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

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

eigenvalues of cartan matrices of principal 3 blocks of finite groups with abelian sylow 3 subgroups abstract let a be the principal 3 block of a finite group g with an abelian sylow 3 subgroup p let c a be the cartan matrix of a and we denote by c a the unique largest eigenvalue of c a the value c a is called the frobenius perron eigenvalue of c a we shall prove that c a is a rational number if and only if a and the principal 3 block of n g p are morita equivalent this generalizes earlier wada s theorem in 2007 where he proves it only for the case that the order of p is nine while we prove it for the case that p is an arbitrary finite abelian 3 group the result presented here uses the classification of finite simple groups

on vertices of simple modules for symmetric groups of small degrees abstract we present some algorithmic methods for the computation of vertices of indecomposable and simple modules over group algebras in prime characteristic furthermore we apply these to the simple modules of the symmetric groups and determine the vertices of all simple modules of the symmetric groups of degree at most 14 and 15 in characteristic 2 and 3 respectively with one exception

vertex operators symmetric functions and the spin group γn abstract this work provides a vertex operator approach to the symmetric group sn and its double covering group n by generalizing a result of frenkel and sato for sn we formulate a correspondence between the space v of certain twisted vertex operators the ring of symmetric functions over q 2 and the space of nontrivial irreducible characters of n under this identification we show that a distinguished orthogonal basis of v corresponds to the set of nontrivial irreducible characters of n where both are parametrized by partitions with odd integer parts the counterpart of this distinguished basis in the ring over q 2 is the set of schur s q functions which are loosely speaking the square roots of the schur functions the nontrivial part of the character table of n is shown to be given by certain matrix coefficients in v

irreducible specht modules are signed young modules recently donkin defined signed young modules as a simultaneous generalization of young and twisted young modules for the symmetric group we show that in odd characteristic if a specht module s is irreducible then s is a signed young module thus the set of irreducible specht modules coincides with the set of irreducible signed young modules this provides evidence for our conjecture that the signed young modules are precisely the class of indecomposable self dual modules with specht filtrations the theorem is false in characteristic two

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

on hilbert divisors of brauer characters abstract the hilbert divisor p a of an irreducible p brauer character of a finite group g carries deep information about respectively the module affording in 8 we conjectured that belongs to a p block of defect 0 if and only if its hilbert divisor is 1 in this note we continue our investigations

the vertices of the simple modules of m12 over a field of characteristic 2 abstract in 9 james determined the modular characters the decomposition and the cartan matrices of the mathieu groups it is the aim of this paper to describe the vertices and green correspondents of the simple modules of m 12 over a field of characteristic 2 it turns out that the defect groups of the blocks are also the vertices of the simple modules in these blocks furthermore there is a simple module with a green correspondent whose socle is not multiplicity free

quantum α determinant cyclic modules of uq gln abstract as a particular one parameter deformation of the quantum determinant we introduce a quantum determinant det q and study the u q gl n cyclic module generated by it we show that the multiplicity of each irreducible representation in this cyclic module is determined by a certain polynomial called the q content discriminant a part of the present result is a quantum counterpart for the result of matsumoto and wakayama s matsumoto m wakayama alpha determinant cyclic modules of gl n c j lie theory 16 2006 393 405 however a new distinguished feature arises in our situation specifically we determine the degeneration of the multiplicities for classical singular points and give a general conjecture for singular points involving semi classical and quantum singularities moreover we introduce a quantum permanent per q and establish another conjecture which describes a reciprocity between the multiplicities of the irreducible summands of the cyclic modules generated respectively by det q and per q

on equivalences between blocks of group algebras reduction to the simple components abstract a conjecture of michel broue states that if d is an abelian sylow p subgroup of a finite group g and h n g d then the principal blocks of g and h are rickard equivalent the structure of groups with abelian sylow p subgroups as determined by p fong and m e harris raises the following question assuming that broue s conjecture holds for the simple components of g under what conditions does it hold for g itself due to the structure of g this problem requires mainly the lifting of rickard complexes to p extensions of the simple components and the construction of complexes over wreath products we give here these reduction steps which may be regarded as a clifford theory of tilting complexes

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

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

eigenvalues of cartan matrices of principal 2 blocks with abelian defect groups abstract let g be a finite group with an abelian sylow 2 subgroup p let c b be the cartan matrix of the principal 2 block b of g we show that the frobenius perron eigenvalue b of c b is a rational integer if and only if b and its brauer correspondent block b of n g p are morita equivalent by using a classification of finite simple groups with an abelian sylow 2 subgroup in this case we can take the brauer character table b of b as a unimodular eigenvector matrix u b of c b over a complete discrete valuation ring r

