on a theorem of benson and curtis iwahori conjectured 5 that the hecke algebra over q of a chevalley group g over a finite field f with respect to a bore1 subgroup is isomorphic to the group algebra over q of the weyl group the analogous statement with q replaced by its algebraic closure was proved by tits 3 ex 27 p 561 using a deformation argument in 11 benson and curtis proved using a case by case analysis that iwahori s conjecture is true over q whenever g is simple of type ze e with the help of springer they proved that when g is of type e or e there exists irreducible representations of the hecke algebra whose character involve 4 and hence iwahori s conjecture over q is false in general the main result of this paper is an explicit construction of an isomorphism of the hecke algebra over q q with the group algebra over q q of the weyl group the construction makes essential use of the polynomials p w of 8 at some point in the proof i have to appeal to the interpretation of p l in terms of multiplicities in verma modules 8 i 51 which has recently been established by brylinski and kashiwara 4 and independently by belinson and bernstein

on the existence of a w graph for an irreducible representation of a coxeter group on montre que toute representation irreductible d un groupe de coxeter fini peut etre fournie par un graphe w

hecke algebras of finite type are cellular let mathcal h be the one parameter hecke algebra associated to a finite weyl group w defined over a ground ring in which bad primes for w are invertible using deep properties of the kazhdan lusztig basis of mathcal h and lusztig s a function we show that mathcal h has a natural cellular structure in the sense of graham and lehrer thus we obtain a general theory of specht modules for hecke algebras of finite type previously a general cellular structure was only known to exist in types an and bn

computational construction of irreducible w graphs for types e6 and e7 abstract the concept of w graph was introduced in the influential paper david kazhdan george lusztig representations of coxeter groups and hecke algebras invent math 53 1979 165 184 of kazhdan and lusztig if the underlying coxeter group is the symmetric group then kazhdan and lusztig showed that every irreducible representation arises from a left cell and hence is given by a w graph this is the optimal picture that can hope for for other types of coxeter groups the representations arising from left cells are no longer irreducible however gyoja a gyoja on the existence of a w graph for an irreducible representation of a coxeter group j algebra 86 1984 422 438 3 proved by a general argument that every irreducible representation of a hecke algebra associated with a finite coxeter group is given by a w graph but this is a pure existence result and the question remained open of how to construct such w graphs explicitly in r b howlett yunchuan yin inducing w graphs math z 244 2003 415 431 we provided a general method for producing w graphs by induction of w graphs from parabolic subgroups and then yunchuan yin irreducible w graphs for type d 4 and d 5 comm algebra 34 2006 547 565 7 we constructed all the irreducible w graphs for type d 4 and d 5 by hand calculation in this paper we will show that the algorithm introduced in r b howlett yunchuan yin inducing w graphs math z 244 2003 415 431 is sufficiently powerful to construct explicit w graphs for all irreducible representations of hecke algebras of type e 6 and e 7 the computer algebra system magma was used for the calculations

irreducible projective modules of the hecke algebras of a finite coxeter group in this paper we classify the irreducible projective modules or equivalently see e g 6 the irreducible principal indecomposable modules of a non semisimple hecke algebra h w e of a finite coxeter group w let w 3 be a finite coxeter system 2 and t an indeterminate q t we denote by h w q the generic hecke iwaho algebra 9 2 lo this means that h w q is the c t algebra with the basis t w we w and the multiplication defined by

on the topology of components of some springer fibers and their relation to kazhdan lusztig theory abstract we describe the irreducible components of springer fibers for hook and two row nilpotent elements of gl n c as iterated bundles of flag manifolds and grassmannians we then relate the topology in particular the intersection homology poincare polynomials of the pairwise intersections of these components with the inner products of the kazhdan lusztig basis elements of irreducible representations of the rational iwahori hecke algebra of type a corresponding to the hook and two row young shapes

