character stabilizer limits relative to a normal nilpotent subgroup in his recent paper 11 dade established the existence of some rather unexpected invariants of certain irreducible characters of finite groups the purpose of the present paper is to state and to provide a relatively easy proof for a somewhat weakened and less general form of dade s result in fact dade in his introduction to 11 explicitly encouraged me to publish this simplified approach to explain and motivate our theorem we need to discuss stabilizer limits let x e irr g we work only with characters over the complex numbers dade on the other hand considers other characteristic zero fields too if ma g and 8 is an irreducible constituent of xm let t z o the inertia group then there is a unique character ve irr ti 0 such that qg x we call q the clzfford correspondent of x with respect to 0 note that by taking m 1 we see that x is one of its own clifford correspondents we can repeat this process and consider clifford correspondents in t for the clifford correspondent q e irr t of x a character arising via any number of such iterations we call a compound clifsord correspondent ccc of x and we denote the set of these objects by ccc x note that if ic eccc x then ic x consider minimal element of ccc x these are the compound clifford correspondents of x which themselves have no proper clifford correspondents in other words they are quasiprimitive following t berger and dade we call these characters the stabilizer limits of x what do the different stabilizer limits for a given x e irr g have in common in general the answer is not much

indirect acquisition of information in quantum mechanics long sequences of successive direct projective measurements or observations of just a few uninteresting physical quantities pertaining to a quantum system such as clicks of some detectors may reveal indirect but precise and unambiguous information on the values of some very interesting observables of the system in this paper the mathematics underlying this claim is developed i e we attempt to contribute to a mathematical theory of indirect and in particular non demolition observations and measurements in quantum mechanics our attempt leads us to make some novel uses of classical notions and results of probability theory such as the algebra of functions measurable at infinity the central limit theorem results concerning relative entropy and its role in the theory of large deviations etc

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

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

dini derivative and a characterization for lipschitz and convex functions on riemannian manifolds dini derivatives in riemannian manifold settings are studied in this paper in addition a characterization for lipschitz and convex functions defined on riemannian manifolds and sufficient optimality conditions for constraint optimization problems in terms of the dini derivative are given

p length and character degrees in p solvable groups abstract let g be a p solvable group where p is a prime we prove that the p length of g is less or equal then the number of distinct irreducible character degrees of g not divisible by p furthermore we prove that the result still holds if we impose some restriction on the field of values of the characters in particular if p 2 we can consider only rational valued characters

quantum stochastic processes as models for state vector reduction an elementary introduction of quantum state valued markovian stochastic processes qsp for n state quantum systems is given it is pointed out that a so called master constraint must be fulfilled for a given master equation a continuous and as a new alternative possibility a discontinuous qsp are derived both are discussed as possible models for state reduction during measurement

iterated stochastic measurements we describe a measurement device principle based on discrete iterations of bayesian updating of system state probability distributions although purely classical by nature these measurements are accompanied with a progressive collapse of the system state probability distribution during each complete system measurement this measurement scheme finds applications in analysing repeated non demolition indirect quantum measurements we also analyse the continuous time limit of these processes either in the brownian diffusive limit or in the poissonian jumpy limit in the quantum mechanical framework this continuous time limit leads to belavkin s equations which describe quantum systems under continuous measurements this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

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

nonsmooth analysis and hamilton jacobi equations on riemannian manifolds we establish some perturbed minimization principles and we develop a theory of subdifferential calculus for functions defined on riemannian manifolds then we apply these results to show existence and uniqueness of viscosity solutions to hamilton jacobi equations defined on riemannian manifolds

large time behavior and convergence rate for quantum filters under standard non demolition conditions a quantum system mathcal s undergoing continuous time measurement is usually described by a jump diffusion stochastic differential equation such an equation is called a quantum filtering equation or quantum stochastic master equation and its solution is called a quantum filter or quantum trajectory this solution describes the evolution of the state of mathcal s in the context of quantum non demolition measurement we investigate the large time behavior of this solution it is rigorously shown that for large time this solution behaves as if a direct von neumann measurement has been performed at time 0 in particular the solution converges to a random pure state which can be directly linked to the wave packet reduction postulate using the theory of girsanov transformation we obtain the explicit rate of convergence towards this random state the problem of state estimation used in experiment is also investigated

