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

a delayed feedback control method for the lattice hydrodynamic model caused by the historic density difference effect abstract a delayed feedback control pattern is brought forward for lattice hydrodynamic model of traffic flow with the consideration of the historic density difference effect we utilize the modern control theory to obtain the linear stability conditions with the consideration of feedback control signal which shows that the stability of traffic flow is closely related to the information of the historic density difference furthermore numerical simulations affirm that the delayed feedback control signal can suppress the traffic congestion efficiently

abelian sylow subgroups in a finite group ii abstract let p 3 5 be a prime we prove that sylow p subgroups of a finite group g are abelian if and only if the class sizes of the p elements of g are all coprime to p this gives a solution to a problem posed by r brauer in 1956 for p 3 5

the development version of the chevie package of gap3 abstract i describe the current state of the development version of the chevie package which allows to transform the theories of coxeter groups reductive algebraic groups complex reflection groups hecke algebras braid monoids etc into actual computations examples are given showing the code to check some results of lusztig

vers une decomposition de jordan des blocs des groupes reductifs finis abstract let g be a reductive algebraic group over an algebraic closure of a prime field f p defined over f q with frobenius endomorphism f let g f be the subgroup of rational points let l be a prime number assume that l is different from p if g f is in duality with g f then by a theorem of m broue and j michel m broue j michel blocs et series de lusztig dans un groupe reductif fini j reine angew math 395 1989 56 67 for any l bloc b of g f there exists a unique g f conjugacy class s of l semi simple elements such that some irreducible representation of b is in the rational lusztig s series associated to s if s 1 b is said to be unipotent if g is not connected with identity component g define the unipotent l blocs of g f as the l blocks that cover some unipotent l bloc of g f assuming l is good for g we construct from g f and s some reductive algebraic group g s f defined over f q and a one to one map from the set of l blocks of g f with associated class s onto the set of unipotent l blocks of g s f such that if b corresponds to b then there is a significant height preserving one to one map from the set of irreducible representations irr b onto the set irr b the respective defect groups of b and b are isomorphic the associated brauer categories are equivalent one do not assume that the center of g is connected the class of s may be isolated in g

brauer s height conjecture for p solvable groups we complete the proof of the height conjecture for p solvable groups using the classification of finite simple groups introduction the height conjecture is the statement that a p block of a finite group has an abelian defect group if and only if all ordinary irreducible characters in the block have height zero while a proof of this conjecture for general finite groups seems remote considerable progress has been made toward proving it for p solvable groups fong 5 proved that all characters in a block with abelian defect group have height zero in a p solvable group and he proved the converse direction for the principal block 5 and for solvable groups in the case that p is the largest prime divisor of the group order 6 recently 24 8 the converse direction has been established for all solvable groups in this paper we prove the converse direction for all p solvable groups assuming the classification of finite simple groups in its general outline this paper resembles 8 where we proved the height conjecture for solvable groups the reader is assumed to have some familiarity with 8 now we state our main results the analogs of the main results of 8 theorem a suppose that n g that g n is p solvable that t e irr n and that p 1 x i w l for all x e irr g i p then the p sylow subgroups of g n are abelian theorem b let b be a p block of a p solvable group with defect group d if every ordinary irreducible character in b has height zero then d is abelian theorem c suppose that n g that g n is p solvable and that w e irr n suppose that e is an integer such that pe l does not divide x i w l for all x e irr g ip then the derived length of a p sylow subgroup of g n is at most 2e 1 theorem d let b and d be as in theorem b if every ordinary irreducible character in b has height at most e then the derived length of d is at most 2e 1 received by the editors january 10 1983 1980 mathematics subject classification primary 20c20 20c 15 1984 american mathematical society 0002 9947 84 1 00 25 per page

solvability of singular second order m point boundary value problems abstract let f 0 1 r 2 r be a function satisfying caratheodory s conditions and 1 t e t l 1 0 1 let i 0 1 a i r i 1 m 2 0 1 2 m 2 1 be given this paper is concerned with the problem of existence of a c 1 0 1 solution for the m point boundary value problem x f t x t x t e t 0 t 1 0 1 x 0 0 x 1 i 1 m 2 a i x i the proof of our main result is based upon the leray schauder continuation theorem

