splendid and perverse equivalences inspired by the works of rickard on splendid equivalences 9 and of chuang and rouquier on perverse equivalences 6 we are here interested in the combination of both i e a splendid perverse equivalence this is naturally the right framework to understand the relations between global and local perverse equivalences between blocks of finite groups as a splendid equivalence induces local derived equivalences via the brauer functor we prove that under certain conditions we have an equivalence between a perverse equivalence between the homotopy category of p permutation modules and local derived perverse equivalences in the case of abelian defect groups

broueʼs abelian defect group conjecture for the sporadic simple janko group j4 revisited abstract we show that the 3 block of the sporadic simple janko group j 4 with defect group c 3 c 3 and the principal 3 block of the alternating group a 8 are puig equivalent answering a question posed in our previous paper to accomplish this we apply computational techniques in particular an explicit version of the brauer construction

complexes de chaines etales et courbes de deligne lusztig abstract in a first part we lift the usual constructions of functors between derived categories of etale sheaves over schemes with a sheaf of algebras to pure derived categories for varieties with finite group actions we recover in a more functorial way rickard s construction we apply this to the case of deligne lusztig varieties and show that broue s conjecture holds for curves the additional ingredient we need is obtained from an easy property of the cohomology of etale covers of the affine line

classifying blocks with abelian defect groups of rank 3 for the prime 2 abstract in this paper we classify all blocks with defect group c 2 n c 2 c 2 up to morita equivalence together with a recent paper of wu zhang and zhou this completes the classification of morita equivalence classes of 2 blocks with abelian defect groups of rank at most 3 the classification holds for blocks over a suitable discrete valuation ring as well as for those over an algebraically closed field the case considered in this paper is significant because it involves comparison of morita equivalence classes between a group and a normal subgroup of index 2 so requires novel reduction techniques which we hope will be of wider interest we note that this also completes the classification of blocks with abelian defect groups of order dividing 16 up to morita equivalence a consequence is that broue s abelian defect group conjecture holds for all blocks mentioned above

equivalences between blocks of cohomological mackey algebras let g be a finite group and k mathcal o k be a p modular system which is large enough let r mathcal o or k there is a bijection between the blocks of the group algebra rg and the central primitive idempotents the blocks of the so called cohomological mackey algebra co mu r g here we introduce the notion of permeable derived equivalence and we prove that a permeable derived equivalence between two blocks of group algebras implies the existence of a derived equivalence between the corresponding blocks of cohomological mackey algebras in particular in the context of broue s abelian defect group conjecture if two blocks are splendidly derived equivalent then the corresponding blocks of cohomological mackey algebras are derived equivalent

scaling properties of scale free evolving networks continuous approach the scaling behavior of scale free evolving networks arising in areas such as communications scientific citations collaborations etc is studied we derive universal scaling relations describing properties of such networks and indicate the limits of their validity we show that the main properties of scale free evolving networks may be described in the framework of a simple continuous approach the simplest models of networks growing according to a mechanism of preferential attachment of links to nodes are used we consider different forms of this preference and demonstrate that the range of preferential attachments producing scale free networks is wide we also obtain scaling relations for networks with nonlinear accelerating growth and describe the temporal evolution of the arising distributions size effects char22 the cutoffs of these distributions char22 introduce restrictions for the observation of power law dependences mainly we discuss the so called degree distribution i e the distribution of the number of connections of nodes a scaling form of the distribution of links between pairs of individual nodes for a growing network of citations is also studied we describe the effects of differences between nodes the aging of nodes changes the exponents of the distributions the appearance of a single node with high fitness changes the degree distribution of a network dramatically if its fitness exceeds some threshold value this node captures a finite part of all links of the network we show that permanent random damage to a growing scale free network char22 a permanent deletion of some links char22 radically changes the values of the scaling exponents results of other kinds of permanent damage are described

universal deformation rings and klein four defect groups in this paper the universal deformation rings of certain modular representations of a finite group are determined the representations under consideration are those which are associated to blocks with klein four defect groups and whose stable endomorphisms are given by scalars it turns out that these universal deformation rings are always subquotient rings of the group ring of a klein four group over the ring of witt vectors