counting objects which behave like irreducible brauer characters of finite groups let n 0 g where g is a finite group so that g acts on irr n the set of irreducible ordinary complex characters of n suppose 8 e irr n is g invariant and write irr g 10 to denote the set of those x e irr g such that x involves and hence is a multiple of 8 how large is irr g io the answer to this question is well known and not very hard to prove analogously to the fact that lirr g i is equal to the number of conjugacy classes of g it turns out that iirr gjb l is equal to the number of certain good classes in the group g n there are two equivalent descriptions of which classes of g n these are one characterization depends on a certain complex twisted group algebra of g n and in this formulation the result which counts irr g 1 t9 is due to i schur s the other formulation of this result due to p x gallagher z defines goodness internally to g it depends on the character theory of g and its subgroups specifically gallagher defines an element of 2 e g n to be o good if some extension of 8 to n g is invariant in c where c n c jg the property of being q good is preserved by conjugacy in gin and gallagher shows that lirr g i 13 l is equal to the number of classes of g in which consist of good elements somewhat less well known is the corresponding result for irreducible brauer characters with respect to some prime p if 8 e ibr n is g invariant we intend to count the elements of ibr g t9 i e the irreducible brauer characters of g whose restrictions to n are multiples of 8 in view of the fact that iibr g equals the number of p regular classes of g it is reasonable to guess that iibr gl f l is the number of e good p regular

irreducible constituents of induced monomial characters abstract an irreducible complex character of a finite group is called monomial if it is induced from a linear i e degree one character of a subgroup let g be a p solvable group n a normal subgroup of g and a monomial character of n of p power degree we present a theorem stating that under certain conditions imposed on g n and p the irreducible characters of g lying over are again monomial this is a converse to a recent theorem of e c dade

actions and invariant character degrees iii point subgroup in general we use the same notation as in 6 and 7 wx wxpart of the proof of theorem a depends on the basic properties of thegajendragadkar p special characters 1 and we assume the reader iswxfamiliar with those however we will repeatedly use a deeper fact anirreducible character a of a hall p subgroup

on normal subgroups and simple modules with a given vertex in a p solvable group abstract let n be a normal subgroup of a p solvable group g and let m be a simple fn module where f is an algebraically closed field of characteristic p let q be any p subgroup of g in this paper we count the number of simple fg modules lying over m and having q as a vertex

subgroups with the character restriction property suppose he g are finite groups and every ordinary irreducible character 0 irr h is the restriction xh of some x e irr g following 2 we will say that h has property cr character restriction in g when this occurs an obvious sufficient condition for cr to hold is that h has a normal complement in g but this is of course not necessary as for instance a cyclic group g of order 4 shows a number of authors have shown however that property cr together with suitable additional hypotheses on h and g does imply the existence of a normal complement for h for instance sah 6 shows this if h is a hall rc subgroup of g and either h is solvable or g is n separable also hawkes and humphreys 2 prove that cr yields a normal complement if g is solvable and h is an f projector for g where 9 is any saturated formation among the results in the present paper are a few more situations where cr yields a normal complement the first of these generalizes sah s result

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

fully ramified subgroups a configuration that occurs often in character theory of solvable groups and therefore has been the object of much research is that of fully ramified sections see 7 6 4 23 a normal subgroup h of g is said to be fully ramified in g if there exists an irreducible character 0 of h such that 0 ed for x some irreducible character of g and xh e0 the purpose of this paper is to study a generalization of this situation which seems to occur quite often that is when h is not known to be normal a technique is developed to study such situations based on the following result

proximal subgradient and a characterization of lipschitz function on riemannian manifolds a characterization of lipschitz behavior of functions defined on riemannian manifolds is given in this paper first it is extended the concept of proximal subgradient and some results of proximal analysis from hilbert space to riemannian manifold setting a technique introduced by clarke stern and wolenski f h clarke r j stern p r wolenski subgradient criteria for monotonicity the lipschitz condition and convexity canad j math 45 1993 1167 1183 for generating proximal subgradients of functions defined on a hilbert spaces is also extended to riemannian manifolds in order to provide that characterization a number of examples of lipschitz functions are presented so as to show that the lipschitz behavior of functions defined on riemannian manifolds depends on the riemannian metric

