decomposition numbers and local properties abstract if g is a finite group and p is a prime we give evidence that the p decomposition matrix encodes properties of p sylow normalizers

generation of finite simple groups by an involution and an element of prime order abstract we prove that every non abelian finite simple group is generated by an involution and an element of prime order

lower bounds for degrees of irreducible brauer characters of finite general linear groups let g denote the finite general linear group gln fq over the finite field with q elements associated to each partition of n there is an irreducible unipotent complex character of g the degree of is a polynomial in q given by green s hook formula 8 p 444 the polynomial is monic of degree b where is the transpose of and for a partition m1 m2 mh 0 of n b denotes n n 1 2 h i 1 imi an easy consequence of green s formula is that

l espace symetrique de drinfeld et correspondance de langlands locale ii we study the geometry and the cohomology of the tamely ramified cover of drinfeld s p adic symmetric space over a p adic field k for this tame level we prove in a purely local way most of a conjecture of harris on the form of the ell adic cohomologies as well as the torsion freeness of the integral cohomology in this paper we also compute the ell adic cohomology of coxeter deligne lusztig variety associated to mathrm gl d and some results of independent interest on the coefficient systems over the bruhat tits building associated to mathrm gl d k have been established

on element orders in covers of finite simple classical groups abstract we show that finite simple symplectic groups of dimension at least 6 finite simple orthogonal groups of dimension at least 7 and finite simple unitary groups of dimension at least 4 except for u 5 2 have the following property if such a group g defined over a field of characteristic p acts on a vector space over a field of positive characteristic other than p then the corresponding semidirect product contains an element whose order is distinct from the order of any element of g

correspondance de jacquet langlands locale et congruences modulo ell let mathrm f be a non archimedean local field of residual characteristic p and ell be a prime number different from p we consider the local jacquet langlands correspondence between ell adic discrete series of mathrm gl n mathrm f and an inner form mathrm gl m mathrm d we show that it respects the relationship of congruence modulo ell more precisely we show that two integral ell adic discrete series of mathrm gl n mathrm f are congruent modulo ell if and only if the same holds for their jacquet langlands transfers to mathrm gl m mathrm d we also prove that the langlands jacquet morphism from the grothendieck group of finite length ell adic representations of mathrm gl n mathrm f to that of mathrm gl m mathrm d defined by badulescu is compatible with reduction mod ell

composite kernel machine regression based on likelihood ratio test for joint testing of genetic and gene environment interaction effect most common human diseases are a result from the combined effect of genes the environmental factors and their interactions such that including gene environment ge interactions can improve power in gene mapping studies the standard strategy is to test the snps one by one using a regression model that includes both the snp effect and the ge interaction however the snp by snp approach has serious limitations such as the inability to model epistatic snp effects biased estimation and reduced power thus in this article we develop a kernel machine regression framework to model the overall genetic effect of a snp set considering the possible ge interaction specifically we use a composite kernel to specify the overall genetic effect via a nonparametric function andwe model additional covariates parametrically within the regression framework the composite kernel is constructed as a weighted average of two kernels one corresponding to the genetic main effect and one corresponding to the ge interaction effect we propose a likelihood ratio test lrt and a restricted likelihood ratio test rlrt for statistical significance we derive a monte carlo approach for the finite sample distributions of lrt and rlrt statistics extensive simulations and real data analysis show that our proposed method has correct type i error and can have higher power than score based approaches under many situations

on the inductive alperin mckay and alperin weight conjecture for groups with abelian sylow subgroups abstract we study the inductive alperin mckay conjecture the inductive isaacs navarro refinement and the inductive blockwise alperin weight conjecture for groups of lie type in the generic case of abelian sylow l subgroups we also show that the alternating groups the suzuki groups and the ree groups satisfy the inductive condition necessary for spath s reduction of the blockwise alperin weight conjecture to the case of simple groups

