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

brauer characters and rationality a finite group g has no non trivial rational valued irreducible p brauer characters if and only if g has no non trivial rational elements of order not divisible by p

the product form for sojourn time distributions in cyclic exponential queues consider a closed cyclic queuing system consisting of m exponential queues the laplace stieltjes transform of the joint distribution of the consecutive sojourn times of a customer at the m queues is determined and shown to have a product form the proof is based on a reversibility argument

simple classical groups of lie type are determined by their character degrees let g be a finite group denote by irr g the set of all irreducible complex characters of g let cd g be the set of all irreducible complex character degrees of g forgetting multiplicities that is cd g 1 irr g and let x1 g be the set of all irreducible complex character degrees of g counting multiplicities let h be a finite nonabelian simple classical group in this paper we will show that if g is a finite group and x1 g x1 h then g is isomorphic to h in particular this implies that the nonabelian simple classical groups of lie type are uniquely determined by the structure of their complex group algebras

symplectic groups symplectic spreads codes and unimodular lattices abstract it is known that the symplectic group sp 2 n p has two complex conjugate irreducible representations of degree p n 1 2 realized over q p provided that p 3 mod 4 in the paper we give an explicit construction of an odd unimodular sp 2 n p 2 invariant lattice p n in dimension p n 1 for any p n 3 mod 4 such a lattice has been constructed by r bacher and b b venkov in the case p n 27 a second main result says that these lattices are essentially unique we show that for n 3 the minimum of p n is at least p 1 2 and at most p n 1 2 the interrelation between these lattices symplectic spreads of f p 2 n and self dual codes over f p is also investigated in particular using new results of u dempwolff and l bader w m kantor and g lunardon we come to three extremal self dual ternary codes of length 28

moments of weil representations of finite special unitary groups abstract we prove an nth moment 1 result for irreducible weil representations of degree q n 1 q 1 of special unitary groups su n q for any odd n 3 and any prime power q

statistics of zero crossings in rough interfaces with fractional elasticity we study numerically the distribution of zero crossings in one dimensional elastic interfaces described by an overdamped langevin dynamics with periodic boundary conditions we model the elastic forces with a riesz feller fractional laplacian of order z 1 2 such that the interfaces spontaneously relax with a dynamical exponent z to a self affine geometry with roughness exponent by continuously increasing from 1 2 macroscopically flat interface described by independent ornstein uhlenbeck processes phys rev 36 823 1930 phrvao0031 899x10 1103 physrev 36 823 to 3 2 super rough mullins herring interface three different regimes are identified i 1 2 1 the effect on p l of short scale smoothening is also analyzed numerically and analytically a tight relation between the mean interval the mean width of the interface and the density of zeros is also reported the results drawn from our analysis of rough interfaces subject to particular boundary conditions or constraints along with discretization effects are relevant for the practical analysis of zeros in interface imaging experiments or in numerical analysis

rational irreducible characters and rational conjugacy classes in finite groups we prove that a finite group g has two rational valued irreducible characters if and only if it has two rational conjugacy classes and determine the structure of any such group along the way we also prove a conjecture of gow stating that any finite group of even order has a non trivial rational valued irreducible character of odd degree

local systems and finite unitary and symplectic groups abstract for powers q of any odd prime p and any integer n 2 we exhibit explicit local systems on the affine line a 1 in characteristic p 0 if 2 n and on the affine plane a 2 if 2 n whose geometric monodromy groups are the finite symplectic groups sp 2 n q when n 3 is odd we show that the explicit rigid local systems on the affine line in characteristic p 0 constructed in 11 do have the special unitary groups su n q as their geometric monodromy groups as conjectured therein and also prove another conjecture of 11 that predicted their arithmetic monodromy groups

quasiprimitive linear groups with quadratic elements abstract a semisimple complex linear transformation with exactly two distinct eigenvalues is called a quadratic element in this paper the finite irreducible linear groups generated by quadratic elements of order 3 for which the multiplicities of the eigenvalues are distinct are determined this is used to obtain bounds on the degree of an irreducible primitive linear group containing such elements there are no irreducible primitive linear groups containing quadratic elements of order 4 with eigenvalues 1 i with distinct multiplicities where i is a primitive fourth root of 1