vertices for characters of p solvable groups suppose that g is a finite p solvable group we associate to every irreducible complex character x irr g of g a canonical pair q where q is a p subgroup of g and irr q uniquely determined by x up to g conjugacy this pair behaves as a green vertex and partitions irr g into families of characters using the pair q we give a canonical choice of a certain p radical subgroup r of g and a character irr r associated to x which was predicted by some conjecture of g r robinson

relative partial characters and relative blocks of p solvable groups abstract suppose that g is a finite p solvable group and let n be a normal subgroup let irr n with inertial group t in g suppose extends to a subgroup r of t containing n such that r n is a sylow p subgroup of the quotient group t n we show that the set irr g of irreducible characters of g lying over can be partitioned into blocks which behave like the classical brauer p blocks we also establish a well behaved one to one correspondence between the blocks of g and certain brauer p blocks of a central extension of t n by a p subgroup

character correspondences and subgroups of operator groups abstract let a and g be finite groups with coprime orders and suppose that a acts on g by automorphisms let g a irr a g irr c g a be the glauberman isaacs correspondence let b a and let irr a g we examine the conjecture that g a is an irreducible constituent of the restriction of g b to c g a and show that it is valid if g is supersolvable then we show when the analog of this conjecture for brauer characters and the uno wolf correspondence holds

counting characters above invariant characters in solvable groups abstract this paper discusses two related questions first given a g invariant character of a normal subgroup n of a solvable group what can we say if the number of characters of g above is in some sense as small as possible isaacs and navarro 5 have shown that under certain assumptions about primes dividing the order of the group one can show that g n must have a very particular structure here we show that these assumptions can be weakened to obtain results about all solvable groups we also discuss a related question about blocks for a prime p and a p block b of g we let k b denote the number of ordinary characters in b it is relatively easy to show that k b is bounded below by k g d which is the number of conjugacy classes of g that intersect the defect group d of b in this paper we ask what can be said if equality is achieved we show that for p solvable groups if k b k g d then b is nilpotent and thus k b irr d in addition we show that this result holds for many blocks of arbitrary finite groups including all blocks of the symmetric groups we also extend a result on fully ramified coprime actions in 5

the minimal number of characters over a normal p subgroup abstract if n is a normal p subgroup of a finite group g and irr n is a g invariant irreducible character of n then the number irr g of irreducible characters of g over is always greater than or equal to the number k p g n of conjugacy classes of g n consisting of p elements in this paper we investigate when there is equality

a canonical class of p degree characters of sn abstract let p be an odd prime we show the existence of a natural correspondence defined by restriction of characters between the irreducible characters of degree coprime to p of the symmetric group s p k and those of the k fold wreath product s p s p this result allows us to identify a candidate set of non exceptional characters for symmetric groups

sylow normalizers and brauer character degrees suppose that g is a finite group in this note we show that a local condition about sylow normalizers is equivalent to a global condition on the degrees of certain irreducible brauer characters of g theorem a let g be a finite p q solvable group and let q sylq g and p sylp g then every irreducible p brauer character of g of q degree has p degree if and only if ng q is contained in some g conjugate of ng p theorem a needs a solvability hypothesis if p 7 then the irreducible p brauer characters of the group g psl 2 27 have degrees 1 13 26 28 if we set q 2 then each q degree is also a p degree

height zero characters and normal subgroups in p solvable groups abstract fix a prime number p and let n be a normal subgroup of a finite p solvable group g let b be a p block of n and suppose b is a p block of g covering b let d be a defect group for the fong reynolds correspondent of b with respect to b and let b be the unique p block of n n g d having defect group d and inducing b suppose further that irr b and let irr 0 b be the set of irreducible characters in b of height zero that lie over we show that the number of characters in irr 0 b is equal to the number of characters in t irr 0 b t where t runs through the inertial group t of b in g this result generalizes a theorem of t okuyama and m wajima which confirms the alperin mckay conjecture for p solvable groups

