation equivalence between two source algebras a b of blocks with a common defect group and same local structure induces an isotypy

duality in g algebras keywords finite group complete discrete valuation ring symmetric g algebra symmetric bilinear form projective ideal symmetric algebra rg lattice unimodular symmetric g algebra almost projective ideal almost split sequences block defect groups virtually irreducible lattices splitting trace lattices reference ctg article 1988 005doi 10 1007 bf01161746 record created on 2008 12 16 modified on 2017 05 12

hyperfocal subalgebras of blocks and computation of characters abstract in terms of puig s theory on blocks it is described how to compute the irreducible characters of any block from the hyperfocal subalgebra of the block and the source algebra and the characters of a block whose hyperfocal subgroup is a direct factor of its defect group are characterized

on group stable commutative separable semisimple subalgebras in a g algebra a of a p group g over a perfect ground field there is a commutative separable semisimple g subalgebra such that any traces on relative projectivity in a can be controlled in the subalgebra and in a residually central sense the maximal such subalgebras form exactly one a g conjugacy class applying to modules the induced module of an indecomposable module of a subnormal subgroup of p primary index is characterized

fully group graded algebras and a theorem of fong abstract it is proved that any algebra fully graded by a finite group over a complete discrete valuation ring with an algebraically closed residue field of characteristic a prime p is morita equivalent to an embedded graded subalgebra which is a crossed product and an explicit way to get a decomposition of unity with a bounded length is shown when the finite group is p solvable a theorem of fong s type for fully graded algebras is obtained

defects of irreducible characters ofp soluble groups abstract we prove a refinement of the p soluble case of robinson s conjectural local characterization of the defect of an irreducible character

the indecomposability of a certain bimodule given by the brauer construction broue s abelian defect conjecture 3 6 2 predicts for a p block of a finite group g with an abelian defect group p a derived equivalence between the block algebra and its brauer correspondent by a result of rickard 11 such a derived equivalence would in particular imply a stable equivalence induced by tensoring with a suitable bimodule and it appears that these stable equivalences in turn tend to be obtained by gluing together morita equivalences at the local levels of the considered blocks see e g 4 6 3 8 3 1 12 4 1 and 13 5 6 a 4 1 this note provides a technical indecomposability result which is intended to verify in suitable circumstances the hypotheses that are necessary to apply gluing results as mentioned above this is used in 7 to show that broue s abelian defect group conjecture holds for nonprincipal blocks of the simple held group and the sporadic suzuki group

superconcentration and randomized dvoretzky s theorem for spaces with 1 unconditional bases abstract let n be a sufficiently large natural number and let b be an origin symmetric convex body in r n in the l position and such that the space r n b admits a 1 unconditional basis then for any e 0 1 2 and for random c e log n log 1 e dimensional subspace e distributed according to the rotation invariant haar measure the section b e is 1 e euclidean with probability close to one this shows that the worst case dependence on e in the randomized dvoretzky theorem in the l position is significantly better than in john s position it is a previously unexplored feature which has strong connections with the concept of superconcentration introduced by s chatterjee in fact our main result follows from the next theorem let b be as before and assume additionally that b has a smooth boundary and e n b n c e n grad b 2 for a small universal constant c 0 where grad b is the gradient of b and n is the standard gaussian measure in r n then for any p 1 c log n the p th power of the norm b p is c log n superconcentrated in the gauss space

finite basis theorems for relatively congruence distributive quasivarieties q is any quasivariety a congruence relation 0 on a mem ber a of q is a q congruence if a 0 g q the set couqa of all q congruences is closed under arbitrary intersection and hence forms a complete lattice cong a q is relatively congruence distributive if conja is distribu tive for every a e q relatively congruence distributive quasivarieties occur naturally in the theory of abstract data types q is finitely generated if it is generated by a finite set of finite algebras the following generalization of baker s finite basis theorem is proved theorem i every finitely generated and relatively congruence distributive quasivariety is finitely based a subquasivariety jz of an arbitrary quasivariety q is called a relative subvariety of q if it is of the form v n q for some variety v i e a base for z can be obtained by adjoining only identities to a base for q theorem ii every finitely generated relative subvariety of a relatively congruence distributive quasivariety is finitely based the quasivariety of generalized equality test algebras is defined and the structure of its members studied this gives rise to a finite algebra whose quasi identities are finitely based while its identities are not connections with logic and the algebraic theory of data types are discussed

poincare s inequalities and talagrand s concentration phenomenon for the exponential distribution we present a simple proof based on modified logarithmic sobolev inequalities of talagrand s concentration inequality for the exponential distribution we actually observe that every measure satisfying a poincare inequality shares the same concentration phenomenon we also discuss exponential integrability under poincare inequalities and its consequence to sharp diameter upper bounds on spectral gaps

on russo s approximate zero one law consider the product measure mu p on 0 1 n when 0 operatorname resp 1 is given weight 1 p operatorname resp p consider a monotone subset a of 0 1 n we give a precise quantitative form to the following statement if a does not depend much on any given coordinate d mu p a dp is large thus in that case there is a threshold effect and mu p a jumps from near 0 to near 1 in a small interval

g algebras jacobson radical and almost split sequences keywords finite group indecomposable kg module sas sequence almost split exact sequences restriction relative jacobson radicals local criterion multiplicity module vertex source indecomposable projective module twisted group algebra induction reference ctg article 1988 006doi 10 1007 bf01393690 record created on 2008 12 16 modified on 2017 05 12