broue s abelian defect group conjecture holds for the harada norton sporadic simple group hn abstract in representation theory of finite groups there is a well known and important conjecture due to m broue he conjectures that for any prime p if a p block a of a finite group g has an abelian defect group p then a and its brauer corresponding block b of the normaliser n g p of p in g are derived equivalent rickard equivalent this conjecture is called broue s abelian defect group conjecture we prove in this paper that broue s abelian defect group conjecture is true for a non principal 3 block a with an elementary abelian defect group p of order 9 of the harada norton simple group hn it then turns out that broue s abelian defect group conjecture holds for all primes p and for all p blocks of the harada norton simple group hn

a two sided q analogue of the coxeter complex m cabanes and j rickard showed in 3 that the alvis curtis character duality of a finite group of lie type is induced in non defining characteristic l by a derived equivalence given by tensoring with a bounded complex x and they further conjecture that this derived equivalence should actually be a homotopy equivalence following a suggestion of r kessar we show here for the special case of principal blocks of general linear groups with abelian sylow l subgroups that this is true by an explicit verification relating the complex x to the coxeter complex of the corresponding weyl group

a ghost ring for the left free double burnside ring and an application to fusion systems abstract for a finite group g we define a ghost ring and a mark homomorphism for the double burnside ring b g g of left free g g bisets in analogy to the case of the burnside ring b g the ghost ring has a much simpler ring structure and after tensoring with q one obtains an isomorphism of q algebras as an application of a key lemma we obtain a very general formula for the brauer construction applied to a tensor product of two p permutation bimodules m and n in terms of brauer constructions of the bimodules m and n over a field of characteristic 0 we determine the simple modules of the left free double burnside algebra and prove semisimplicity results for the bifree double burnside algebra these results carry over to results about biset functor categories finally we apply the ghost ring and mark homomorphism to fusion systems on a finite p group we extend a remarkable bijection due to ragnarsson and stancu between saturated fusion systems and certain idempotents of the bifree double burnside algebra over z p to a bijection between all fusion systems and a larger set of idempotents in the bifree double burnside algebra over q

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

the dade group of a metacyclic p group the dade group d p of a finite p group p formed by equivalence classes of endo permutation modules is a finitely generated abelian group its torsion free rank equals the number of conjugacy classes of non cyclic subgroups of p and it is conjectured that every non trivial element of its torsion subgroup d t p has order 2 or also 4 in case p 2 the group d t p is closely related to the injectivity of the restriction map res t p rightarrow prod e t e where e runs over elementary abelian subgroups of p and t p denotes the group of equivalence classes of endo trivial modules which is still unknown for almost extra special groups p odd as metacyclic p groups have no almost extra special section we can verify the above conjecture in this case finally we compute the whole dade group of a metacyclic p group

broue s conjecture holds for principal 3 blocks with elementary abelian defect group of order 9 abstract in representation theory of finite groups there is a well known and important conjecture due to m broue he conjectures that for any prime p if a finite group g has an abelian sylow p subgroup p then the derived categories of the principal p blocks of g and of the normalizer ng p of p in g are equivalent we prove in this paper that broue inage kus conjecture holds for the principal 3 block of an arbitrary finite group g with an elementary abelian sylow 3 subgroup p of order 9 by using initiated works for the case where g is simple which are due to puig okuyama waki miyachi and the authors the result depends on the classification of finite simple groups

endo monomial modules over p groups and their classification in the abelian case abstract the class of endo monomial modules is introduced generalizing endo permutation modules they can be classified by determining the monomial dade group and for abelian p groups this group is proven to be isomorphic to the ordinary dade group of endo permutation modules

self equivalences of stable module categories let p be an abelian p group e a cyclic p group acting freely on p and k an algebraically closed field of characteristic p 0 in this work we prove that every self equivalence of the stable module category of k p rtimes e comes from a self equivalence of the derived category of k p rtimes e work of puig and rickard allows us to deduce that if a block b with defect group p and inertial quotient e is rickard equivalent to k p rtimes e then they are splendidly rickard equivalent that is broue s original conjecture implies rickard s refinement of the conjecture in this case all of this follows from a general result concerning the self equivalences of the thick subcategory generated by the trivial module

equivalences between blocks of p local mackey algebras let g be a finite group and k o k be a p modular system let r o or k there is a bijection between the blocks of the group algebra and the blocks of the so called p local mackey algebra mu 1 r g let b be a block of rg with abelian defect group d let b be its brauer correspondant in n g d it is conjectured by broue that the blocks rgb and rng d b are derived equivalent here we look at equivalences between the corresponding blocks of p local mackey algebras we prove that an analogue of the broue s conjecture is true for the p local mackey algebras in the following cases for the principal blocks of p nilpotent groups and for blocks with defect 1

