orbit invariants and an application to the baby monster abstract we prove that the minimal base size for the permutation action of the sporadic simple baby monster group b on the cosets of its 7th and 8th maximal subgroup in decreasing order of size is 3 and 2 respectively motivated by the large sizes of these permutation actions we develop new computational methods to prove that an orbit is regular and to show that two orbits are disjoint

solvability of linear local and nonlocal robin problems over c ω abstract let r n be a bounded lipschitz domain we first consider an elliptic boundary value problem with general robin boundary conditions the boundary conditions can be either local or nonlocal depending on the conditions imposed on the elliptic operator we prove that this boundary value problem is uniquely solvable and moreover we show that such weak solution is holder continuous on we also prove that a realization of the associated differential operator with generalized local or nonlocal robin boundary conditions generates an analytic c 0 semigroup of angle 2 over c we conclude by applying the elliptic regularity theory to solve the corresponding cauchy problem over c

on the multiplicity free actions of the sporadic simple groups abstract we introduce a database containing the character tables of the endomorphism rings of the multiplicity free permutation modules of the sporadic simple groups their automorphism groups their schur covers and their bicyclic extensions we describe the techniques used to compile the data and present a couple of applications to orbital graphs

the proper dissipative extensions of a dual pair let a and widetilde a be dissipative operators on a hilbert space mathcal h and let a widetilde a form a dual pair i e a subset widetilde a resp widetilde a subset a we present a method of determining the proper dissipative extensions widehat a of this dual pair i e a subset widehat a subset widetilde a provided that mathcal d a cap mathcal d widetilde a is dense in mathcal h applications to symmetric operators symmetric operators perturbed by a relatively bounded dissipative operator and more singular differential operators are discussed finally we investigate the stability of the numerical range of the different dissipative extensions

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

alternating forms and self adjoint operators abstract let k be a field and n 1 an integer we study the action of the symplectic group over k on the set of alternating forms on k 2 n we show that the action on pairs of forms can be interpreted in terms of the conjugation action on self adjoint operators and obtain some old and new results using this interpretation in particular for each k and n we determine the smallest base size for the action of pgl 2 n k on the set of non degenerate skew symmetric matrices over the field k modulo scalars

anatomy of the monster ii abstract we summarise much of what is currently known about the subgroup structure of the fischer griess monster introduction the publication of the atlas of finite groups 1 henceforth referred to as the atlas marked the end of a chapter in the history of group theory in more ways than one the classification of finite simple groups had been completed and using new computational techniques the last of the sporadic groups had been constructed and enough progress had been made in classifying their maximal subgroups to make the rest seem like a tidying up job since the publication of the atlas these computational methods have been pushed further the classification of the maximal subgroups for fi 22 th fi 23 j 4 and fi 24 was completed by kleidman linton parker and wilson in 7 8 9 10 12 13 14 this means that appendix 2 of the atlas of brauer characters 11 when collated with the atlas itself completes the listing for all sporadic groups except the baby monster and monster for the baby monster four new maximal subgroups s 6 l 3 4 2 2 s 6 s 6 4 l 2 31 and l 2 49 2 3 were given in appendix 2 of 11 and wilson has since found four more m 11 l 3 3 l 2 17 2 and l 2 ll 2 for references see 23 24 25 with work on the 2 locals by meierfrankenfeld and shpektorov 15 an end to the baby monster problem is in sight with a lot of work also having been done on maximal subgroups of generic groups this leaves only the monster unfortunately the degree of its smallest representation over any field 196882 is too big for present day computational techniques to make any headway

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

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

green functions and character sheaves purely combinatorial way using earlier ideas of p hall in the general case they were introduced 3 in terms of l adic cohomology as values of certain virtual representations of r o t is a maximal torus defined over fq at unipotent elements the characters of rg 0 at arbitrary elements were then expressible in a simple form in terms of green functions of g and of smaller groups one drawback of the definition of 3 was that it did not allow one to compute the green functions except in some simple cases another definition of green functions was proposed by springer first in terms of trigonometric sums on the lie algebra and later in geometrical terms 17 this definition was applicable only in large characteristic and in that case it was identified with the definition in 3 by springer 17 and kazhdan 10 in 12 i observed that green functions for gln can be redefined in terms of intersection cohomology this led me to a new definition of green functions

