on centers of blocks with one simple module abstract let g be a finite group and let b be a non nilpotent block of g with respect to an algebraically closed field of characteristic 2 suppose that b has an elementary abelian defect group of order 16 and only one simple module the main result of this paper describes the algebra structure of the center of b this is motivated by a similar analysis of a certain 3 block of defect 2 in kessar 2012 15

elements of specified order in simple algebraic groups in this paper we let g be a simple algebraic group and r be a natural number and consider the codimension in g of the variety of elements g g satisfying g r 1 we shall obtain a lower bound for this codimension which is independent of characteristic and show that it is attained if g is of adjoint type

on the subgroup structure of classical groups in this paper we establish a result on the subgroup structure of classical groups over algebraically closed elds and use this to give a new proof of a fundamental theorem of m aschbacher on subgroups of nite classical groups let v be a nite dimensional vector space over an algebraically closed eld k and let g be one of the classical algebraic groups sl v sp v or so v our result is a reduction theorem concerning the subgroups of g we de ne a certain collection c of natural proper subgroups of g and prove that any closed nite or in nite subgroup of g either lies in a member of c or is roughly speaking a simple group acting irreducibly on v aschbacher s result is an analogous reduction theorem for subgroups of nite classical groups we obtain this as a relatively easy consequence of our main result by taking xed points under the action of a frobenius morphism using a standard process involving lang s theorem the proof of the main result uses elementary linear algebra together with a few basic facts from the theory of algebraic groups various complications which arise in the nite group setting in as become much more straightforward in the algebraic group setting in particular questions involving extension elds do not occur and issues of conjugacy are easily settled when we descend to nite invent math 134 427 453 1998

character sheaves v this paper is part of a series s 131 devoted to the study of a class g of irreducible perverse sheaves called character sheaves on a connected reductive algebraic group g the numbering of chapters sections and references will continue that of s 131 this paper is a step towards the classification of character sheaves on g one of the main results is the following one under certain assumptions there is a natural surjective map with finite fibers from g to the set of all pairs 9 c up to conjugacy by the weyl group where 9 is a tame local system on the maximal torus and c is a two sided cell in the stabilizer w of 9 in the weyl group the assumptions made on g are

semiclassical asymptotics of eigenvalues for schrodinger operators with magnetic fields abstract we consider schrodinger operators with magnetic fields on a two dimensional compact manifold or on r 2 the purpose is to study the semiclassical asymptotics of the eigenvalues by two different methods we obtain some facts on the harmonic oscillators under uniform magnetic fields and we also discuss them

oscillatory integrals with quadratic phase function on a real abstract wiener space abstract stochastic oscillatory integrals with quadratic phase function are studied in the case where amplitude functions are multiple wiener integrals we show that a principle of stationary phase holds when kernels of multiple wiener integrals are of trace class on the way we establish a new criterion for a wiener functional to be analytic

fixed point ratios in actions of finite classical groups iii abstract this is the first in a series of four papers on fixed point ratios in actions of finite classical groups our main result states that if g is a finite almost simple classical group and is a faithful transitive non subspace g set then either fpr x x g 1 2 for all elements x g of prime order or g is one of a small number of known exceptions here fpr x denotes the proportion of points in which are fixed by x in this introductory note we present our results and describe an application to the study of minimal bases for primitive permutation groups a further application concerning monodromy groups of covers of riemann surfaces is also outlined

analysis of wiener functionals malliavin calculus and its applications to heat kernels on donne un expose facilement accessible du calcul de malliavin du calcul differentiel de dimension infinie sur l espace de wiener

the loewy structure of certain fixpoint algebras part i abstract we investigate the loewy structure of the fixpoint algebra of the group algebra of the additive group of a finite field f under the action of a subgroup of the multiplicative group of f

symmetric local algebras with 5 dimensional center we prove that a symmetric split local algebra whose center is 5 dimensional has dimension 5 or 8 this implies that the defect groups of a block of a finite group containing exactly fivbe irreducible frobenius characters and exactly one irreducible brauer character have order 5 or are nonabelian of order 8

symmetric local algebras and small blocks of finite groups one of the major problems in modular representation theory is the deter mination of the numbers k b of ordinary and z b of modular irreducible characters for the block b of a finite group g with defect group d not much is known about the opposite problem of classifying the defect group by the numbers k b and 1 b of course there is brauer s result that k b 1 is equivalent to d 1 the next two cases have been solved in a recent paper of j brandt 11 he showed as a corollary of more general results that k b 2 or 1 b 1 and k b 3 imply 1 d i k b the case i b 2 and k b 3 seems to be of a quite different nature therefore in this paper we restrict our attention to the case f b 1 and extend brandt s result to the case k b 4 it turns out that this theorem follows from some general properties of symmetric local algebras so we state our result in this more general context 1 from blocks to symmetric local algebras we are going to prove the following theorem a theorem let b be a block of the finite group g with defect group d if z b 1 and k b 4 then idi k b the possible defect groups are therefore cyclic or a klein four group dade has determined the general structure of a block with cyclic defect group and brauer of a block with klein four defect group however we will not use their results we first show how this theorem follows from a more general theorem on symmetric local algebras for us a symmetric algebra is a pair a a where 190