on heights of characters of finite groups abstract in this paper we first state a conjecture on the lower bound of the maximal height of characters in a p block of a finite group then we show that our conjecture holds for all blocks of covering groups of a sporadic simple group for all blocks of a quasi simple group g with g z g isomorphic to a 6 a 7 or a simple group of lie type with an exceptional covering group for all blocks of a symmetric group as well as for all blocks of finite general linear or unitary groups for the proof of the case with symmetric groups it relates to an open question of olsson on the existence of t core partitions where t 3 is an integer as a byproduct our investigation on heights of characters of the symmetric groups and of the general linear or unitary groups also gives evidence for the isaacs moreto navarro tiep conjecture claiming that the number of distinct irreducible character degrees of a sylow p subgroup p of an arbitrary finite group g is at most one more than the number of irreducible character degrees of g that are multiples of p

nilpotent blocks of quasisimple groups for odd primes we investigate the nilpotent blocks of positive defect of the quasisimple groups for odd primes in particular it is shown that every nilpotent block of a quasisimple group has abelian defect groups a conjecture of puig concerning the recognition of nilpotent blocks is also shown for these groups

conjectures of alperin and broue for 2 blocks with elementary abelian defect groups of order 8 using the classification of finite simple groups we prove alperin s weight conjecture and the character theoretic version of broue s abelian defect group conjecture for 2 blocks of finite groups with an elementary abelian defect group of order 8

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

brauer s height zero conjecture for quasi simple groups abstract we show that brauer s height zero conjecture holds for blocks of finite quasi simple groups this result is used in navarro spath s reduction of this conjecture for general groups to the inductive alperin mckay condition for simple groups

a novel lattice hydrodynamic model accounting for driver s memory effect and the difference of optimal velocity on curved road abstract a novel lattice hydrodynamic model is put forward which takes the effects of driver s memory and difference of optimal velocity into account on curved road linear stability analysis for the novel model is discussed and the stability condition is deduced through nonlinear analysis the mkdv equation is deduced to explore the evolution of jams near the vertex the exact solution of the mkdv equation is also derived the influence of the above two factors on traffic stability are investigated by numerical examples both numerical and analytical results demonstrate that memory effect and the difference of optimal velocity on curved road can ease traffic jams efficiently

characters of relative p degree over normal subgroups let z be a normal subgroup of a finite group g let irr z be an irreducible complex character of z and let p be a prime number if p does not divide the integers 1 1 for all irr g lying over then we prove that the sylow p subgroups of g z are abelian this theorem which generalizes the gluck wolf theorem to arbitrary finite groups is one of the principal obstacles to proving the celebrated brauer height zero conjecture

defect zero p blocks for finite simple groups we classify those finite simple groups whose brauer graph or decomposition matrix has a p block with defect 0 completing an investigation of many authors the only finite simple groups whose defect zero p blocks remained unclassified were the alternating groups an here we show that these all have a p block with defect 0 for every prime p 5 this follows from proving the same result for every symmetric group sn which in turn follows as a consequence of the t core partition conjecture that every non negative integer possesses at least one t core partition for any t 4 for t 17 we reduce this problem to lagrange s theorem that every non negative integer can be written as the sum of four squares the only case with t 17 that was not covered in previous work was the case t 13 this we prove with a very different argument by interpreting the generating function for t core partitions in terms of modular forms and then controlling the size of the coefficients using deligne s theorem nee the weil conjectures we also consider congruences for the number of p blocks of sn proving a conjecture of garvan that establishes certain multiplicative congruences when 5 p 23 by using a result of serre concerning the divisibility of coefficients of modular forms we show that for any given prime p and positive integer m the number of p blocks with defect 0 in sn is a multiple of m for almost all n we also establish that any given prime p divides the number of p modularly irreducible representations of sn for almost all n 1996 american mathematical society

the decomposition of lusztig induction in classical groups abstract we give a short combinatorial proof of asai s decomposition formula for lusztig induction of unipotent characters in groups of classical type relying solely on the mackey formula