a large deviations perspective on the efficiency of multilevel splitting we analyze the performance of a multilevel splitting method for rare event simulation related to one recently proposed in the telecommunications literature this method splits promising paths into subpaths at intermediate levels to increase the number of observations of a rare event in our previous paper 1997 we gave sufficient conditions in specific classes of models for this method to be asymptotically optimal here we focus on necessary conditions in a general setting we show through a variety of results the importance of choosing the intermediate thresholds in a way consistent with the most likely path to a rare set both when the number of levels is fixed and when it increases with the rarity of the event in the latter case we give very general necessary conditions based on large deviations rate functions these indicate that even when the intermediate levels are chosen appropriately the method will frequently fail to be asymptotically optimal we illustrate the conditions with examples

shift exponent and breakdown of universality for the two layer ising model on a square lattice abstract by using a function for a spin 3 2 ising model we investigate the critical phenomena of a general two layer ising model of square lattice we find that the value of the shift exponent is 1 73 0 04 for the system with the same intra layer interaction and 0 5 for the system with different intra layer interactions we also find an evidence that the value of critical exponent exceeds 1 in some region of interaction parameter space where the degeneracy of the ground state is fourfold

james conjecture for hecke algebras of exceptional type i in this paper and a second part to follow we complete the programme initiated more than 15 years ago of determining the decomposition numbers and verifying james conjecture for iwahori hecke algebras of exceptional type the new ingredients which allow us to achieve this aim are the fact recently proved by the rst author that all hecke algebras of nite type are cellular in the sense of graham lehrer and the explicit determination of w graphs for the irreducible generic representations of hecke algebras of type e7 and e8 by howlett and yin thus we can reduce the problem of computing decomposition numbers to a manageable size where standard techniques e g parker s meataxe and its variations can be applied in this part we describe the theoretical foundations for this procedure

the breuil mezard conjecture when l neq p let l and p be primes let f qp be a finite extension with absolute galois group gf let f be a finite field of characteristic l and let w gf gln f be a continuous representation let r be the universal framed deformation ring for if l d p then the breuil mezard conjecture as recently formulated by emerton and gee relates the mod l reduction of certain cycles in r to the mod l reduction of certain representations of gln of we state an analogue of the breuil mezard conjecture when l p and we prove it whenever l 2 using automorphy lifting theorems we give a local proof when l is quasibanal for f and is tamely ramified we also analyze the reduction modulo l of the types defined by schneider and zink

rare event restart simulation of two stage networks abstract repetitive simulation trials after reaching thresholds restart is a widely applied accelerated simulation technique that allows the evaluation of extremely low probabilities in this method a number of simulation retrials are performed when the process enters regions of the state space where the chance of occurrence of the rare event is higher formulas for evaluating the optimal number of regions and retrials as well as guidelines for obtaining suitable importance functions were provided in previous papers nevertheless further investigations are required to apply these guidelines to practical cases in this paper we obtain efficient importance functions for networks with two stages and different nodes in each stage some approximations are used to derive the formulas the goodness of such approximations is supported by the simulation results obtained the methodology applied can be extended to other networks

on the decomposition numbers of the ree groups f42 q2 in non defining characteristic abstract we compute the l modular decomposition matrices of the simple ree groups f 4 2 q 2 where q 2 2 2 n 1 and n 0 for all primes l 3 up to some entries in the unipotent and one entry in the non unipotent characters we determine the smallest degree of a nontrivial irreducible l modular representation of f 4 2 q 2 for all primes l 3 and also obtain results on the 3 modular decomposition matrices of f 4 2 q 2

decomposition matrices for low rank unitary groups we study the decomposition matrices of the unipotent blocks of finite special unitary groups sun q for unitary primes larger than n up to few unknown entries we give a complete solution for n 2 10 we also prove a general result for two column partitions when divides q 1 this is achieved using projective modules coming from the adic cohomology of deligne lusztig varieties