newton methods for nonsmooth convex minimization connections among lagrangian riemannian newton and sqp methods this paper studies newton type methods for minimization of partly smooth convex functions sequential newton methods are provided using local parameterizations obtained from lagrangian theory and from riemannian geometry the hessian based on the lagrangian depends on the selection of a dual parameter g by revealing the connection to riemannian geometry a natural choice of g emerges for which the two newton directions coincide this choice of g is also shown to be related to the least squares multiplier estimate from a sequential quadratic programming sqp approach and with this multiplier sqp gives the same search direction as the newton methods

primitive characters of subgroups ofm groups one of the hardest areas in the character theory of solvable groups continues to be the monomial groups a finite group is said to be an m group or monomial if all of its irreducible characters are monomial that is to say induced from linear characters two are still the main problems on m groups are hall subgroups of m groups monomial under certain oddness hypothesis are normal subgroups of m groups monomial in both cases there is evidence that this could be the case the primitive characters of the subgroups in question are the linear characters this is the best result up to date 4 6 recently some idea appears to be taking form in 14 t okuyama proved that if g is an m group and p is a sylow p subgroup of g then n p p is an m group some time later and more generally i m lsaacs shows that if h is a hall subgroup of an m group then n s h h is also an m group 91 since it is well known that normal hall subgroups of m groups are m groups in this case the strong hypothesis of normality makes the problem immediate perhaps it is not only true that hall subgroups of m groups are m groups but something stronger that hall subgroup normalizers also are in this direction we prove theorem a let g be an m group and let h be a hall subgroup o f g i f c e i r r nc h is primitive then c is linear

bounds on the number and sizes of conjugacy classes in finite chevalley groups with applications to derangements we present explicit upper bounds for the number and size of conjugacy classes in finite chevalley groups and their variations these results have been used by many authors to study zeta functions associated to representations of finite simple groups random walks on chevalley groups the final solution to the ore conjecture about commutators in finite simple groups and other similar problems in this paper we solve a strong version of the boston shalev conjecture on derangements in simple groups for most of the families of primitive permutation group representations of finite simple groups the remaining cases are settled in two other papers of the authors and applications are given in a third

generalized gradients and characterization of epi lipschitz sets in riemannian manifolds abstract in this paper a notion of generalized gradient on riemannian manifolds is considered and a subdifferential calculus related to this subdifferential is presented a characterization of the tangent cone to a nonempty subset s of a riemannian manifold m at a point x is obtained then these results are applied to characterize epi lipschitz subsets of complete riemannian manifolds

primitive characters of maximal subgroups of solvable groups abstract let g be an m group and suppose that h is a maximal subgroup of odd index and that irr h is primitive it was shown by mark lewis that 1 2 divides g h we show in this situation that either 1 2 g h or 1 1 a number of related results are established for groups that are not necessarily m groups

vertices and normal subgroups in solvable groups abstract for a finite group g and a fixed prime p one can attach to each irreducible brauer character of g a p subgroup q called the vertex of that is unique up to conjugacy in this paper we examine the behavior of the vertices of with respect to normal subgroups when g is assumed to be p solvable for arbitrary finite p solvable groups we develop a correspondence between the set of brauer characters of g with vertex q and the set of brauer characters of a certain subgroup of g with vertex q moreover in the case that g has odd order we extend a result of navarro regarding the behavior with respect to normal subgroups of a correspondence of brauer characters of p degree to a result regarding the behavior with respect to normal subgroups of a correspondence of brauer characters of arbitrary degree

finite time consensus on strongly convex balls of riemannian manifolds with switching directed communication topologies abstract it is known that a consensus problem on any connected complete riemannian manifold can be transformed into the one on its strongly convex balls via the compression decompression along geodesics from the viewpoint of interior metrics this paper mainly provides a consensus protocol for strongly convex geodesic balls in which the communication can be switching and directed with the aid of nonsmooth analysis tools on riemannian manifolds our analysis shows that all dynamical points involved can achieve consensus in finite time meanwhile the corresponding global algorithm is given with its application to the consensus problem of rotation attitudes as well as a case simulation to demonstrate and verify our proposed techniques

irreducible characters and normal subgroups in groups of odd order abstract if b is a p block of a normal subgroup n of a finite group g of odd order and b is its brauer correspondent in n n q where q is a defect group of b then for any p block b of g over b there exists a natural height preserving bijection from the set of irreducible complex characters of b lying over height zero characters onto the set of irreducible complex characters of the harris knorr correspondent b of b over b lying over height zero characters