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

levy s stochastic area and the principle of stationary phase abstract let w 0 h 0 0 be the 2 dimensional classical pinned wiener space over 0 1 a localization phenomenon will be shown for stochastic oscillatory integrals with levy s stochastic area as phase function

2 blocks with minimal nonabelian defect groups abstract we study numerical invariants of 2 blocks with minimal nonabelian defect groups these groups were classified by redei see redei 1947 41 if the defect group is also metacyclic then the block invariants are known see sambale 43 in the remaining cases there are only two infinite families of interesting defect groups in all other cases the blocks are nilpotent we prove brauer s k b conjecture and olsson s conjecture for all 2 blocks with minimal nonabelian defect groups for one of the two families we also show that alperin s weight conjecture and dade s conjecture are satisfied this paper is a part of the author s phd thesis

finite subgroups of algebraic groups generalizing a classical theorem of jordan to arbitrary characteristic we prove that every finite subgroup of gln over a field of any characteristic p possesses a subgroup of bounded index which is composed of finite simple groups of lie type in characteristic p a commutative group of order prime to p and a p group while this statement can be deduced from the classification of finite simple groups our proof is self contained and uses methods only from algebraic geometry and the theory of linear algebraic groups we believe that our results can serve as a viable substitute for classification in a range of applications in various areas of mathematics

on wiener functionals of order 2 associated with soliton solutions of the kdv equation abstract the eigenvalues and eigenvectors of the hilbert schmidt operators corresponding to the wiener functionals of order 2 which give a rise of soliton solutions of the kdv equation are determined two explicit expressions of the stochastic oscillatory integral with such wiener functional as phase function are given one is of infinite product type and the other is of levy s formula type as an application the asymptotic behavior of the stochastic oscillatory integral will be discussed

blocks with transitive fusion systems suppose that all nontrivial subsections of a p block b are conjugate where p is a prime by using the classification of the finite simple groups we prove that the defect groups of b are either extraspecial of order p 3 with p in 3 5 or elementary abelian

products of conjugacy classes in finite and algebraic simple groups abstract we prove the arad herzog conjecture for various families of finite simple groups if a and b are nontrivial conjugacy classes then a b is not a conjugacy class we also prove that if g is a finite simple group of lie type and a and b are nontrivial conjugacy classes either both semisimple or both unipotent then a b is not a conjugacy class we also prove a strong version of the arad herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes as a consequence we obtain a complete classification of pairs of centralizers in a simple algebraic group which have dense product a special case of this has been used by prasad to prove a uniqueness result for tits systems in quasi reductive groups our final result is a generalization of the baer suzuki theorem for p elements with p 5

base sizes for simple groups and a conjecture of cameron let g be a permutation group on a finite set a base for g is a subset b with pointwise stabilizer in g that is trivial we write b g for the smallest size of a base for g in this paper we prove that b g 6 if g is an almost simple group of exceptional lie type and is a primitive faithful g set an important consequence of this result when combined with other recent work is that b g 7 for any almost simple group g in a non standard action proving a conjecture of cameron the proof is probabilistic and uses bounds on fixed point ratios

bounding the number of characters in a block of a finite group abstract we present a strong upper bound on the number k b of irreducible characters of a p block b of a finite group g in terms of local invariants more precisely the bound depends on a chosen major b subsection u b its normalizer n g u b in the fusion system and a weighted sum of the cartan invariants of b in this way we strengthen and unify previous bounds given by brauer wada kulshammer wada hethelyi kulshammer sambale and the present author

on the number of irreducible characters in a 2 block both of these conjectures have been verified for d cyclic by dade 9 or 1 d 1 pz by brauer and feit 7 for proof see brauer 3 ii in addition the first conjecture has been verified if the inertial index of the block is 1 see brauer 4 iv while the second conjecture has been partly verified for psolvable groups by fong in one of several discussions i had with r brauer in 1974 he suggested to find a verification of these conjectures in the case of a principal 2 block with h z z as defect group without involving the classification of groups with this sylow 2 subgroup from which the conjectures may quite easily be checked to hold in 181 ward was forced to do this for groups of ree type anyway but the computations were quite involved and the very special properties of these groups including their order were used heavily in theorem 2 of section 2 we prove that the first conjecture holds in general for any defect group of order less than or equal to 8 while theorem 8 of section 3 states that the second conjecture hold if the block in addition is assumed to be principal we shall not depend in our work on the classification of these groups in particular we give an alternate and shorter proof of the result in 181 mentioned above without using any other property