generalization of likelihood ratio tests under nonstandard conditions in this paper we analyze the statistic which is the difference in the values of an estimating function evaluated at its local maxima on two different subsets of the parameter space assuming that the true parameter is in each subset but possibly on the boundary our results extend known methods by covering a large class of estimation problems which allow sampling from nonidentically distributed random variables specifically the existence and consistency of the local maximum estimators and asymptotic properties of useful hypothesis tests are obtained under certain law of large number and central limit type assumptions other models covered include those with general log likelihoods and or covariates as an example the large sample theory of two way nested random variance components models with covariates is derived from our main results

on the decomposition numbers of the finite general linear groups ii let g gl q q a prime power and let r be an odd prime not dividing q let s be a semisimple element of g of order prime to r and assume that r divides qdeg x 1 for all elementary divisors a of s relating representations of certain hecke algebras over symmetric groups with those of g we derive a full classification of all modular irreducible modules in the r block bh of g with semisimple part s the decomposition matrix d of bs may be partly described in terms of the decomposition matrices of the symmetric groups corresponding to the hecke algebras above moreover d is lower unitriangular this applies in particular to all r blocks of g if r divides q 1 thus in this case the r decomposition matrix of g is lower unitriangular introduction the modular representation theory of finite groups of lie type defined over a field of characteristic p is naturally divided into the cases of equal characteristic r p and unequal characteristic r p this paper begins the study of decomposition numbers in the unequal characteristic case of the finite general linear groups when r 2 let g be the full linear group of degree n over gf q and 2 r be a prime not dividing q let s c g be semisimple and consider the geometric conjugacy class s g of irreducible characters corresponding to the g conjugacy class of s using basic properties of the deligne lusztig operators introduced in 7 we will determine a parabolic subgroup p ps of g and a cuspidal irreducible character of x of p such that s g is just the set of constituents of xg the induced character let r r k be an r modular system for g and m a kp module affording x then end kg m is called the hecke algebra of m compare 5 and has been calculated in a more general context by r b howlett and g i lehrer in 12 it turns out that end kg m k w the hecke algebra of w where w wj denotes the weyl group of cg s the centralizer of s in g compare 1 2 4 13 in particular k w has a basis tw iw c w such that the structure constants of the multiplication are all contained in r c k so let r w be the r order in k w generated by wtiw ew and r w r r r w choosing a special rp lattice s in m and using the classification of the r blocks of rg given by p fong and b srinivasan in received by the editors july 25 1983 and in revised form october 18 1983 may 23 1984 and september 1 1984 1980 mathematics subject classification primary 20c20 secondary 20g40 this work was supported by the deutsche forschungsgemeinschaft 1985 american mathematical society 0002 9947 85 1 00 25 per page

stable basic sets for finite special linear and unitary groups abstract in this paper we show using deligne lusztig theory and kawanaka s theory of generalised gelfand graev representations that the decomposition matrix of the special linear and unitary group in non defining characteristic can be made unitriangular with respect to a basic set that is stable under the action of automorphisms

low dimensional representations of special linear groups in cross characteristics the low dimensional projective irreducible representations in cross characteristics of the projective special linear group mbox psl n q are investigated if n geq 3 and n q neq 3 2 3 4 4 2 4 3 all such representations of the first degree which is q n q q 1 kappa with kappa 0 or 1 and the second degree which is q n 1 q 1 come from weil representations we show that the gap between the second and the third degree is roughly q 2n 4

rational brauer characters it is known that every finite group of even order has a non trivial complex irreducible character which is rational valued we prove the modular version of this result if p is an odd prime and g is any finite group of even order then g has a non trivial irreducible p brauer character which is rational valued

the structure of rank 3 permutation modules for o2n 2 and um 2 acting on nonsingular points abstract we study the odd characteristic structure of permutation modules for the rank 3 natural actions of o 2 n 2 n 3 and u m 2 m 4 on nonsingular points of their standard modules