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 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

source algebras of blocks sources of simple modules and a conjecture of feit abstract we verify a finiteness conjecture of feit on sources of simple modules over group algebras for various classes of finite groups related to the symmetric groups

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

generalized group characters and complex oriented cohomology theories let bg be the classifying space of a finite group g given a multiplicative cohomology theory e the assignment g 7i e bg is a functor from groups to rings endowed with induction transfer maps in this paper we investigate these functors for complex oriented cohomology theories e using the theory of complex representations of finite groups as a model for what one would like to know an analogue of artin s theorem is proved for all complex oriented e the abelian subgroups of g serve as a detecting family for e bg modulo torsion dividing the order of g when e is a complete local ring with residue field of characteristic p and associated formal group of height n we construct a character ring of class functions that computes 1 e bg the domain of the characters is gn p the set of n tuples of elements in g each of which has order a power of p a formula for induction is also found the ideas we use are related to the lubin tate theory of formal groups the construction applies to many cohomology theories of current interest completed versions of elliptic cohomology e n theory etc the nth morava k theory euler characteristic for bg is computed to be the number of g orbits in gn p for various groups g including all symmetric groups we prove that k n bg concentrated in even degrees our results about e bg extend to theorems about e eg gx where x is a finite g cw complex

a categorification of twisted heisenberg algebras abstract we categorify a quantized twisted heisenberg algebra associated to a finite subgroup of sl 2 c

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

on schur algebras and related algebras ii this paper is a continuation of 7 the purpose of 7 was to set up in the context of reductive algebraic groups a theory of generalized schur algebras a special case of our construction gives the schur algebras s n r studied extensively in lo we concentrate here on the schur algebras themselves and show that our construction yields new results in their representation theory by means of the schur functor see 10 chap 6 we obtain new results in the representation theory of the symmetric groups the main results are as follows the schur algebras s n r defined over a principal ideal domain r have finite global dimension see also 1 i over a field or complete valuation ring r every finitely generated projective s n r module has a weyl module filtration we prove the irreducibility of the socle of certain weyl modules see also 15 we describe a connection between weights of projective indecomposable modules for the schur algebras in terms of the involution on irreducible representations of symmetric groups given by tensoring with the sign representation we show that over a complete valuation ring or a field every young module for a symmetric group has a specht series in particular the projective indecomposables have specht series finally we show that for r n and a complete valuation ring or field r the blocks of the schur algebra s n r are given by the rule embodied in nakayama s conjecture the notations and conventions are as in the first part of this work 7

the broue conjecture for the faithful 3 blocks of 4 m22 abstract we verify broue s conjecture for the faithful 3 blocks of defect 2 of the non split central extension of the sporadic simple mathieu group m 22 by a cyclic group of order 4 the proof is based on a strategy due to okuyama and rickard where a stable equivalence is lifted to a derived equivalence the stable equivalence in turn is provided by exploiting a result due to puig to handle this particular example next to theoretical investigations we apply a whole bunch of computational tools

twisted orbifold k theory we use equivariant methods to define and study the orbifold k theory of an orbifold x adapting techniques from equivariant k theory we construct a chern character and exhibit a multiplicative decomposition for k orb x in particular showing that it is additively isomorphic to the orbifold cohomology of x a number of examples are provided we then use the theory of projective representations to define the notion of twisted orbifold k theory in the presence of discrete torsion an explicit expression for this is obtained in the case of a global quotient

vertices sources and green correspondents of the simple modules for the large mathieu groups we investigate the simple modules for the sporadic simple mathieu groups m22 m23 and m24 as well as those of the automorphism group the covering groups and the bicyclic extensions of m22 in characteristics 2 and 3 we determine the vertices and sources as well as the green correspondents of these simple modules we also find two 3 blocks with elementary abelian defect groups of order 9 in these groups which are morita equivalent to their brauer correspondents

vertices of specht modules and blocks of the symmetric group abstract this paper studies the vertices in the sense defined by j a green of specht modules for symmetric groups the main theorem gives for each indecomposable non projective specht module a large subgroup contained in one of its vertices a corollary of this theorem is a new way to determine the defect groups of symmetric groups the main theorem is also used to find the green correspondents of a particular family of simple specht modules as a corollary this gives a new proof of the brauer correspondence for blocks of the symmetric group the proof of the main theorem uses the brauer homomorphism on modules as developed by m broue together with combinatorial arguments using young tableaux