an extended macro traffic flow model accounting for the driver s bounded rationality and numerical tests abstract in this paper we propose a macro traffic flow model to explore the effects of the driver s bounded rationality on the evolutions of traffic waves which include shock and rarefaction waves and small perturbation and on the fuel consumption and emissions that include co hc and no x during the evolution process the numerical results illustrate that considering the driver s bounded rationality can prominently smooth the wavefront of the traffic waves and improve the stability of traffic flow which shows that the driver s bounded rationality has positive impacts on traffic flow but considering the driver s bounded rationality reduces the fuel consumption and emissions only at the upstream of the rarefaction wave while enhances the fuel consumption and emissions under other situations which shows that the driver s bounded rationality has positive impacts on the fuel consumption and emissions only at the upstream of the rarefaction wave while negative effects on the fuel consumption and emissions under other situations in addition the numerical results show that the driver s bounded rationality has little prominent impact on the total fuel consumption and emissions during the whole evolution of small perturbation

the stabilization effect of the density difference in the modified lattice hydrodynamic model of traffic flow abstract a modified lattice hydrodynamic model of traffic flow is proposed by introducing the density difference between the leading and the following lattice the stability condition of the modified model is obtained through the linear stability analysis the results show that considering the density difference leads to the stabilization of the system the burgers equation and mkdv equation are derived to describe the density waves in the stable and unstable regions respectively numerical simulations show that considering the density difference not only could stabilize traffic flow but also makes the lattice hydrodynamic model more realistic

positive solutions of multi point boundary value problems at resonance abstract we establish new results on the existence of positive solutions for some multi point boundary value problems at resonance our results are based on a recent leggett williams norm type theorem due to o regan and zima we also derive a new result for a three point problem previously studied by several authors

lattice hydrodynamic modeling of two lane traffic flow with timid and aggressive driving behavior abstract in this paper a new two lane lattice hydrodynamic traffic flow model is proposed by considering the aggressive or timid characteristics of driver s behavior the effect of driver s characteristic on the stability of traffic flow is examined through linear stability analysis it is shown that for both the cases of lane changing or without lane changing the stability region significantly enlarges reduces as the proportion of aggressive timid drivers increases to describe the propagation behavior of a density wave near the critical point nonlinear analysis is conducted and mkdv equation representing kink antikink soliton is derived the effect of anticipation parameter with more aggressive timid drivers is also investigated and found that it has a positive negative effect on the stability of two lane traffic flow dynamics simulation results are found consistent with the theoretical findings which confirm that the driver s characteristics play a significant role in a two lane traffic system

brauer s height zero conjecture for two primes let p and q be two primes we propose that brauer s height zero conjecture for the principal p blocks of finite groups can naturally be extended from the perspective of q we prove one direction of this new conjecture and show the reverse direction assuming that the inductive alperin mckay condition holds for the finite simple groups

on symmetric quotients of symmetric algebras we investigate symmetric quotient algebras of symmetric algebras with an emphasis on finite group algebras over a complete discrete valuation ring o using elementary methods we show that if an ordinary irreducible character of a finite group g gives rise to a symmetric quotient over o which is not a matrix algebra then the decomposition numbers of the row labelled by are all divisible by the characteristic p of the residue field of o

multiple positive solutions of resonant and non resonant nonlocal boundary value problems abstract we study the existence of positive solutions for equations of the form u t f t u t a e t 0 1 when f t u 2 u 0 for u 0 for some constant 0 and for the perturbed equation u t 2 u t h t u t a e t 0 1 when h t u 0 subject to various non local boundary conditions in particular we establish existence and multiplicity of positive solutions for non perturbed boundary value problems at resonance by considering equivalent non resonant perturbed problems with the same boundary conditions

nilpotent and abelian hall subgroups in finite groups the research of the first second fourth fifth and sixth authors was supported by the prometeo generalitat valenciana proyectos mtm2010 15296 mtm2010 19938 c03 02 fundacio bancaixa p11b2010 47 and fondos feder the third author gratefully acknowledges financial support by erc advanced grant 291512 the seventh author was supported by the nsa grant h98230 13 1 0229 the eighth author gratefully acknowledges the support of the nsf grants dms 0901241 and dms 1201374

a novel lattice hydrodynamic model considering the optimal estimation of flux difference effect on two lane highway abstract the optimal estimation of flux difference effect is introduced to construct a new lattice hydrodynamic model for two lane highway through linear stability analysis it is found that the new consideration plays an important influence upon the stability of two lane traffic flow moreover the mkdv equation near the critical point is derived from nonlinear analysis which describes the propagation behavior of traffic jam in two lane traffic system simulation results show that the stability of two lane traffic system can be increased and the emergence of traffic jams are effectively relieved by the optimal estimation of flux difference effect which is in good agreement with theoretical analysis