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

on quotients of hom functors and representations of finite general linear groups i abstract this is a second article on quotients of hom functors and their applications to the representation theory of finite general linear groups in a nondescribing characteristic after some general results on quotients of hom functors and their connection to the harish chandra theory these constructions are used to obtain a full classification of the l modular irreducible representations of gl n q for some prime power q which is not divisible by the prime l and to explain some facts on their harish chandra series and decomposition numbers

a proof of the mullineux conjecture a partition of a positive integer n is a sequence 1 2 m 0 of integers such that i n for a positive integer p a partition 1 2 m or its young diagram is called p regular if it does not have p or more equal parts i e if there does not exist t m p 1 with t t 1 t p 1 let f be a field of characteristic p 0 it is well known that irreducible representations of the symmetric group sn over f are naturally parametrized by pregular partitions of n cf for example 9 12 if is such a partition we denote the corresponding irreducible module by d let sgnn be the one dimensional sign representation of sn over f i e sgnn f as a vector space and g f sgn g f for any g sn f f here sgn g is just the sign of the permutation g it is clear that for any irreducible d the tensor product d sgnn is also irreducible the problem usually called the problem of mullineux is to find the p regular partition such that d sgnn d put d sgnn dn in this way a bijection bn on the set pn of p regular partitions of n is defined for each positive integer n and the problem is

representations lisses modulo l de gl m d let f be a non archimedean locally compact field of residue characteristic p let d be a finite dimensional central division f algebra and let r be an algebraically closed field of characteristic different from p we classify all smooth irreducible representations of gl m d with coefficients in r in terms of multisegments generalizing works by zelevinski tadic and vigneras we prove that any irreducible r representation of gl m d has a unique supercuspidal support and thus get two classifications one by supercuspidal multisegments classifying representations with a given supercuspidal support and one by aperiodic multisegments classifying representations with a given cuspidal support these constructions are made in a purely local way with a substantial use of type theory

on the modular composition factors of the steinberg representation abstract let g be a finite group of lie type and st k be the steinberg representation of g defined over a field k we are interested in the case where k has prime characteristic l and st k is reducible tinberg has shown that the socle of st k is always simple we give a new proof of this result in terms of the hecke algebra of g with respect to a borel subgroup and show how to identify the simple socle of st k among the principal series representations of g furthermore we determine the composition length of st k when g gl n q or g is a finite classical group and l is a so called linear prime

p parts of brauer character degrees abstract let g be a finite group and let p be an odd prime under certain conditions on the p parts of the degrees of its irreducible p brauer characters we prove the solvability of g as a consequence we answer a question proposed by b huppert in 1991 if g has exactly two distinct irreducible p brauer character degrees then is g solvable we also determine the structure of non solvable groups with exactly two irreducible 2 brauer character degrees

semiparametric regression of multidimensional genetic pathway data least squares kernel machines and linear mixed models we consider a semiparametric regression model that relates a normal outcome to covariates and a genetic pathway where the covariate effects are modeled parametrically and the pathway effect of multiple gene expressions is modeled parametrically or nonparametrically using least squares kernel machines lskms this unified framework allows a flexible function for the joint effect of multiple genes within a pathway by specifying a kernel function and allows for the possibility that each gene expression effect might be nonlinear and the genes within the same pathway are likely to interact with each other in a complicated way this semiparametric model also makes it possible to test for the overall genetic pathway effect we show that the lskm semiparametric regression can be formulated using a linear mixed model estimation and inference hence can proceed within the linear mixed model framework using standard mixed model software both the regression coefficients of the covariate effects and the lskm estimator of the genetic pathway effect can be obtained using the best linear unbiased predictor in the corresponding linear mixed model formulation the smoothing parameter and the kernel parameter can be estimated as variance components using restricted maximum likelihood a score test is developed to test for the genetic pathway effect model variable selection within the lskm framework is discussed the methods are illustrated using a prostate cancer data set and evaluated using simulations