scattering matrices and dirichlet to neumann maps abstract a general representation formula for the scattering matrix of a scattering system consisting of two self adjoint operators in terms of an abstract operator valued titchmarsh weyl m function is proved this result is applied to scattering problems for different self adjoint realizations of schrodinger operators on unbounded domains schrodinger operators with singular potentials supported on hypersurfaces and orthogonal couplings of schrodinger operators in these applications the scattering matrix is expressed in an explicit form with the help of dirichlet to neumann maps

a faber krahn inequality for robin problems in any space dimension we prove a faber krahn inequality for the first eigenvalue of the laplacian with robin boundary conditions asserting that amongst all lipschitz domains of fixed volume the ball has the smallest first eigenvalue we prove the result in all space dimensions using ideas from m h bossel c r acad sci paris ser i math 302 1986 47 50 where a proof for smooth domains in the plane was given the method does not involve the use of symmetrisation arguments the results also imply variants of the cheeger inequality for the first eigenvalue

eventually and asymptotically positive semigroups on banach lattices abstract we develop a theory of eventually positive c 0 semigroups on banach lattices that is of semigroups for which for every positive initial value the solution of the corresponding cauchy problem becomes positive for large times we give characterisations of such semigroups by means of spectral and resolvent properties of the corresponding generators complementing existing results on spaces of continuous functions this enables us to treat a range of new examples including the square of the laplacian with dirichlet boundary conditions the bi laplacian on l p spaces the dirichlet to neumann operator on l 2 and the laplacian with non local boundary conditions on l 2 within the one unified theory we also introduce and analyse a weaker notion of eventual positivity which we call asymptotic positivity where trajectories associated with positive initial data converge to the positive cone in the banach lattice as t this allows us to discuss further examples which do not fall within the above mentioned framework among them a network flow with non positive mass transition and a certain delay differential equation

bases for primitive permutation groups and a conjecture of babai abstract a base of a permutation group g is a sequence b of points from the permutation domain such that only the identity of g fixes b pointwise we show that primitive permutation groups with no alternating composition factors of degree greater than d and no classical composition factors of rank greater than d have a base of size bounded above by a function of d this confirms a conjecture of babai

verification of the ordinary character table of the baby monster abstract we prove the correctness of the character table of the sporadic simple baby monster group that is shown in the atlas of finite groups

the 5 modular characters of the sporadic simple harada norton group hn and its automorphism group hn 2 abstract we determine the 5 modular character table of the sporadic simple harada norton group hn and its automorphism group hn 2 using the meat axe and condensation

presentations on standard generators for classical groups abstract for each family of finite classical groups and their associated simple quotients we provide an explicit presentation on a specific generating set of size at most 8 since there exist efficient algorithms to construct this generating set in any copy of the group our presentations can be used to verify claimed isomorphisms between representations of the classical group the presentations are available in magma

domain perturbation for elliptic equations subject to robin boundary conditions abstract we prove domain perturbation theorems for linear and nonlinear elliptic equations under robin boundary conditions the theory allows very singular perturbation of domains in particular it includes cutting holes parts degenerating to a set of measure zero such as the dumbbell problem or wildly oscillating boundaries in the last case we show that the limiting problem is the dirichlet problem

chern simons forms on principal superfiber bundles a graded weil homomorphism is defined for principal superfiber bundles and the related transgression or chern simons forms are introduced as an example of the application of these concepts a superextension of the dirac monopole is discussed

eigenvalue inequalities for the laplacian with mixed boundary conditions abstract inequalities for the eigenvalues of the negative laplacian subject to mixed boundary conditions on polyhedral and more general bounded domains are established the eigenvalues subject to a dirichlet boundary condition on a part of the boundary and a neumann boundary condition on the remainder of the boundary are estimated in terms of either dirichlet or neumann eigenvalues the results complement several classical inequalities between dirichlet and neumann eigenvalues due to polya payne levine and weinberger friedlander and others