nonlinear analysis of a new lattice hydrodynamic model with the consideration of honk effect on flux for two lane highway abstract influence of the honk on traffic flux is taken to construct a new lattice model for two lane freeway the linear stability analysis is applied to obtain the stability condition involving the honk effect on traffic flux in two lane system nonlinear analysis is implemented to get the mkdv equation for the phase transition of two lane lattice traffic flow numerical simulation is worked out to authenticate that the impact of honk on traffic flux strengthens the stability of traffic flow and suppresses the traffic congestion effectively under lane changing in two lane lattice traffic flow which is accordance with the analytic results

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

modified kdv equation for jamming transition in the continuum models of traffic abstract continuum models of traffic are proposed to describe the jamming transition in traffic flow on a highway they are the simplified versions of the hydrodynamic model of traffic two continuum models are presented one is described by the partial differential equations and the other is the discrete lattice version the linear stability theory and the nonlinear analysis are applied to the continuum models the modified korteweg de vries equation kdv near the critical point is derived using the reduction perturbation method it is shown that the jamming transition and the density wave in the congested traffic flow are described by the modified kdv equation the solutions of the kdv equations obtained from the two models are compared with that of the optimal velocity model car following model

existence of solutions to functional boundary value problem of second order nonlinear differential equation abstract in this paper a functional boundary value problem is studied based on mawhin s coincidence degree theory some existence theorems are obtained in the case of non resonance and the cases of dim ker l 1 and dim ker l 2 at resonance the results not only generalize and improve some known results of multi point and integral boundary value problems but also give some existence theorems for boundary value problems that all their boundary value conditions are relied on both x and x

a new lattice model accounting for multiple optimal current differences anticipation effect in two lane system abstract this paper extends a two lane lattice hydrodynamic traffic flow model to take into account the driver s anticipation effect in sensing the multiple optimal current differences based on the proposed model we derive analytically the effect of driver s anticipation of multiple optimal current differences on the instability of traffic dynamics the phase diagrams have been plotted and discussed that the stability region enhances with anticipation effect in sensing multiple optimal current differences through simulation it is found that the oscillation of density wave around critical density decreases with an increase in lattice number and anticipation time for transient and steady state the simulation results are in good agreement with the theoretical analysis which show that considering the driver s anticipation of multiple optimal current differences in two lane lattice model stabilizes the traffic flow and suppresses the traffic jam efficiently

stabilization effect of traffic flow in an extended car following model based on an intelligent transportation system application an extended car following model is proposed by incorporating an intelligent transportation system in traffic the stability condition of this model is obtained by using the linear stability theory the results show that anticipating the behavior of more vehicles ahead leads to the stabilization of traffic systems the modified korteweg de vries equation the mkdv equation for short near the critical point is derived by applying the reductive perturbation method the traffic jam could be thus described by the kink antikink soliton solution for the mkdv equation from the simulation of space time evolution of the vehicle headway it is shown that the traffic jam is suppressed efficiently with taking into account the information about the motion of more vehicles in front and the analytical result is consonant with the simulation one

quasi isolated blocks and brauer s height zero conjecture this paper has two main results firstly we complete the parametrisation of all p blocks of nite quasi simple groups by nding the so called quasi isolated blocks of exceptional groups of lie type for bad primes this relies on the explicit decomposition of lusztig induction from suitable levi subgroups our second major result is the proof of one direction of brauer s long standing height zero conjecture on blocks of nite groups using the

jamming transitions and the modified korteweg de vries equation in a two lane traffic flow abstract the two lattice models are presented to simulate the traffic flow on a two lane highway they are the lattice versions of the hydrodynamic model of traffic the one model a is described by the differential difference equation where time is a continuous variable and space is a discrete variable and the other model b is the difference equation in which both time and space variables are discrete the jamming transitions among the freely moving phase the coexisting phase and the uniform congested phase are studied by using the nonlinear analysis and the computer simulation the modified korteweg de vries mkdv equations are derived from the lattice models near the critical point the traffic jam is described by a kink antikink solution obtained from the mkdv equation it is found that the critical point the coexisting curve and the neutral stability line decrease with increasing the rate of lane changing also the computer simulation is performed for the model b it is shown that the coexisting curves obtained from the mkdv equation are consistent with the simulation result