symmetric group blocks of small defect has a specht module s and a p modular irreducible module d the decomposition number s d is de ned to be the composition multiplicity of d in s the following facts areknown about blocks of weight w a if w 0 or 1 then all the decomposition numbers of the block are 0 or 1 b if w 2 and p 2 then all the decomposition numbers of the block are 0 or 1 s2 c if w 4 then some decomposition numbers of a block can be greater than 1 even if p w moreover if w 0 1 or 2 then there is a known method for determining the decompositionnumbers r the situation for the case w 3 is still not properly understood in particular thedecomposition number s 2p 2 2p 2 p 1 1 d 3p 3 2p 1 is yet to be determined for p 5 this is just one of a collection of decomposition numbers forweight 3 which we are unable to evaluate our investigation of blocks of weight 3 grew out of an attempt to improve upon the earlier resultsof martin and russell mr by explicitly calculating the decomposition numbers this led us todiscover various errors and omissions in mr which place in doubt the claim made there that whenp w all the decomposition numbers are 0 or 1 note incidentally that if the decompositionnumbers for a given block are known to be 0 or 1 then the decomposition numbers can in principle be determined by applying schaper s theorem sch jm1 2 basic resultsby using a result of brundan and kleshchev we are able to improve upon the presentation ofseveral of the basic techniques used in s1 s2 mr for estimating decomposition numbers in orderto state these results recall that the diagram of a partition is the set of nodes i j 1 j

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

ising model with two spin interactions and three spin interactions on a square lattice abstract an ising model with two spin interactions in one direction and four spin interactions on the smallest plaquettes is investigated on the square lattice the part of the two spin interactions is essential to determine the nature of the phase transition of the model the system with nearest neighbour interactions has no phase transition the one with next nearest neighbour interactions has a spontaneous magnetization and belongs to the same class of universality as the eight vertex model the one with third neighbour interactions has no spontaneous magnetization but belongs to the same class of universality as onsager s lattice and so on

importance functions for restart simulation of general jackson networks abstract restart is an accelerated simulation technique that allows the evaluation of extremely low probabilities in this method a number of simulation retrials are performed when the process enters regions of the state space where the chance of occurrence of the rare event is higher these regions are defined by means of a function of the system state called the importance function guidelines for obtaining suitable importance functions and formulas for the importance function of two stage networks were provided in previous papers in this paper we obtain effective importance functions for restart simulation of jackson networks where the rare set is defined as the number of customers in a particular target node exceeding a predefined threshold although some rough approximations and assumptions are used to derive the formulas of the importance functions they are good enough to estimate accurately very low probabilities for different network topologies within short computational time

critical exponents of the two layer ising model the symmetric two layer ising model tlim is studied by the corner transfer matrix renormalization group method the critical points and critical exponents are calculated it is found that the tlim belongs to the same universality class as the ising model the shift exponent is calculated to be 1 773 which is consistent with the theoretical prediction of 1 75 with 1 3 deviation

the layered ising model mean field and interfacial approximations the critical temperature for layered ising models on the s c lattice is calculated using the recently proposed transfer matrix version of a mean field approximation for two layered case the present results tc j 3 20755 5 and 3 62279 2 for the open and periodic boundary conditions respectively improve considerably on the previous estimation obtained by the high temperature series expansion this model is also studied using the interfacial approximation i e the muller hartmann zittartz method we show that in this method the behaviour of interface in the weak interlayer coupling limit is similar to that of interacting quantum particles for the strong interlayer interaction limits some results for the two layered case are also obtained

constrained variational approach to second order phase transitions application to a two layer ising film within the framework of the transfer matrix formalism a variational method is elaborated with a constraint related to the fundamental role of the correlation length near a critical point the method is applied to a two layer ferromagnetic film made of two horizontal ising planes interacting through a vertical coupling the critical curve of the model as well as the correlation length above and near tc are calculated the critical exponent nu obtained agrees with the expected two dimensional exact value for the two particular values of the vertical coupling for which there are numerical estimates of tc through series analysis or the monte carlo method an essential agreement with these estimates is found analytical expressions are given in the weak and strong vertical coupling regimes