blow up and global existence of solutions to an inhomogeneous parabolic system abstract we study the critical phenomenon of a weakly coupled inhomogeneous parabolic system which is largely unexplored even though the homogeneous problem has been widely studied the conditions on the powers of the nonlinear terms are new compared with the well known homogeneous case the method appears to be different from the standard ones in the literature an application to the lame emden system is also given

deformations of complexes for finite dimensional algebras abstract let k be a field and let be a finite dimensional k algebra we prove that every bounded complex v of finitely generated modules has a well defined versal deformation ring r v which is a complete local commutative noetherian k algebra with residue field k we also prove that nice two sided tilting complexes between and another finite dimensional k algebra preserve these versal deformation rings additionally we investigate stable equivalences of morita type between self injective algebras in this context we apply these results to the derived equivalence classes of the members of a particular family of algebras of dihedral type that were introduced by erdmann and shown by holm to be not derived equivalent to any block of a group algebra

equivariant splendid rickard equivalences abstract broue s conjecture says that if a finite group h has an abelian sylow p subgroup p then the principal blocks of h and n h p are splendidly rickard equivalent in this paper broue s conjecture is reduced to simple groups with a stability condition a structure theorem on finite groups with abelian sylow p subgroups due to kimmerle and sandling is used

blocks with defect group z2n z2n z2m abstract in this paper we prove that a block algebra with defect group z 2 n z 2 n z 2 m where n 2 and m is arbitrary is morita equivalent to its brauer correspondent

broueʼs abelian defect group conjecture holds for the double cover of the higman sims sporadic simple group abstract in the representation theory of finite groups there is a well known and important conjecture due to broue saying that for any prime p if a p block a of a finite group g has an abelian defect group p then a and its brauer corresponding block b of the normaliser n g p of p in g are derived equivalent we prove in this paper that broue s abelian defect group conjecture and even rickard s splendid equivalence conjecture are true for the faithful 3 block a with an elementary abelian defect group p of order 9 of the double cover 2 hs of the higman sims sporadic simple group it then turns out that both conjectures hold for all primes p and for all p blocks of 2 hs

coxeter orbits and brauer trees abstract we study the cohomology with modular coefficients of deligne lusztig varieties associated to coxeter elements under some torsion free assumption on the cohomology we derive several results on the principal l block of a finite reductive group g f q when the order of q modulo l is assumed to be the coxeter number these results include the determination of the planar embedded brauer tree of the block as conjectured by hiss lubeck and malle 1995 in 25 and the derived equivalence predicted by the geometric version of broue s conjecture broue and malle 1993 7

self tilting complexes yield unstable modules let g be a group and let r be a commutative ring let t rp icr rg be the group of isomorphism classes of standard self equivalences of the derived category of bounded complexes of rg modules the subgroup hdr g of t rp icr rg consisting of self equivalences xing the trivial rg module acts on the cohomology ring h g r the action is functorial with respect to r the self equivalences which are splendid in a sense dened by j rickard act natural with respect to transfer and restriction to centralizers of p subgroups in case r is a eld of characteristic p in the present paper we prove that this action of self equivalences on h g r commutes with the action of the steenrod algebra and study the behaviour of the action of splendid self equivalences with respect to lannes t functor introduction in an earlier joint paper 9 with rapharouquier i dened the group t rp icr a of standard self equivalences of a derived module category d b a for an r algebra a which is projective as an r module for any a module m let hdm a be the subgroup of t rp icr a which is formed by the self equivalences mapping m to an isomorphic copy then in an earlier paper 11 i showed that under some hypothesis on m the group hdm a acts in a natural way on the ext algebra ext a m m in case of a being a group algebra rg with r being a eld of characteristic p and g being a nite group then j rickard denes in 8 what is a splendid equivalence by some technical conditions basically by asking that the homogeneous components of a tilting complex are p permutation modules induced from diagonal p subgroups and by some invertibility condition in the homotopy category these splendid equivalences induce self equivalences of the derived categories of centralizers of p subgroups by the brauer construction in 12 i showed that then for m r being the trivial module the action of those splendid equivalences commute with transfer and restriction from and to the cohomology rings of centralizers of p subgroups in the present paper we enlarge these properties still further the action of self equivalences of the bounded derived category d b rg on h g r commutes with the action of the steenrod algebra on h g r for any prime p as consequence of the above statements any cohomology ring h g fp denes a functor h g fp p hd p b0 p g from the modules over the group of derived self equivalences of the principal block of the group ring fp hd p fpg xing the trivial module to the category of unstable modules up and similarly in the opposite direction obviously we may restrict to splendid self equivalences we shall describe the composition of lannes t functor with the rst functor and the im age of free unstable modules by the second moreover by a result of lannes tv h g fp decomposes into direct product of cohomology rings as unstable modules we shall prove that this decomposition is also a decomposition as modules over the action of splendid self equivalences this will give evidence that splendid equivalences are the correct objects to study in this context the paper is organized as follows section 1 recalls the necessary denitions and properties of even s norm map as it is used here and the denition of the steenrod operation in section 2 it is shown that the normalized part of the outer automorphism group of the