cross characteristic representations of odd characteristic symplectic groups and unitary groups in ls landazuri and seitz gave lower bounds for irreducible representations of chevalley groups in non defining characteristic when referring to irreducible representations for quasisimple groups g we will assume that the modules are nontrivial on f g see also sz gpps ho for some improvements on these bounds these results have proved to be useful in many applications in particular they have been used to classify the maximal subgroups of classical groups containing an element of prime order acting irreducibly on a subspace of large dimension cf gpps and to show that low dimensional modules in characteristic p for groups with no normal p subgroup are semisimple see gu it is also important to identify the modules which have dimension close to the smallest possible dimension and to prove that there are no irreducible modules with dimension in a certain range above it this was done in gpps and gt for sln q further improvements were obtained by brundan and kleshchev brk hiss and malle hm have obtained results similar to gt for unitary groups in this paper we consider the groups g sp2n q with n 2 and q p odd and g sun q with n 3 landazuri and seitz ls had already shown that the minimal dimension d of any irreducible module in the nondefining characteristic has dimension exactly q 1 2 for the symplectic case and q 1 q 1 for the unitary case it was proved in gpps that aside from some small exceptions every irreducible kg module in a nondefining characteristic has dimension d d 1 or at least dimension 2d in characteristic 0 tiep and zalesskii tz1 using deligne lusztig

probability judgments for continuous quantities linear combinations and calibration expert judgment elicitation is often required in probabilistic decision making and the evaluation of risk one measure of the quality of probability distributions given by experts is calibration the faithfulness of the probabilities in an empirically verifiable sense a method of measuring calibration for continuous probability distributions is presented here a discussion of the impact of using linear rules for combining such judgments is given and an empirical demonstration is given using data collected from experts participating in a large scale risk study it is shown by theoretical argument that combining well calibrated distributions of individual experts using linear rules can only result in reducing calibration in contrast it is demonstrated both by example and empirically that an equally weighted linear combination of experts who tend to be overconfident can produce distributions that are better calibrated than the experts individual distributions using data from training exercises it is shown that the improvement in calibration is rapid as the number of experts is increased from one to five or six but there is only modest improvement from increasing the number of experts beyond that point

hall higman type theorems for semisimple elements of finite classical groups minimum polynomials of semisimple elements of prime power order p a of finite classical groups in nontrivial irreducible cross characteristic representations are studied in particular an analogue of the hall higman theorem is established which shows that the degree of such a polynomial is at least p a 1 p 1 with a few explicit exceptions

on 2 brauer characters of odd degree a natural bijection is constructed between the odd degree irreducible 2 brauer characters of certain distinguished groups and the characters of mathbf n g p p where p in mathrm syl 2 g in these cases and more we show that these 2 brauer characters are liftable to irreducible complex characters of g

asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions abstract large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described specifically the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics are derived these results generalize the work of moran 1971 chant 1974 and chernoff 1954 some of chant s results are shown to be incorrect the approach used in deriving these results follows from comments made by moran and chant the problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate gaussian distribution from a sample of size 1 in this representation the gaussian random variable corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the normalized maximum likelihood estimator thus the limiting distribution of the maximum likelihood estimator is the same as the distribution of the projection of the gaussian random v

low dimensional representations of special unitary groups abstract we prove gap results for the low dimensional representation of unitary groups in nondefining characteristics more precisely we give a lower bound for the third smallest degree of a nontrivial absolutely irreducible representation of the groups mentioned above which is almost in the order of magnitude of the square of the two smallest such representations as a corollary we obtain the uniqueness of weil representations in all cross characteristics for the unitary groups our approach uses some results on ordinary representations which can be proved using deligne lusztig theory and which might be of independent interest as well as information on decomposition matrices

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

representations of finite special linear groups in non defining characteristic we classify irreducible modules over the finite special linear group sln q in the non defining characteristic l describe restrictions of irreducible modules from gln q to sln q classify complex irreducible characters of sln q irreducible modulo l and discuss unitriangularity of the l decomposition matrix for sln q

the blocks and weights of finite special linear and unitary groups abstract this paper has two main parts first we give a classification of the l blocks of finite special linear and unitary groups sl n q in the non defining characteristic l 3 second we describe how the l weights of sl n q can be obtained from the l weights of gl n q when l gcd n q and verify the alperin weight conjecture for sl n q under the condition l gcd n q as a step to establish the alperin weight conjecture for all finite groups we prove the inductive blockwise alperin weight condition for any unipotent l block of sl n q if l gcd n q