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

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

asymptotic expansions of the sojourn time distribution functions of jobs in closed product form queuing networks striking progress has been made recently in obtaining expressions for the sojourn time distribution function stdf of a job at a c server first come first serve fcfs center in a closed product form queuing network these results have more recently been extended and expressions have been obtained for the joint distribution function df of the sojourn times of a job at a sequence of single server fcfs centers lying on an overtake free path however these formulas present considerable computational problems in the case of large closed queuing networks in this paper asymptotic techniques developed by mitra and mckenna for the calculation of the partition function of large product form closed queuing networks are applied to the sojourn time problem asymptotic expansions are obtained for the stdf of a job at c server fcfs center in closed product form queuing networks similar expansions are obtained for the joint df of the sojourn times of a job at a sequence of single server fcfs centers lying on an overtake free path in addition integral expressions are obtained for the stdf of a job at a single server fcps center in a closed product form queuing network in which all the centers are load independent these integral expressions also yield useful asymptotic expansions finally integral expressions are also obtained for the joint df of the sojourn times of a job at the centers of an overtake free path in such a network

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

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

the non coprime k gv problem abstract let v be a finite faithful completely reducible fg module for a finite field f and a finite group g in various cases explicit linear bounds in v are given for the numbers of conjugacy classes k g v and k g of the semidirect product gv and of the group g respectively these results concern the so called non coprime k g v problem

a reducible weil representation of sp 4 realized by differential operators in the space of smooth functions on h2 s1 this work presents a novel way to obtain the associated romanovski functions rn m x with n m in the three separate regions in terms of n and m we obtain the raising and lowering relations with respect to the both indices simultaneously in the three regions then a reducible weil representation of the real lie algebra sp 4 is realized in the space of complex valued smooth functions on h2 s1 by differential forms for the cartan weyl basis its invariant subspace is the second rare instance of the highest weight irreducible representation of sp 4 all whose weight spaces are one dimensional

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

some characterizations of the weil representations of the symplectic and unitary groups abstract let g be a finite symplectic or unitary group we characterize the weil representations of g via their restriction to a standard subgroup then we complete the determination of complex representations of g with specific minimal polynomials of certain elements by showing that they coincide with the weil representations

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

representations of the symmetric group which are irreducible over subgroups let f be an algebraically closed field of characteristic p and n be the symmetric group on n letters in this paper we classify all pairs g d where d is an irreducible f n module of dimension greater than 1 and g is a proper subgroup of n such that the restriction d g is irreducible provided p 3

generalized survival in step fluctuations the properties of the generalized survival probability that is the probability of not crossing an arbitrary location r during relaxation have been investigated experimentally via scanning tunneling microscope observations and numerically the results confirm that the generalized survival probability decays exponentially with a time constant tau s r the distance dependence of the time constant is shown to be tau s r tau s0 exp r w t where w2 t is the material dependent mean squared width of the step fluctuations the result reveals the dependence on the physical parameters of the system inherent in the prior prediction of the time constant scaling with r l alpha with l the system size and alpha the roughness exponent the survival behavior is also analyzed using a contrasting concept the generalized inside survival s in t r which involves fluctuations to an arbitrary location r further from the average numerical simulations of the inside survival probability also show an exponential time dependence and the extracted time constant empirically shows r w lambda behavior with lambda varying over 0 6 to 0 8 as the sampling conditions are changed the experimental data show similar behavior and can be well fit with lambda 1 0 for t 300 k and 0 5 lambda 1 for t 460 k over this temperature range the ratio of the fixed sampling time to the underlying physical time constant and thus the true correlation time increases by a factor of approximately 10 3 preliminary analysis indicates that the scaling effect due to the true correlation time is relevant in the parameter space of the experimental observations

weil representations via abstract data and heisenberg groups a comparison abstract let b be a ring not necessarily commutative having an involution and let u 2 m b be the unitary group of rank 2m associated to a hermitian or skew hermitian form relative to when b is finite we construct a weil representation of u 2 m b via heisenberg groups and find its explicit matrix form on the bruhat elements as a consequence we derive information on generalized gauss sums on the other hand there is an axiomatic method to define a weil representation of u 2 m b and we compare the two weil representations thus obtained under fairly general hypotheses when b is local not necessarily finite we compute the index of the subgroup of u 2 m b generated by its bruhat elements besides the independent interest this subgroup and index are involved in the foregoing comparison of weil representations