broueʼs abelian defect group conjecture holds for the sporadic simple conway group co3 in the representation theory of finite groups there is a well known and important conjecture due to m broue he conjectures that for any prime p if a p block a of a finite group g has an abelian defect group p then a and its brauer corresponding block an of the normaliser ng p of p in g are derived equivalent rickard equivalent this conjecture is called strong version of broue s abelian defect group conjecture in this paper we prove that the strong version of broue s abelian defect group conjecture is true for the non principal 2 block a with an elementary abelian defect group p of order 8 of the sporadic simple conway group co3 this result completes the verification of the strong version of broue s abelian defect group conjecture for all primes p and for all p blocks of co3

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

trivial source modules in blocks with cyclic defect groups we shall present a method to get trivial source modules easily just by looking at values of ordinary characters at non identity p elements in finite groups instead of doing huge calculation the method is only for a case where defect groups are cyclic nevertheless it works well at least when we want to prove broue s abelian defect group conjecture for blocks which have elementary abelian defect groups of order p2

diagonal p permutation functors abstract let k be an algebraically closed field of positive characteristic p and let f be an algebraically closed field of characteristic 0 we consider the f linear category f p p k of finite groups in which the set of morphisms from g to h is the f linear extension f t h g of the grothendieck group t h g of p permutation k h k g bimodules with twisted diagonal vertices the f linear functors from f p p k to f mod are called diagonal p permutation functors they form an abelian category f p p k we study in particular the functor f t sending a finite group g to the grothendieck group f t g of p permutation kg modules and show that f t is a semisimple object of f p p k equal to the direct sum of specific simple functors parametrized by isomorphism classes of pairs p s of a finite p group p and a generator s of a p subgroup acting faithfully on p this leads to a precise description of the evaluations of these simple functors in particular we show that the simple functor indexed by the trivial pair 1 1 is isomorphic to the functor sending a finite group g to f k 0 k g where k 0 k g is the grothendieck group of projective kg modules

on morita and derived equivalences for cohomological mackey algebras by results of the second author a source algebra equivalence between two p blocks of finite groups induces an equivalence between the categories of cohomological mackey functors associated with these blocks and a splendid derived equivalence between two blocks induces a derived equivalence between the corresponding categories ofcohomological mackey functors the main result of this paper proves a partial converse an equivalence resp rickard equivalence between the categories of cohomological mackey functors of two blocks of finite groups induces a permeable morita resp derived equivalence between the two block algebras

derived categories and deligne lusztig varieties ii this paper is a continuation and a completion of boro1 we extend the jordan decomposition of blocks we show that blocks of finite groups of lie type in non describing characteristic are morita equivalent to blocks of subgroups associated to isolated elements of the dual group the key new result is the invariance of the part of the cohomology in a given modular series of deligne lusztig varieties associated to a given levi subgroup under certain variations of parabolic subgroups we also show that the equivalence arises from a splendid rickard equivalence even in the setting of boro1 the finer homotopy equivalence was unknown as a consequence the equivalence preserves defect groups and categories of subpairs we finally determine when deligne lusztig induced representations of tori generate the derived category of representations

acceleratingly growing scale free networks with tunable degree exponents abstract in this paper we analytically study the probabilistic accelerating network m j gagen j s mattick phys rev e 72 2005 016123 in its accelerating regimes by using mean field theory in the growing network the number of links added with each new node is a nonlinearly increasing function a n t where n t is the number of nodes present at time t it is found that the network appears to have a power law degree distribution for large degree with tunable degree exponents ranging from 3 0 to theoretically infinity and the degree exponent depends only on the parameter as 1 2 1 the analytical results are found to be in good agreement with those obtained by extensive numerical simulations