2 blocks with abelian defect groups we give a classification up to morita equivalence of 2 blocks of quasi simple groups with abelian defect groups as a consequence we show that donovan s conjecture holds for elementary abelian 2 groups and that the entries of the cartan matrices are bounded in terms of the defect for arbitrary abelian 2 groups we also show that a block with defect groups of the form c2m c2m for m 2 has one of two morita equivalence types and hence is morita equivalent to the brauer correspondent block of the normaliser of a defect group this completes the analysis of the morita equivalence types of 2 blocks with abelian defect groups of rank 2 from which we conclude that donovan s conjecture holds for such 2 groups a further application is the completion of the determination of the number of irreducible characters in a block with abelian defect groups of order 16 the proof uses the classification of finite simple groups

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

on the brauer feit bound for abelian defect groups we improve the brauer feit bound on the number of irreducible characters in a p block for abelian defect groups by making use of halasi and podoski every coprime linear group admits a base of size two http arxiv org abs 1212 0199v1 7 and kessar and malle ann math 178 2 321 384 11 we also prove brauer s k b conjecture for 2 blocks with abelian defect groups of rank at most 5 and 3 blocks and 5 blocks with abelian defect groups of rank at most 3

equivariance and extendibility in finite reductive groups with connected center we show that several character correspondences for finite reductive groups g are equivariant with respect to group automorphisms under the additional assumption that the linear algebraic group associated to g has connected center the correspondences we consider are the so called jordan decomposition of characters introduced by lusztig and the generalized harish chandra theory of unipotent characters due to broue malle michel in addition we consider a correspondence giving character extensions due to the second author in order to verify the inductive mckay condition from isaacs malle navarro for the non abelian finite simple groups of lie types 3 mathsf d 4 mathsf e 8 mathsf f 4 2 mathsf f 4 and mathsf g 2

a new two lane lattice model by considering predictive effect in traffic flow abstract in this paper a new hydrodynamic lattice model is proposed by considering the predictive effect in traffic flow and the optimal current difference effect ocde on two lane unidirectional traffic system the model is further analyzed theoretically by the means of stability analysis the influence of the predictive effect with the consideration of ocde is analyzed on traffic stream stability through linear stability analysis in a two lane system when lane changing is allowed the mkdv equation is derived by using nonlinear stability analysis and the density wave in terms of the kink antikink wave is obtained around the critical point it is observed that the predictive effect plays an important role to enhance the stability of traffic flow in a two lane system numerical simulation verifies the theoretical predictions which confirms that the traffic jam can be suppressed more effectively by taking predictive effect in a traffic system when lane changing is permitted

on the existence of a three point boundary value problem at resonance in rn abstract we apply a coincidence degree theorem due to mawhin to obtain a sufficient condition for the existence of at least one solution for a class of nonlinear second order differential equations in r n associated with three point boundary conditions

on the number of simple modules in a block of a finite group abstract we prove that if b is a p block with non trivial defect group d of a finite p solvable group g then l b p r where r is the sectional rank of d we remark that there are infinitely many p blocks b with non abelian defect groups and l b p r 1 we conjecture that the inequality l b p r holds for an arbitrary p block with defect group of sectional rank r we show this to hold for a large class of p blocks of various families of quasi simple and nearly simple groups

sp6 2a is good for the mckay alperin weight and related local global conjectures abstract the so called local global conjectures in the representation theory of finite groups relate the representation theory of g to that of certain proper subgroups such as the normalizers of particular p groups recent results by several authors reduce some of these conjectures to showing that a certain collection of stronger conditions holds for all finite simple groups here we show that g sp 6 2 a is good for these reductions for the mckay conjecture the alperin weight conjecture and their blockwise versions

on cartan matrices and lower defect groups for covering groups of symmetric groups we determine the elementary divisors of the cartan matrices of spin p blocks of the covering groups of the symmetric groups when p is an odd prime as a consequence we also compute the determinants of these cartan matrices and in particular we confirm a conjecture by brundan and kleshchev that these determinants depend only on the weight but not on the sign of the block