the distribution of queuing network states at input and output instants queuing networks are studied at selected points in the steady state namely at the moments when jobs of a given class arrive into a given node either from the outside or from other nodes and at the moments when jobs of a given class leave a given node either for the outside or for other nodes the processes defined by these points are known to be in general non potsson interdependent and serially correlated therefore the relation between the distribution of the system state embedded at those moments and the steady state or random point distribution is not obvious a priori for a large class of networks having product form equihbrium distribnttons it is shown that a if the given job class belongs to an open subchain the state distributions at input pomts output points and random points are identical and b if the job class belongs to a closed subchain the distribution at input and output points ts the same as the steady state distribution of a network with one less job in that subchain

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 weil representation of a unitary group associated to a ramified quadratic extension of a finite local ring we find all irreducible constituents of the weil representation of a unitary group um a of rank m associated to a ramified quadratic extension a of a finite commutative local and principal ring r of odd characteristic we show that this weil representation is multiplicity free with monomial ir reducible constituents we also find the number of these constituents and describe them in terms of clifford theory with respect to a congruence sub group we find all character degrees in the special case when r is a field

minimum polynomials of the elements of prime order in representations of quasi simple groups abstract we determine the irreducible representations of quasi simple groups in which some element of prime order p has less than p distinct eigenvalues let p be a prime greater than 2 let c denote the field of complex numbers gl n c the group of all n n matrices over c let g gl n c be a finite irreducible subgroup z g the center of g let p 2 be a prime we call g an n p group if it contains a matrix g such that g p is scalar g has at most p 1 distinct eigenvalues and g does not belong to a proper normal subgroup of g we assume p 2 as no n 2 group exist for n 1 this paper is a major step toward the determination of all n p groups this will serve for recognition of finite linear groups containing a given matrix with the above property for some p the bulk of the work is to determine quasi simple n p groups this is done in the current paper and the general case will be dealt with in a subsequent work

spatial survival probability for one dimensional fluctuating interfaces in the steady state we report numerical and analytic results for the spatial survival probability for fluctuating one dimensional interfaces with edwards wilkinson or kardar parisi zhang dynamics in the steady state our numerical results are obtained from analysis of steady state profiles generated by integrating a spatially discretized form of the edwards wilkinson equation to long times we show that the survival probability exhibits scaling behavior in its dependence on the system size and the sampling interval used in the measurement for both steady state and finite initial conditions analytic results for the scaling functions are obtained from a path integral treatment of a formulation of the problem in terms of one dimensional brownian motion a deterministic approximation is used to obtain closed form expressions for survival probabilities from the formally exact analytic treatment the resulting approximate analytic results provide a fairly good description of the numerical data

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

surjective word maps and burnside s p aq b theorem we prove surjectivity of certain word maps on finite non abelian simple groups more precisely we prove the following if n is a product of two prime powers then the word map x y mapsto x ny n is surjective on every finite non abelian simple group if n is an odd integer then the word map x y z mapsto x ny nz n is surjective on every finite quasisimple group these generalize classical theorems of burnside and feit thompson we also prove asymptotic results about the surjectivity of the word map x y mapsto x ny n that depend on the number of prime factors of the integer n

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

even unimodular lattices associated with the weil representation of the finite symplectic group soit q une puissance d un nombre premier impair p et soit g le groupe symplectique sp 2n q quand n est pair et q p est un nombre premier avec p 3 mod 4 il existe un treillis integral rationnel g invariant l dans m qui supporte une forme unimodulaire definie positive symetrique paire m est un qg module et q le corps des rationnels