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

reduction of inner product representations of unitary groups a direct method for the reduction of inner products of irreducible representations irreps of unitary groups has been proposed using the duality between the permutation and unitary groups a canonical tensor basis set has been used to obtain a closed expression for the clebsch gordan coefficients of u n this expression involves the subduction coefficients arising in the outer product reduction of sn1 sn2 sn1 n2 of the permutation groups the symmetrization coefficients of u n and matrix elements of the standard representation of sn the expression holds good for an inner product reduction of irreps of u n and is independent of n the method has been illustrated with examples

extension theory and kreĭn type resolvent formulas for nonsmooth boundary value problems for a strongly elliptic second order operator a on a bounded domainr n it has been known for many years how to interpret the general closed l2 realizations of a as representing boundary conditions generally nonlocal when the domain and coe cients are smooth the purpose of the present paper is to extend this representation to nonsmooth domains and coecients including the case of hol der c 3 2 smoothness in such a way that pseudodierential methods are still avail able for resolvent construc tions and ellipticity considerations we show how it can be done for domains with

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

spectral theory for perturbed krein laplacians in nonsmooth domains abstract we study spectral properties for h k the krein von neumann extension of the perturbed laplacian v defined on c 0 where v is measurable bounded and nonnegative in a bounded open set r n belonging to a class of nonsmooth domains which contains all convex domains along with all domains of class c 1 r r 1 2 in particular in the aforementioned context we establish the weyl asymptotic formula j n k j 2 n v n n 2 o n 1 2 2 as where v n n 2 n 2 1 denotes the volume of the unit ball in r n and k j j n are the non zero eigenvalues of h k listed in increasing order according to their multiplicities we prove this formula by showing that the perturbed krein laplacian i e the krein von neumann extension of v defined on c 0 is spectrally equivalent to the buckling of a clamped plate problem and using an abstract result of kozlov from the mid 1980s our work builds on that of grubb in the early 1980s who has considered similar issues for elliptic operators in smooth domains and shows that the question posed by alonso and simon in 1980 pertaining to the validity of the above weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting we also study certain exterior type domains r n k n 3 with k r n compact and vanishing bessel capacity b 2 2 k 0 to prove equality of friedrichs and krein laplacians in l 2 d n x that is c 0 has a unique nonnegative self adjoint extension in l 2 d n x

strict inequality of robin eigenvalues for elliptic differential operators on lipschitz domains abstract on a bounded lipschitz domain we consider two selfadjoint operator realizations of the same second order elliptic differential expression subject to robin boundary conditions where the coefficients in the boundary conditions are functions we prove that inequality between these functions on the boundary implies strict inequality between the eigenvalues of the two operators provided that the inequality of the functions in the boundary conditions is strict on an arbitrarily small nonempty open set

boundary condition varying circle billiards and gratings the dirichlet singularity waves in a two dimensional domain with robin mixed boundary conditions that vary smoothly along the boundary exhibit unexpected phenomena if the variation includes a d point where the boundary condition is dirichlet vanishing wavefunction a variety of arguments indicate that the system is singular for a circle billiard the boundary condition fails to determine a discrete set of levels so the spectrum is continuous for a diffraction grating defined by periodically varying boundary conditions on the edge of a half plane the phase of a diffracted beam amplitude remains undetermined in both cases the wavefunction on the boundary has a singularity at a d point described by the polylogarithm function

a local strategy to decide the alperin and dade conjectures abstract we present a new strategy which exploits both the maximal andp local subgroup structure of a given finite simple group in order to decide the alperin and dade conjectures for this group we demonstrate the computational effectiveness of this approach by using it to verify these conjectures for the conway simple group co2

a remark on schatten von neumann properties of resolvent differences of generalized robin laplacians on bounded domains in this note we investigate the asymptotic behavior of the s numbers of the resolvent difference of two generalized self adjoint maximal dissipative or maximal accumulative robin laplacians on a bounded domain with smooth boundary for this we apply the recently introduced abstract notion of quasi boundary triples and weyl functions from extension theory of symmetric operators together with krein type resolvent formulae and well known eigenvalue asymptotics of the laplace beltrami operator on it is shown that the resolvent difference of two generalized robin laplacians belongs to the schatten von neumann class of any order p for which

robin boundary value problems on arbitrary domains we develop a theory of generalised solutions for elliptic boundary value problems subject to robin boundary conditions on arbitrary domains which resembles in many ways that of the dirichlet problem in particular we establish lp lq estimates which turn out to be the best possible in that framework we also discuss consequences to the spectrum of robin boundary value problems finally we apply the theory to parabolic equations