on prime power order elements of general linear groups abstract we will classify the absolutely irreducible subgroups g gl d q which are not realizable modulo scalars over any proper subfield of gf q which are nearly simple and which contain elements of prime power order greater than d 3

restricted likelihood ratio lack of fit tests using mixed spline models penalized regression spline models afford a simple mixed model representation in which variance components control the degree of non linearity in the smooth function estimates this motivates the study of lack of fit tests based on the restricted maximum likelihood ratio statistic which tests whether variance components are 0 against the alternative of taking on positive values for this one sided testing problem a further complication is that the variance component belongs to the boundary of the parameter space under the null hypothesis conditions are obtained on the design of the regression spline models under which asymptotic distribution theory applies and finite sample approximations to the asymptotic distribution are provided test statistics are studied for simple as well as multiple regression models

representations of the general linear groups which are irreducible over subgroups we classify all triples g v h such that sl n q leq g leq gl n q v is a representation of g of dimension greater than one over an algebraically closed field bbb f of characteristic prime to q and h is a proper subgroup of g such that the restriction v downarrow h is irreducible this problem is a natural part of the aschbacher scott program on maximal subgroups in finite classical groups

non abelian composition factors of m rational groups in this thesis we discuss several problems in the representation theory of finite groups of lie type in chapter 2 we will give essential background material that will be useful for the entirety of the thesis we will investigate the construction of groups of lie type as well as their representations we will define the field of values of a character afforded by a representation and state useful results concerning these fields in chapter 3 we examine zsigmondy primes and their existence a necessary ingredient in proving our main results in chapters 4 and 5 we describe our main results in the ordinary and modular cases which we now summarize a finite group g is said to be m rational for a fixed positive integer m if q x q m for any irreducible character x irr g in 1976 r gow studied the structure of solvable rational groups i e m 1 and found that the possible composition factors of a solvable rational group are cyclic groups of prime order p 2 3 5 22 just over a decade later w feit and g seitz classified the possible non abelian composition factors of non solvable rational groups in 2008 j thompson found an upper bound of p 13 for the order of the possible cyclic composition factors of an arbitrary rational group and conjectured that the bound can be improved to p 5 more recently j mckay posed the question of determining the structure of quadratic rational groups i e m 2 j tent studied the cyclic composition factors of solvable quadratic rational groups in 2013 in chapter 4 we answer mckay s question concerning non abelian composition factors and generalize our results to non solvable m rational groups modular character theory was founded by r brauer in the 1930 s and has been useful in proving historical results including the classification of finite simple groups in chapter 5 we prove the modular version of our results though our conclusions are similar to those found in the complex case the methods for proving the results are typically much more complicated

combining probability forecasts summary linear pooling is by far the most popular method for combining probability forecasts however any non trivial weighted average of two or more distinct calibrated probability forecasts is necessarily uncalibrated and lacks sharpness in view of this linear pooling requires recalibration even in the ideal case in which the individual forecasts are calibrated towards this end we propose a beta transformed linear opinion pool for the aggregation of probability forecasts from distinct calibrated or uncalibrated sources the method fits an optimal non linearly recalibrated forecast combination by compositing a beta transform and the traditional linear opinion pool the technique is illustrated in a simulation example and in a case study on statistical and national weather service probability of precipitation forecasts

on the inductive alperin mckay condition for simple groups of type a abstract as a sequel to 6 we verify the so called inductive am condition introduced in 22 for simple groups of type a and blocks with maximal defect this is part of the program set up to verify the alperin mckay conjecture through its reduction to a problem on quasi simple groups see 23 but also the missing direction of brauer s height zero conjecture see 21

rank 3 permutation modules of the finite classical groups abstract the cross characteristic permutation modules for the actions of the finite classical groups on singular 1 spaces of their natural modules are studied the composition factors and submodule lattices are determined