the principal 3 blocks of four and five dimensional projective special linear groups in non defining characteristic abstract in representation theory of finite groups there is a well known and important conjecture due to m broue he has conjectured that for any prime p if a finite group g has an abelian sylow p subgroup p then the principal p blocks of g and the normalizer ng p of p in g are derived equivalent let q be a power of a prime such that q 2 or 5 mod 9 in this paper we show that broue s conjecture is true for p 3 and for g psl4 q and g psl5 q in these cases g has elementary abelian sylow 3 subgroups of order 9 what we prove here is the following in the case g psl4 q all the principal 3 blocks of g are morita even puig equivalent independently of infinitely many q in the case g psl5 q all the principal 3 blocks of g are morita even puig equivalent to the principal 3 block of ng p independently of infinitely many q

group graded basic morita equivalences abstract we introduce group graded basic morita equivalences between algebras determined by blocks of normal subgroups and by using the extended brauer quotient we show that they induce graded basic morita equivalences at local levels

rouquier blocks in chevalley groups of type e abstract in finite chevalley groups of type e we shall find a rouquier like block algebra in non defining characteristic we construct a morita equivalence between this block and a principal block of a local subgroup as an application of the equivalence we shall determine some decomposition numbers in type e

accelerating hyperaccelerating and decelerating networks many growing networks possess accelerating statistics where the number of links added with each new node is an increasing function of network size so the total number of links increases faster than linearly with network size in particular biological networks can display a quadratic growth in regulator number with genome size even while remaining sparsely connected these features are mutually incompatible in standard treatments of network theory which typically require that every new network node possesses at least one connection to model sparsely connected networks we generalize existing approaches and add each new node with a probabilistic number of links to generate either accelerating hyperaccelerating or even decelerating network statistics in different regimes under preferential attachment for example slowly accelerating networks display stationary scale free statistics relatively independent of network size while more rapidly accelerating networks display a transition from scale free to exponential statistics with network growth such transitions explain for instance the evolutionary record of single celled organisms which display strict size and complexity limits

a remark on rickard complexes abstract in this paper we characterize a rickard complex which induces a rickard equivalence between the block algebras of a block b and its brauer correspondent and whose vertices have the same order as defect groups of the block b the homology of such a rickard complex vanishes at all degree but degree q and the homology at degree q induces a basic morita equivalence between the block algebras in the sense of puig

splendid morita equivalences for principal 2 blocks with dihedral defect groups given a dihedral 2 group p of order at least 8 we classify the splendid morita equivalence classes of principal 2 blocks with defect groups isomorphic to p to this end we construct explicit stable equivalences of morita type induced by specific scott modules using brauer indecomposability and gluing methods we then determine when these stable equivalences are actually morita equivalences and hence automatically splendid morita equivalences finally we compute the generalised decomposition numbers in each case

an algorithmic approach to perverse derived equivalences broue s conjecture for ω8 2 abstract following craven and rouquier s method 8 to tackle broue s abelian defect group conjecture we present two algorithms implementing that procedure in the case of principal blocks of defect d c l c l for a prime l the first describes a stable equivalence between b 0 g and b 0 n g d and the second tries to lift such a stable equivalence to a perverse derived equivalence as an application we show that broue s conjecture holds for the principal 5 block of the simple group 8 2

irreducible morphisms of categories of complexes abstract we study irreducible morphisms of complexes in particular we show that the irreducible morphisms having one finite irreducible submorphism fall into three canonical forms and we give necessary and sufficient conditions for a given morphism of that type to be irreducible our characterization of the above mentioned type of irreducible morphisms of complexes characterizes also some class of irreducible morphisms of the derived category d for a finite dimensional k algebra where k is a field

the mckay conjecture and brauer s induction theorem let g be an arbitrary finite group the mckay conjecture asserts that g and the normaliser ng p of a sylow p subgroup p in g have the same number of characters of degree not divisible by p that is of p degree we propose a new refinement of the mckay conjecture which suggests that one may choose a correspondence between the characters of p degree of g and ng p to be compatible with induction and restriction in a certain sense this refinement implies in particular a conjecture of isaacs and navarro we also state a corresponding refinement of the broue abelian defect group conjecture we verify the proposed conjectures in several special cases

on p permutation equivalences between rickard equivalences and isotypies broue and rickard defined in their landmark papers from 1990 and 1996 the notions of an isotypy and a splendid equivalence between p blocks of finite groups here we define a notion of equivalence which we call a p permutation equivalence that implies an isotypy and is implied by a splendid equivalence moreover we study properties of p permutation equivalences