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

characters bilinear forms and solvable groups abstract we prove a number of results about the ordinary and brauer characters of finite solvable groups in characteristic 2 by defining and using the concept of the extended nucleus of a real irreducible character in particular we show that the isaacs canonical lift of a real irreducible brauer character has frobenius schur indicator 1 we also show that the principal indecomposable module corresponding to a real irreducible brauer character affords a quadratic geometry if and only if each extended nucleus is a split extension of a nucleus

bounds on the number of lifts of a brauer character in a p solvable group abstract the fong swan theorem shows that for a p solvable group g and brauer character ibr p g there is an ordinary character irr g such that 0 where 0 denotes restriction to the p regular elements of g this still holds in the generality of separable groups where ibr p g is replaced by i g for i g let l irr g 0 in this paper we give a lower bound for the size of l in terms of the structure of the normal nucleus of and if g is assumed to be odd and p we give an upper bound for l in terms of the vertex subgroup for

character heights and p length in blocks of p solvable groups abstract we give the lower bounds 2 l and 2 l 2 2 l 4 for the number of irreducible characters of height zero and positive height respectively in a p block of a p solvable group where l is the p length of an associated p solvable group we also prove some results on extensions of linear characters in p subgroups of p solvable groups

on the inductive blockwise alperin weight condition for classical groups abstract recently there has been substantial progress on the alperin weight conjecture as a step to establish the alperin weight conjecture for all finite groups we prove the inductive blockwise alperin weight condition for simple groups of classical type under some additional assumption

self dual modules for finite groups of odd order abstract let g be a finite group of odd order and let f be a finite field suppose that v is an fg module which carries a g invariant non degenerate bilinear form which is symmetric or symplectic we show that v contains a self perpendicular submodule if and only if the characteristic polynomials of some specified elements of g regarded as linear transformations of v are precisely squares this result can be applied to the study of monomial characters if the form on v is symplectic and self dual group codes if the form is symmetric

nonsmooth analysis on smooth manifolds we study infinitesimal properties of nonsmooth nondifferentiable functions on smooth manifolds the eigenvalue function of a matrix on the manifold of symmetric matrices gives a natural example of such a nonsmooth function a subdifferential calculus for lower semicontinuous functions is developed here for studying constrained optimization problems nonclassical problems of calculus of variations and generalized solutions of first order partial differential equations on manifolds we also establish criteria for monotonicity and invariance of functions and sets with respect to solutions of differential inclusions

complex obtuse random walks and their continuous time limits we study a particular class of complex valued random variables and their associated random walks complex obtuse random variables they generalize to the complex case the real valued obtuse random variables introduced in equations de structure pour des martingales vectorielles seminaire de probabilites xxviii p 256278 lecture notes in math vol 1583 springer berlin 1994 in order to understand the structure of normal martingales in mathbb r n the extension to the complex case is motivated by quantum statistical mechanics in particular for characterizing those quantum baths acting as classical noises the extension of obtuse random variables to the complex case is far from obvious and makes use of very interesting algebraical structures we show that complex obtuse random variables are characterized by a 3 tensor which admits certain symmetries we show that these symmetries are the exact 3 tensor analogue of the normal character for 2 tensors i e matrices that is a necessary and sufficient condition for being diagonalizable in some orthonormal basis we discuss the passage to the continuous time limit for these random walks and show that they converge in distribution to normal martingales in mathbb c n we show that the 3 tensor associated to these normal martingales encodes their behavior in particular the diagonalization directions of the 3 tensor indicate the directions of the space where the martingales behaves like a diffusion and those where it behaves like a poisson process we finally prove the convergence in the continuous time limit of the corresponding multiplication operators on the canonical fock space with an explicit expression in terms of the associated 3 tensor again