random effects models for longitudinal data models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit multivariate models with general covariance structure are often difficult to apply to highly unbalanced data whereas two stage random effects models can be used easily in two stage models the probability distributions for the response vectors of different individuals belong to a single family but some random effects parameters vary across individuals with a distribution specified at the second stage a general family of models is discussed which includes both growth models and repeated measures models as special cases a unified approach to fitting these models based on a combination of empirical bayes and maximum likelihood estimation of model parameters and using the em algorithm is discussed two examples are taken from a current epidemiological study of the health effects of air pollution

equivalent blocks of finite general linear groups in non describing characteristic abstract j chuang r kessar and j rickard have proved broue s abelian defect group conjecture for many symmetric groups we adapt the ideas of kessar and chuang towards finite general linear groups represented over non describing characteristic we then describe morita equivalences between certain p blocks of gl n q with defect group c p c p as q varies see theorem 2 here p and q are coprime this generalizes work of s koshitani and m hyoue who proved the same result for principal blocks of gl n q when p 3 1 in a different way

degrees of rational characters of finite groups abstract a classical theorem of john thompson on character degrees states that if the degree of any complex irreducible character of a finite group g is 1 or divisible by a prime p then g has a normal p complement in this paper we consider fields of values of characters and prove some improvements of this result

variance components testing in the longitudinal mixed effects model this article discusses the asymptotic behavior of likelihood ratio tests for nonzero variance components in the longitudinal mixed effects linear model described by laird and ware 1982 biometrics 38 963 974 our discussion of the large sample behavior of likelihood ratio tests for nonzero variance components is based on the results for nonstandard testing situations by self and liang 1987 journal of the american statistical association 82 605 610

on a conjecture of g e wall abstract we prove that if g is a finite almost simple group having socle of lie type of rank r then the number of maximal subgroups of g is at most c r 2 3 g where c is an absolute constant this verifies a conjecture of wall for groups of sufficiently large rank using this we prove that any finite group g has at most 2 c g 3 2 maximal subgroups

the inductive blockwise alperin weight condition for psl n q and psu n q with cyclic outer automorphism groups abstract in this paper we prove that if n q is such chosen that the outer automorphism groups of psl n q and psu n q are cyclic psl n q and psu n q satisfy the inductive blockwise alperin weight condition for any prime l different from the defining characteristic to do this it suffices to study actions of automorphisms on the brauer characters and weights of gl n q and gu n q for l and to show that the blockwise bijection between brauer characters and weights given in 1 3 4 and 5 is equivariant

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

lie type like groups abstract a finite simple group of lie type in defining characteristic p has exactly two p blocks the principal block and a block of defect zero consisting of the steinberg character whose degree is the p part of the order of the group in this paper we characterize finite groups g which have exactly the principal p block and a p block of defect zero consisting of an irreducible character of degree g p

the irreducible representations of the alternating group which remain irreducible in characteristic p let p be an odd prime and an the alternating group of degree n we determine which ordinary irreducible representations of u n remain irreducible in characteristic p verifying the author s conjecture from 2010 given the preparatory work done in 2010 our task is to determine which self conjugate partitions label specht modules for the symmetric group in characteristic p having exactly two composition factors this is accomplished through the use of the robinson brundan kleshchev i restriction functors together with known results on decomposition numbers for the symmetric group and additional results on the mullineux map and homomorphisms between specht modules

modular representations as a possible basis of finite physics the approach in which physical systems are described by the elements of a linear space over a finite field and operators of physical quantities by linear operators in this space is discussed the mathematical formulation of the correspondence between such a description and the conventional one is given for a case when the characteristic of the finite field is sufficiently large the above correspondence is considered in the examples of finite analogs of representations of so 2 3 so 1 4 and osp 1 4 algebras finite analogs of representations of infinite dimensional algebras are briefly discussed

the irreducible brauer characters of the finite special linear groups in non describing characteristics abstract the irreducible brauer characters ofsln q are investigated for primeslnot dividingq they are described in terms of a set of ordinary characters ofsln q whose reductions modulolare a generating set of the additive group of generalized brauer characters and the decomposition numbers of this set these decomposition numbers are described by combinatorial means in terms of the decomposition numbers ofgln q the latter have been investigated in great detail the last 15 years and are known completely forn 10