broue s conjecture for the nonprincipal block of sl 2 q with full defect abstract m broue gives an important conjecture which is called broue s abelian defect group conjecture this conjecture says that a p block where p is a prime number of a finite group with an abelian defect group is derived equivalent to its brauer correspondent in the normalizer of the defect group in this paper we prove that this conjecture is true for the nonprincipal block of sl 2 p n for a positive integer n

splendid equivalences derived categories and permutation modules we introduce the notion of a splendid equivalence between blocks of finite groups which is a strengthening of the notion of an equivalence of derived categories such an equivalence is always induced by a two sided tilting complex whose terms are summands of permutation modules a feature that is compatible with the belief that these complexes should be related to the adic cohomology of deligne lusztig varities in the case of reductive groups over finite fields we show that a splendid equivalence has good consequences over and above those that follow from a derived equivalence in particular it gives for principal blocks a structural explanation of brou s concept of an isotypy a compatible family of perfect isometries we give several examples including examples of morita equivalences that are splendid in an unexpectedly non trivial way

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

the derived categories of some blocks of symmetric groups and a conjecture of broue abstract a well known conjecture of broue in the representation theory of finite groups involves equivalences of derived categories of blocks the aim of this paper is to verify this conjecture for defect 2 blocks of symmetric groups actually we prove for these blocks a refinement of broue s conjecture due to rickard

on central idempotents in the brauer algebra abstract we provide a method for constructing central idempotents in the brauer algebra using the splitting of short exact sequences of bimodules from this we determine certain primitive central idempotents by working over a suitable integral ring we hence demonstrate an efficient method of constructing pieces of the representation theory of the brauer algebra over artinian rings from the integral case

on the glauberman and watanabe correspondences for blocks of finite p solvable groups if g is a finite p solvable group for some prime p a a solvable subgroup of the automorphism group of g of order prime to g such that a stabilises a p block b of g and acts trivially on a defect group p of b then there is a morita equivalence between the block b and its watanabe correspondent w b of c g a given by a bimodule m with vertex p and an endo permutation module as source which on the character level induces the glauberman correspondence and which is an isotypy by watanabe s results

broue s abelian defect group conjecture for the held group and the sporadic suzuki group abstract in representation theory of finite groups there is a well known and important conjecture due to m broue he conjectures that for any prime p if a p block a of a finite group g has an abelian defect group p then a and its brauer corresponding p block b of ng p are derived equivalent we demonstrate in this paper that broue s conjecture holds for non principal 3 blocks a with elementary abelian defect group p of order 9 of the simple held group and the sporadic simple suzuki group

isometries and extra special sylow groups of order p3 abstract a new type of isometry between the sets of irreducible characters of blocks is introduced and a conjecture is stated using the classifications of finite simple groups and saturated fusion systems over an extra special group p of order p 3 and exponent p we check that the conjecture is affirmatively answered for the principal blocks of finite groups with sylow p subgroup p

perverse equivalences and broueʼs conjecture abstract we give a new approach to the construction of derived equivalences between blocks of finite groups based on perverse equivalences in the setting of broue s abelian defect group conjecture we provide in particular local and global perversity data describing the principal blocks and the derived equivalences for a number of finite simple groups with sylow subgroups elementary abelian of order 9 we also examine extensions to automorphism groups in a general setting

universal deformation rings and dihedral defect groups let k be an algebraically closed field of characteristic 2 and let w be the ring of infinite witt vectors over k suppose g is a finite group and b is a block of kg with dihedral defect group d which is morita equivalent to the principal 2 modular block of a finite simple group we determine the universal deformation ring r g v for every kg module v which belongs to b and has stable endomorphism ring k it follows that r g v is always isomorphic to a subquotient ring of wd moreover we obtain an infinite series of examples of universal deformation rings which are not complete intersections

on stable equivalences with endopermutation source abstract we show that a bimodule between block algebras which has a fusion stable endopermutation module as a source and which induces morita equivalences between centralisers of nontrivial subgroups of a defect group induces a stable equivalence of morita type this is the converse to a theorem of puig the special case where the source is trivial has long been known by many authors the earliest instance of a result deducing a stable equivalence of morita type from local morita equivalences with possibly nontrivial endopermutation source is due to puig in the context of blocks with abelian defect groups with a frobenius inertial quotient the present note is motivated by an application due to biland to blocks of finite groups with structural properties known to hold for hypothetical minimal counterexamples to the z p theorem