a singular non local problem at resonance abstract we study a second order non local problem using the coincidence degree theory we show the existence of a solution whose derivative is singular at the right end point of the interval which is a new result for any resonant non local problem under the non local condition we find a general way to ensure that the differential operator is a fredholm operator of index zero

full velocity difference model for a car following theory in this paper we present a full velocity difference model for a car following theory based on the previous models in the literature to our knowledge the model is an improvement over the previous ones theoretically because it considers more aspects in car following process than others this point is verified by numerical simulation then we investigate the property of the model using both analytic and numerical methods and find that the model can describe the phase transition of traffic flow and estimate the evolution of traffic congestion

rationality of finite group representations and local subgroups abstract we show that local fields of values and local schur indices of irreducible characters in a block of a finite group are controlled by local subgroups a method for computing certain schur indices and an application for the prime 2 are given

on restriction of characters to defect groups abstract we put forward a blockwise version of a recent conjecture of 6 on finite groups let b be a p block of a finite group g with defect group d let irr b be a character with positive height in this note we conjecture that the number of distinct linear constituents of the restriction d is 0 or at least p we prove that this is indeed the case for various classes of finite groups and p blocks

computational proof of the mackey formula for q 2 let g be a connected reductive group defined over a finite field with q elements we prove that the mackey formula for the lusztig induction and restriction holds in g whenever q 2 or g does not have a component of type e

a note on restriction of characters of alternating groups to sylow subgroups abstract we restrict irreducible characters of alternating groups of degree divisible by p to their sylow p subgroups and study the number of linear constituents

on the morita frobenius numbers of blocks of finite reductive groups abstract we show that the morita frobenius number of the blocks of the alternating groups the finite groups of lie type in defining characteristic and the ree and suzuki groups is 1 we also show that the morita frobenius number of almost all of the unipotent blocks of the finite groups of lie type in non defining characteristic is 1 and that in the remaining cases it is at most 2

perfect isometries and murnaghan nakayama rules acknowledgement the authors wish to thank the referee for a very careful and precise reading of several earlier versions of this manuscript they are grateful for the number quality and helpfulness of comments and suggestions received

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

large varvec p prime orbits for varvec p p nilpotent linear groups let g be a p nilpotent linear group on a finite vector space v of characteristic p suppose that g v is odd let p be a sylow p subgroup of g we show that there exist vectors v 1 and v 2 in v such that c g v 1 cap c g v 2 p a striking conjecture of malle and navarro offers a simple global criterion for the nilpotence in the sense of broue and puig of a p block of a finite group our result implies that this conjecture holds for groups of odd order

on unipotent blocks and their ordinary characters there is a great deal of evidence suggesting that block theory for finite reductive groups should lead to an enrichment of the harish chandra theory for those groups see s 5 1 fs2 p2 3 4 h 6 1 10 an e generalized harish chandra theory for e a positive integer is introduced in bmm i a the context is as follows let g be a connected reductive group defined over lfq with associated frobenius endomorphism f let e 1 an integer when l is an f stable levi subgroup of g one denotes by r and re the lusztig functor and its adjoint a unipotent e pair l l is defined by an e split levi subgroup l of g and a unipotent character z d l 1 one writes li z1 l2 j 2 if and only if l1 c l 2 and zl l2 l 1 2 l 4 0 see 3 l the notion of unipotent e cuspidal pairs and the resulting partition of g g v 1 are defined as in the classical case corresponding to e 1 if f is a prime not dividing q and e is the order of q mod f it seems natural in connection with blocks to ask if the partition of the unipotent characters 8 g v l resulting from e theory coincides with f blocks the blocks of g v which are defined by unipotent characters are called unipoteut f blocks we prove

2 blocks whose defect group is homocyclic and whose inertial quotient contains a singer cycle abstract we consider a block b of a finite group with defect group d c 2 m n and inertial quotient e containing a singer cycle an element of order 2 n 1 this implies e e f where e c 2 n 1 f c n and e acts transitively on the elements in d of order 2 we classify the basic morita equivalence classes of b over a complete discrete valuation ring o when m 1 b is basic morita equivalent to the principal block of one of s l 2 2 n f d e or j 1 where j 1 occurs only when n 3 when m 1 b is basic morita equivalent to d e