generalized gelfand graev representations of exceptional simple algebraic groups over a finite field i introduction let ffi be an adjoint simple algebraic group of type e n 6 7 8 f 4 or g2 over an algebraically closed field k of characteristic p 0 we assume that 5 has a fixed fq rational structure for a finite subfield fq of k the purpose of this paper is to determine explicitly the values of the irreducible characters of g ffi fq at the unipotent elements when p is good 38 i 4 3 in this part i we shall do this only for large p and the full result will be proved in the forthcoming part ii after submitting the first version of the present paper the author was informed by lusztig that he solved analogous problems for classical groups using his theory of character sheaves and that he can also recover the result of this paper mentioned above our method depends upon 1 a parametrization of the irreducible characters of g and the determination of their multiplicities in deligne lusztig virtual representations lusztig 25 2 the computation of the green functions of g in large characteristic springer 36 7 1 shoji 33 beynon and spaltenstein 3 4 3 the determination of the values of generalized gelfand graev characters 20 of g in good characteristic see section 3 by 1 and 2 the uniform parts of the irreducible characters are already known explicitly in large characteristic this implies that in large characteristic our problem for groups of type e 6 o r e 7 has already been solved it should be mentioned here that the whole character table of the g2 group is also known by chang and ree 8 hence in the g2 case this paper gives just another approach to a part of the result of chang and ree if g is of type g2 f4 or e 8 the restriction of an irreducible character r of g to the set of unipotent elements can be written as a sum of its uniform part with a function of the form

l espace symetrique de drinfeld et correspondance de langlands locale i we study the geometry and the cohomology of the tamely ramified cover of drinfeld s p adic symmetric space over a p adic field k for this tame level we prove in a purely local way most of a conjecture of harris on the form of the ell adic cohomologies as well as the torsion freeness of the integral cohomology in this paper we also compute the ell adic cohomology of coxeter deligne lusztig variety associated to mathrm gl d and some results of independent interest on the coefficient systems over the bruhat tits building associated to mathrm gl d k have been established

an analytic method for evaluating the performance of aggregation rules for probability densities it is shown how infinite sequences of densities with defined properties can be used to evaluate the expected performance of mathematical aggregation rules for elicited densities the performance of these rules is measured through the average variance calibration and average brier score of the aggregates a general result for the calibration of the arithmetic average of densities from well calibrated independent experts is given arithmetic and geometric aggregation rules are compared using sequences of normal densities sequences are developed that exhibit dependence among experts and lack of calibration the impact of correlation number of experts and degree of calibration on the performance of the aggregation is demonstrated

eigenvalues of unipotent elements in cross characteristic representations of finite classical groups abstract let h be a finite classical group g be a unipotent element of h of order s and be an irreducible representation of h with dim 1 over an algebraically closed field of characteristic coprime to s we show that almost always all the s roots of unity occur as eigenvalues of g and classify all the triples h g for which this does not hold in particular we list the triples for which 1 is not an eigenvalue of g we also give estimates of the asymptotic behavior of eigenvalue multiplicities when the rank of h grows and s is fixed

growth of cross characteristic representations of finite quasisimple groups of lie type abstract in this paper we give a bound to the number of conjugacy classes of maximal subgroups of any almost simple group whose socle is a classical group of lie type the bound is 2 n 5 2 n log 2 log 2 q where n is the dimension of the classical socle and q is the size of the defining field to obtain the bound we first bound the number of projective cross characteristic representations of simple groups of lie type as a function of the representation degree these bounds are computed for different families of groups separately in the computation we use information on conjugacy class numbers minimal character degrees and gaps between character degrees

cross characteristic representations of even characteristic symplectic groups we classify the small irreducible representations of sp 2n q with q even in odd characteristic this improves even the known results for complex representations the smallest representation for this group is much larger than in the case when q is odd this makes the problem much more difficult