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 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

cartan matrices and morita equivalence for blocks of the symmetric groups theorem 1 the blocks of the symmetric groups with a given weight w 0 over a field of characteristic p can be collected into families according to the shapes of their cores each family consists of morita equivalent blocks the number of such families is at most flf 1 i l w 1 11 and each family contains a block of some g where n is less than or equal to p2 p 1 2 w 1 2 4 wp

on some trivial source specht modules abstract the paper presented here focuses on the classification of trivial source specht modules we completely classify the trivial source specht modules labelled by hook partitions we also classify the trivial source specht modules labelled by two part partitions in the odd characteristic case moreover in the even characteristic case we prove a result for the classification of the trivial source specht modules labelled by partitions with 2 weight 2 which justifies a conjecture of 16

rationality of eigenvalues of cartan matrices in finite groups abstract we study the case in which eigenvalues and elementary divisors of a cartan matrix of a p block b of a finite group coincide in several cases we prove the coincidence occurs if and only if the perron frobenius eigenvalue of the cartan matrix is equal to the order of a defect group of b

the duistermaat heckman formula with application to circle actions and poincare q polynomials in twisted equivariant k theory in this paper we deduce the sketch of proof of the duistermaat heckman formula and investigate how the known duistermaat heckman result could be specialized to the symplectic structure on the orbit space the theorems of localization in equivariant cohomology not only provide us with beautiful mathematical formulas and stimulate achievements in algorithmic computations but also promote progress in theoretical and mathematical physics we present the elliptic genera and the characteristic q series for the circle actions and twisted equivariant k theory with the case of the symmetric group of n symbols separately analyzed we show that the poincare q polynomials admit presentation in terms of the patterson selberg or the ruelle type spectral functions in this paper we deduce the sketch of proof of the duistermaat heckman formula and investigate how the known duistermaat heckman result could be specialized to the symplectic structure on the orbit space the theorems of localization in equivariant cohomology not only provide us with beautiful mathematical formulas and stimulate achievements in algorithmic computations but also promote progress in theoretical and mathematical physics we present the elliptic genera and the characteristic q series for the circle actions and twisted equivariant k theory with the case of the symmetric group of n symbols separately analyzed we show that the poincare q polynomials admit presentation in terms of the patterson selberg or the ruelle type spectral functions

on blocks with trivial source simple modules abstract motivated by an observation in danz and kulshammer 2009 3 we determine the source algebra and therefore all the structure of the blocks without essential brauer pairs where the simple modules of all the brauer corespondents have trivial sources

eigenvector matrices of cartan matrices for finite groups abstract let c b be the cartan matrix of a p block b of a finite group g we show that there is a unimodular eigenvector matrix u b of c b over a discrete valuation ring r if all eigenvalues of c b are integers when b is a cyclic block a tame block a p block of a p solvable group or the principal 3 block with elementary abelian defect group of order 9

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

twisted vertex representations and spin characters we establish a new group theoretic realization of the basic representations of the twisted affine and twisted toroidal algebras in the same spirit of our new approach to the mckay correspondence our vertex operator construction provides a unified description to the character tables for the spin cover of the wreath product of the twisted hyperoctahedral groups and an arbitrary finite group

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

random point fields associated with certain fredholm determinants i fermion poisson and boson point processes we introduce certain classes of random point fields including fermion and boson point processes which are associated with fredholm determinants of certain integral operators and study some of their basic properties limit theorems correlation functions palm measures etc also we propose a conjecture on an analogue of the determinant and permanent

equivariant k theory generalized symmetric products and twisted heisenberg algebra for a space x acted on by a finite group the product space xn affords a natural action of the wreath product n n sn the direct sum of equivariant k groups were shown earlier by the author to carry several interesting algebraic structures in this paper we study the k groups of n equivariant clifford supermodules on xn we show that is a hopf algebra and it is isomorphic to the fock space of a twisted heisenberg algebra twisted vertex operators make a natural appearance the algebraic structures on x when is trivial and x is a point specialize to those on a ring of symmetric functions with the schur q functions as a linear basis as a by product we present a novel construction of k theory operations using the spin representations of the hyperoctahedral groups

alpha determinant cyclic modules and jacobi polynomials for positive integers n and l we study the cyclic u gl n module generated by the l th power of the determinant det x this cyclic module is isomorphic to the n th tensor space s l n n of the symmetric l th tensor space of n for all but finitely many exceptional values of if is exceptional then the cyclic module is equivalent to a proper submodule of s l n n i e the multiplicities of several irreducible subrepresentations in the cyclic module are smaller than those in s l n n the degeneration of each isotypic component of the cyclic module is described by a matrix whose size is given by a kostka number and whose entries are polynomials in with rational coefficients in particular we determine the matrix completely when n 2 in this case the matrix becomes a scalar and is essentially given by a classical jacobi polynomial moreover we prove that these polynomials are unitary in the appendix we consider a variation of the spherical fourier transformation for o nl o n l as a main tool for analyzing the same problems and describe the case where n 2 by using the zonal spherical functions of the gelfand pair o 2l o 2 l