super twistor realization of unitary super representations of u p q n this paper is concerned with demonstrating that the twistor realization of the ladder representation of u p q can be generalized to super representations of u p q n this is achieved by considering the generalization of twistor elementary states to a super algebraic category unitarity of these super representations follows from the positive definiteness of a super twistor scalar product constructed in this paper although generalizations of the ladder representations have been well studied by other means it will be shown that the super twistor generalization is especially natural and merits special investigation

determination of basis for the irreducible representations of the unitary group for u p q u p u q a direct procedure is outlined for determining the basis spanning finite dimensional irreducible representations of u p q adapted to the subgroup u p u q using a tableau based analysis it is shown that the realization of the semimaximal states follows readily from a knowledge of the matrix elements of the generators ei 1 of u p u q u p q

nonlocal robin laplacians and some remarks on a paper by filonov on eigenvalue inequalities abstract the aim of this paper is twofold first we characterize an essentially optimal class of boundary operators which give rise to self adjoint laplacians in l 2 d n x with nonlocal and local robin type boundary conditions on bounded lipschitz domains r n n n n 2 second we extend friedlander s inequalities between neumann and dirichlet laplacian eigenvalues to those between nonlocal robin and dirichlet laplacian eigenvalues associated with bounded lipschitz domains following an approach introduced by filonov for this type of problems

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

the maximal subgroups of the chevalley groups g2 q with q odd the ree groups 2g2 q and their automorphism groups

critical points of the regular part of the harmonic green function with robin boundary condition abstract in this paper we consider the green function for the laplacian in a smooth bounded domain r n with robin boundary condition g b x g 0 on and its regular part s x y where b 0 is smooth we show that in general as the robin function r x s x x has at least 3 critical points moreover in the case b const we prove that r has critical points near non degenerate critical points of the mean curvature of the boundary and when b const there are critical points of r near non degenerate critical points of b

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

a bound for the eigenvalue counting function for krein von neumann and friedrichs extensions for an arbitrary open nonempty bounded set rn n n and sufficiently smooth coefficients a b q we consider the closed strictly positive higher order differential operator a 2m a b q in l2 defined on w02m 2 associated with the differential expression 2m a b q j k 1n i j bj aj k i k bk q m m n and its krein von neumann extension ak 2m a b q in l2 denoting by n ak 2m a b q 0 the eigenvalue counting function corresponding to the strictly positive eigenvalues of ak 2m a b q we derive the bound n ak 2m a b q cvn 2 n 1 2m2m n n 2m n 2m 0 where c c a b q 0 with c in 0 0 is connected to the eigenfunction expansion of the self adjoint operator a 2m a b q in l2 rn defined on w2m 2 rn corresponding to 2m a b q here vn n 2 n 2 2 denotes the euclidean volume of the unit ball in rn our method of proof relies on variational considerations exploiting the fundamental link between the krein von neumann extension and an underlying abstract buckling problem and on the distorted fourier transform defined in terms of the eigenfunction transform of a 2 a b q in l2 rn we also consider the analogous bound for the eigenvalue counting function for the friedrichs extension af 2m a b q in l2 of a 2m a b q no assumptions on the boundary of are made

on base sizes for almost simple primitive groups abstract let g sym be a finite almost simple primitive permutation group with socle g 0 a subset of is a base for g if its pointwise stabilizer is trivial the base size of g denoted b g is the minimal size of a base we say that g is standard if g 0 a n and is an orbit of subsets or partitions of 1 n or if g 0 is a classical group and is an orbit of subspaces or pairs of subspaces of the natural module for g 0 the base size of a standard group can be arbitrarily large in general whereas the situation for non standard groups is rather more restricted indeed we have b g 7 for every non standard group g with equality if and only if g is the mathieu group m 24 in its natural action on 24 points in this paper we extend this result by classifying the non standard groups with b g 6 the main tools include recent work on bases for actions of simple algebraic groups together with probabilistic methods and improved fixed point ratio estimates for exceptional groups of lie type