unipotent classes in maximal subgroups of exceptional algebraic groups abstract let g be a simple algebraic group of exceptional type over an algebraically closed field and m be a maximal closed connected reductive subgroup of g for each unipotent m class we determine the g class in which it lies

topological generation of exceptional algebraic groups abstract let g be a simple algebraic group over an algebraically closed field k and let c 1 c t be non central conjugacy classes in g in this paper we consider the problem of determining whether there exist g i c i such that g 1 g t is zariski dense in g first we establish a general result which shows that if is an irreducible subvariety of g t then the set of tuples in generating a dense subgroup of g is either empty or dense in in the special case c 1 c t by considering the dimensions of fixed point spaces we prove that this set is dense when g is an exceptional algebraic group and t 5 assuming k is not algebraic over a finite field in fact for g g 2 we only need t 4 and both of these bounds are best possible as an application we show that many faithful representations of exceptional algebraic groups are generically free we also establish new results on the topological generation of exceptional groups in the special case t 2 which have applications to random generation of finite exceptional groups of lie type in particular we prove a conjecture of liebeck and shalev on the random r s generation of exceptional groups

cartan matrices and brauerʼs k b conjecture ii abstract this paper continues sambale 2011 28 we show that the methods developed there also work for odd primes in particular we prove brauer s k b conjecture for defect groups which contain a central cyclic subgroup of index at most 9 as a consequence the k b conjecture holds for 3 blocks of defect at most 3 in the second part of the paper we illustrate the limits of our methods by considering an example then we use the work of kessar koshitani and linckelmann 13 and thus the classification to show that the k b conjecture is satisfied for 2 blocks of defect 5 except for the extraspecial defect group d 8 d 8 as a byproduct we also obtain the block invariants of 2 blocks with minimal nonmetacyclic defect groups some proofs rely on computer computations with gap the gap group 2008 10

cartan matrices and brauer s k b conjecture abstract it is well known that the cartan matrix of a block of a finite group cannot be arranged as a direct sum of smaller matrices in this paper we address the question if this remains true for equivalent matrices the motivation for this question comes from the work of kulshammer and wada 2002 10 which contains certain bounds for the number of ordinary characters in terms of cartan invariants as an application we prove such a bound in the special case where the determinant of the cartan matrix coincides with the order of the defect group in the second part of the paper we show that brauer s k b conjecture holds for 2 blocks under some restrictions on the defect group for example the k b conjecture holds for 2 blocks if the corresponding defect group is a central extension of a metacyclic group by a cyclic group the same is true if the defect group contains a central cyclic subgroup of index 8 in particular the k b conjecture holds for 2 blocks with defect at most 4 the paper is a part of the author s phd thesis

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

lie algebras generated by extremal elements abstract we study lie algebras generated by extremal elements i e elements spanning inner ideals over a field of characteristic distinct from 2 there is an associative bilinear form on such a lie algebra we study its connections with the killing form any lie algebra generated by a finite number of extremal elements is finite dimensional the minimal numbers of extremal generators for the lie algebras of type a n n 1 b n n 3 c n n 2 d n n 4 e n n 6 7 8 f 4 and g 2 are shown to be n 1 n 1 2 n n 5 5 and 4 in the respective cases these results are related to group theoretic ones for the corresponding chevalley groups

character separation and principal covering abstract we investigate the separation of irreducible characters by blocks at different primes and the covering of irreducible characters by blocks viewed as sets of characters these notions are used to prove results on the group structure the covering of all characters of a group by principal blocks is only possible when already one principal block suffices or the generalized fitting subgroup has a very special structure

block separations and inclusions abstract we investigate the separation of characters by blocks at different primes and the inclusions of q blocks in p blocks viewed as sets of characters and use these notions to prove results on the structure of the corresponding groups in particular we provide a new criterion for the nilpotence of a finite group g based on the separation by principal blocks and we show that a condition on block unions has strong structural consequences

classical groups probabilistic methods and the 2 3 generation problem we study the probability that randomly chosen elements of prescribed type in a finite simple classical group g generate g in particular we prove a conjecture of kantor and lubotzky in this area the probabilistic approach is then used to determine the finite simple classical quotients of the modular group psl2 z up to finitely many exceptions

analytic functions cauchy formula and stationary phase on a real abstract wiener space abstract a new complexification of a real abstract wiener space will be introduced and some analogs of the algebra of analytic functions on finite dimensional euclidean space will be considered analytic functions on the original space their holomorphic prolongation to the complexified space and holomorphic functions and a cauchy formula on the complexified space the cauchy formula is a key tool to study probabilistic quantities via deformation of the contour integration namely it will be applied to establish i an explicit representation of stochastic oscillatory integrals with quadratic phase function and ii a stationary phase estimation of the integrals further the later estimation is applicable to study gevrey type smoothness of density functions an integration by parts formula on a totally real submanifold in the complexified space is also studied