the decomposition numbers of the hecke algebra of type letw be the finite coxeter group of typef4 andhr q be the associated hecke algebra with parameter a prime powerq defined over a valuation ringr in a large enough extension field ofq with residue class field of characteristicr in this paper ther modular decomposition numbers ofhr q are determined for allq andr such thatr does not divideq the methods of the proofs involve the study of the generic hecke algebra of typef4 over the ringa u1 2 u 1 2 of laurent polynomials in an indeterminateu1 2 and its specializations onto the ring of integers in various cyclotomic number fields substancial use of computers and computer program systems gap maple meat axe has been made

schubert calculus and torsion explosion the author observes that certain numbers occurring in schubert calculus for sl also occur as entries in intersection forms controlling decompositions of soergel bimodules in higher rank these numbers grow exponentially this observation gives many counter examples to the expected bounds in lusztig s conjecture on the characters of simple rational modules for sl over fields of positive characteristic the examples also give counter examples to the james conjecture on decomposition numbers for symmetric groups

specht modules labelled by hook bipartitions i abstract brundan kleshchev and wang equip the specht modules s over the cyclotomic khovanov lauda rouquier algebra h n with a homogeneous z graded basis in this paper we begin the study of graded specht modules labelled by hook bipartitions n m 1 m in level 2 of h n which are precisely the hecke algebras of type b with quantum characteristic at least three we give an explicit description of the action of the khovanov lauda rouquier algebra generators 1 n 1 on the basis elements of s n m 1 m introducing certain specht module homomorphisms we construct irreducible submodules of these specht modules and thereby completely determine the composition series of specht modules labelled by hook bipartitions

obtaining critical point and shift exponent for the anisotropic two layer ising and potts models cellular automata approach abstract using probabilistic cellular automata with the glauber algorithm we have precisely calculated the critical points for the anisotropic two layer ising and potts models k x k y k z of the square lattice where k x and k y are the nearest neighbor interactions within each layer in the x and y directions respectively and k z is the inter layer coupling a general equation is obtained as a function of the inter and intra layer interactions for both the two layer ising and potts models separately where k z k x and k y k x furthermore the shift exponent for the two layer ising and potts models is calculated it was demonstrated that in the case of 1 for the two layer ising model the value of 1 756 0 0078 supports the scaling theories prediction that however for the unequal intra layer couplings for the two layer ising model and also in the case of both equal and unequal intra layer interactions for the two layer potts model our results are different from those obtained from the scaling theories finally an equation is obtained for the shift exponent as a function of intra layer couplings for the two layer ising and potts models

kazhdan lusztig cells and the murphy basis let h be the iwahori hecke algebra associated with s n the symmetric group on n symbols this algebra has two important bases the kazhdan lusztig basis and the murphy basis we establish a precise connection between the two bases allowing us to give for the first time purely algebraic proofs for a number of fundamental properties of the kazhdan lusztig basis and lusztig s results on the a function

bounds on the dimension of ext for finite groups of lie type abstract let g be a finite group of lie type defined in characteristic p and let k be an algebraically closed field of characteristic r 0 we will assume that r p so we are in the non defining characteristic case let v be a finite dimensional irreducible left kg module in 2011 guralnick and tiep found bounds on the dimension of h 1 g v in non defining characteristic which are independent of v the aim of this paper is to generalize the work of gurlanick and tiep we assume that g is split and use methods of modular harish chandra theory to find bounds on the dimension of ext k g 1 y v where y and v are irreducible kg modules we then use dipper and du s algorithms to illustrate our bounds in a series of examples

w graph ideals abstract we introduce a concept of a w graph ideal in a coxeter group the main goal of this paper is to describe how to construct a w graph from a given w graph ideal the principal application of this idea is in type a where it provides an algorithm for the construction of w graphs for specht modules