elliptic differential operators on lipschitz domains and abstract boundary value problems this paper consists of two parts in the first part which is of more abstract nature the notion of quasi boundary triples and associated weyl functions is developed further in such a way that it can be applied to elliptic boundary value problems on non smooth domains a key feature is the extension of the boundary maps by continuity to the duals of certain range spaces which directly leads to a description of all self adjoint extensions of the underlying symmetric operator with the help of abstract boundary values in the second part of the paper a complete description is obtained of all self adjoint realizations of the laplacian on bounded lipschitz domains as well as kre n type resolvent formulas and a spectral characterization in terms of energy dependent dirichlet to neumann maps these results can be viewed as the natural generalization of recent results by gesztesy and mitrea for quasi convex domains in this connection we also characterize the maximal range spaces of the dirichlet and neumann trace operators on a bounded lipschitz domain in terms of the dirichlet to neumann map the general results from the first part of the paper are also applied to higher order elliptic operators on smooth domains and particular attention is paid to the second order case which is illustrated with various examples

a class of quasi linear parabolic and elliptic equations with nonlocal robin boundary conditions abstract let p 1 n r n a bounded w 1 p extension domain and let be an upper d ahlfors measure on with d n p n we show in the first part that for every p 2 n n 2 n 1 n a realization of the p laplace operator with nonlinear generalized nonlocal robin boundary conditions generates a nonlinear strongly continuous submarkovian semigroup on l 2 and hence the associated first order cauchy problem is well posed on l q for every q 1 in the second part we investigate existence uniqueness and regularity of weak solutions to the associated quasi linear elliptic equation more precisely global a priori estimates of weak solutions are obtained

regular orbits of sporadic simple groups abstract given a finite group g and a faithful irreducible fg module v where f has prime order does g have a regular orbit on v this problem is equivalent to determining which primitive permutation groups of affine type have a base of size 2 let g be a covering group of an almost simple group whose socle t is sporadic and let v be a faithful irreducible fg module where f has prime order dividing g we classify the pairs g v for which g has no regular orbit on v and determine the minimal base size of g in its action on v to obtain this classification for each non trivial g g z g we compute the minimal number of t conjugates of g generating t g

enumerating big orbits and an application b acting on the cosets of fi23 we describe a novel technique to handle big permutation domains for large groups it is applied to the multiplicity free action of the sporadic simple baby monster group on the cosets of its maximal subgroup fi23 to determine the character table of the associated endomorphism ring

green s functions for elliptic and parabolic systems with robin type boundary conditions abstract the aim of this paper is to investigate green s function for parabolic and elliptic systems satisfying a possibly nonlocal robin type boundary condition we construct green s function for parabolic systems with time dependent coefficients satisfying a possibly nonlocal robin type boundary condition assuming that weak solutions of the system are locally holder continuous in the interior of the domain and as a corollary we construct green s function for elliptic system with a robin type condition also we obtain gaussian bound for robin green s function under an additional assumption that weak solutions of robin problem are locally bounded up to the boundary we provide some examples satisfying such a local boundedness property and thus have gaussian bounds for their green s functions

constructions of fischer s baby monster over fields of characteristic not 2 abstract in this paper we describe the computer construction of the representations of fischer s baby monster simple group in 4371 dimensions over fields of 3 and 5 elements as applications we construct representations for the thompson group which will assist in determining much of the 3 modular and 5 modular character tables

the dirichlet to neumann operator via hidden compactness abstract we show that to each symmetric elliptic operator of the form a k a k l l c on a bounded lipschitz domain r d one can associate a self adjoint dirichlet to neumann operator on l 2 which may be multi valued if 0 is in the dirichlet spectrum of a to overcome the lack of coerciveness in this case we employ a new version of the lax milgram lemma based on an indirect ellipticity property that we call hidden compactness we then establish uniform resolvent convergence of a sequence of dirichlet to neumann operators whenever the underlying coefficients converge uniformly and the second order limit operator in l 2 has the unique continuation property we also consider semigroup convergence

variations on a theme of jost and pais abstract we explore the extent to which a variant of a celebrated formula due to jost and pais which reduces the fredholm perturbation determinant associated with the schrodinger operator on a half line to a simple wronski determinant of appropriate distributional solutions of the underlying schrodinger equation generalizes to higher dimensions in this multi dimensional extension the half line is replaced by an open set r n n n n 2 where has a compact nonempty boundary satisfying certain regularity conditions our variant involves ratios of perturbation determinants corresponding to dirichlet and neumann boundary conditions on and invokes the corresponding dirichlet to neumann map as a result we succeed in reducing a certain ratio of modified fredholm perturbation determinants associated with operators in l 2 d n x n n to modified fredholm determinants associated with operators in l 2 d n 1 n 2 applications involving the birman schwinger principle and eigenvalue counting functions are discussed