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

character degrees and random walks in finite groups of lie type for a finite group h let irr h denote the set of irreducible characters of h and define the zeta function zeta h t sum chi in irr h chi 1 t for real t 0 we study the asymptotic behaviour of zeta h t for finite simple groups h of lie type and also of a corresponding zeta function defined in terms of conjugacy classes applications are given to the study of random walks on simple groups and on base sizes of primitive permutation groups

group representations and lattices this paper began as an attempt to understand the euclidean lattices that were recently constructed using the theory of elliptic curves by elkies e and shioda shl sh2 from the point of view of group representations the main idea appears in a note of thompson th2 if one makes strong irreducibility hypotheses on a rational representation v of a finite group g then the g stable euclidean lattices in v are severely restricted unfortunately these hypotheses are rarely satisfied when v is absolutely irreducible over q but there are many examples where the ring endg v is an imaginary quadratic field or a definite quaternion algebra these representations allow us to construct some of the mordell weil lattices considered by elkies and shioda as well as some interesting even unimodular lattices that do not seem to come from the theory of elliptic curves in 1 we discuss lattices and hermitian forms on y7 and in 2 4 the strong irreducibility hypotheses we wish to make in 5 we show how our hypotheses imply the existence of a finite number up to isomorphism of euclidean z g lattices l in v with endg l a maximal order in endg v we give some examples with dim v 8 in 6 and in 7 9 discuss the invariants of l such as the dual lattice and theta function the rest of the paper is devoted to examples in most g is a finite group of lie type and v is obtained as an irreducible summand of the weil representation of g some of the representation theoretic problems left open by this paper are to find all examples of pairs g v satisfying the strong irreducibility hypotheses of 2 4 and to determine the invariants shortest nonzero vector theta function thompson series of the g lattices l so effortlessly constructed inside v

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

lower bounds for the minima of certain symplectic and unitary group lattices we obtain lower bounds for the minimal norms in several families of integral lattices associated to the weil representations of finite symplectic and unitary groups ultimately these bounds come from elkies bounds for a certain family of mordell weil lattices of elliptic curves and from the minima of the barnes wall lattices

the involution width of finite simple groups abstract for a finite group generated by involutions the involution width is defined to be the minimal k n such that any group element can be written as a product of at most k involutions we show that the involution width of every non abelian finite simple group is at most 4 this result is sharp as there are families with involution width precisely 4

the time for a round trip in a cycle of exponential queues in 1 chow obtained the steady state distribution of the cycle time of a customer in a closed tandem queue composed of two exponential fifo servers and populated by an arbitrary fixed number of customers the present paper gives the generalization of chow s result by means of a different proof to the case of arbitrarily many exponential servers

crossing intervals of non markovian gaussian processes we review the properties of time intervals between the crossings at a level m of a smooth stationary gaussian temporal signal the distribution of these intervals and the persistence are derived within the independent interval approximation iia these results grant access to the distribution of extrema of a general gaussian process exact results are obtained for the persistence exponents and the crossing interval distributions in the limit of large m in addition the small time behavior of the interval distributions and the persistence is calculated analytically for any m the iia is found to reproduce most of these exact results and its accuracy is also illustrated by extensive numerical simulations applied to non markovian gaussian processes appearing in various physical contexts

universal asymptotic statistics of maximal relative height in one dimensional solid on solid models we study the probability density function p h m l of the maximum relative height h m in a wide class of one dimensional solid on solid models of finite size l for all these lattice models in the large l limit a central limit argument shows that for periodic boundary conditions p h m l takes a universal scaling form p h m l ensuremath sim sqrt 12 w l ensuremath 1 f mathbf h m sqrt 12 w l mathbf with w l the width of the fluctuating interface and f x the airy distribution function for one instance of these models corresponding to the extremely anisotropic ising model in two dimensions this result is obtained by an exact computation using the transfer matrix technique valid for any lg0 these arguments and exact analytical calculations are supported by numerical simulations which show in addition that the subleading scaling function is also universal up to a nonuniversal amplitude and simply given by the derivative of the airy distribution function f ensuremath x

extensions of unipotent characters and the inductive mckay condition abstract we prove that unipotent characters of groups of lie type extend to their inertia groups in the full automorphism group we then apply this in order to show that certain groups of lie type are good for the prime p 3 or p 2 this deals with all the exceptions not covered by our general approach for verifying the mckay conjecture for all groups of lie type in non defining characteristic