on discrete time linear systems with almost periodic kernels the paper considers several different stability notions for discrete time linear time varying systems specified by almost periodic kernels various input output stability notions are analysed including those in which the signal spaces consist of bounded power and asymptotically almost periodic functions further it is proved that every stable time varying discrete time system commuting with a polynomial in the shift operator is periodic an example shows that this result does not hold for unstable time varying systems

corrections and extensions of optimal control of linear systems with almost periodic input by g da prato and a ichikawa g da prato and a ichikawa consider in their 1987 paper the optimal control problem for linear systems with almost periodic inputs in order to get their results they use the space b ap 2 h the completion of the space of almost periodic functions in the quadratic mean they used that b ap 2 u inputs give almost periodic outputs by means of a scalar example we show that this does not hold in general we prove that one of their main results still holds furthermore we extend this result to all forcing terms in b ap 2 h and show that if the forcing term is almost periodic then the optimal control is almost periodic as well

on the 2 decomposition numbers of steinberg s triality groups d43 q q odd abstract we compute the 2 modular decomposition numbers of steinberg s triality groups d 4 3 q where q is a power of an odd prime except for two decomposition numbers in the steinberg character and classify all absolutely irreducible representations of d 4 3 q in non defining characteristic up to a certain degree

on decomposition matrices for graded algebras abstract we study the relations between the decomposition matrix of a module over a graded algebra and the decomposition matrix of its restriction to the grading subalgebra we show that under appropriate hypotheses if one of the decomposition matrices is stable unitriangular then so is the other a particular case is when one of them is square unitriangular then both are using these results we describe the decomposition matrices of the hecke algebras of the complex reflection groups g de e n with respect to the ones of the ariki koike algebras or the decomposition matrices of q schur algebras of type dn with respect to the ones of q schur algebras of type bn

decomposition numbers of so7 q and sp6 q abstract we complete the l modular decomposition numbers of the unipotent characters in the principal block of the special orthogonal groups so 7 q and the symplectic groups sp 6 q for all prime powers q and all odd primes l different from the defining characteristic

h sampled data synthesis and related numerical issues in this paper we present a new compact derivation of state space formulae for the so called discretisation based solution of the h sampled data control problem our approach is based on the established technique of continuous time lifting which is used to isometrically map the continuous time linear periodically time varying sampled data problem to a discretetime linear time invariant problem state space formulae are derived for the equivalent discrete time problem by solving a set of two point boundary value problems the formulae accommodate a direct feed through term from the disturbance inputs to the controlled outputs of the original plant and are simple requiring the computation of only a single matrix exponential it is also shown that the resultant formulae can be easily re structured to give a numerically robust algorithm for computing the state space matrices

on system gains for linear and nonlinear systems abstract system gains and bounds for system gains are determined for stable linear and nonlinear systems in different signal setups which include l p signal setups and certain persistent signal generalizations of the l 2 and l 1 signal setups these results show that robust h and l 1 control generalize to very versatile persistent signal settings relationships between different system gains are also derived finally an application of nonlinear system gain bounds is given by establishing induced l modelling error bounds for a class of generalized piecewise linear systems approximated with simpler linear time invariant models

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

quantum weyl reciprocity and tilting modules quantum weyl reciprocity relates the representation theory of hecke algebras of type a with that of q schur algebras this paper establishes that weyl reciprocity holds integrally i e over the ring q q 1 of laurent polynomials and that it behaves well under base change a key ingredient in our approach involves the theory of tilting modules for q schur algebras new results obtained in that direction include an explicit determination of the ringel dual algebra of a q schur algebra in all cases in particular in the most interesting situation the ringel dual identifies with a natural quotient algebra of the hecke algebra

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

tracking periodic signals with the overtaking criterion the overtaking criterion is suggested to evaluate the tracking of periodic signals on an infinite horizon existence uniqueness and a feedback form are established the asymptotic dynamics of the solution and its relation to the asymptotic performance of the finite time optimizers are examined

scopes reduction and morita equivalence classes of blocks in finite classical groups ii abstract we extend our work of j algebra 230 2000 378 423 on unipotent blocks of classical groups to blocks of orthogonal groups containing characters labelled by degenerate symbols we also show that for classical groups in the linear prime case there are only finitely many morita types of unipotent blocks with a given defect group

sectional rank and cohomology abstract donovan s conjecture implies a bound on the dimensions of cohomology groups in terms of the size of a sylow p subgroup and we give a proof of a stronger bound in terms of sectional p rank for dim h 1 g v we also prove a reduction theorem for higher cohomology

binomial determinants paths and hook length formulae abstract we give a combinatorial interpretation for any minor or binomial determinant of the matrix of binomial coefficients this interpretation involves configurations of nonintersecting paths and is related to young tableaux and hook length formulae

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

a new approach for analysis and synthesis of time varying systems in this paper new techniques are developed for the analysis of linear time varying ltv systems these lead to a formally simple treatment of robust control problems for ltv systems allowing methods more usually restricted to time invariant systems to be employed in the time varying case as an illustration of this methodology the so called h sub spl infin synthesis problem is solved for ltv systems

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

on donovan s conjecture abstract this article contains reduction theorems for some weaker variants of donovan s conjecture which supposes that for every finite group d of prime order p there are only finitely many morita equivalence classes of p blocks of finite groups having defect groups isomorphic to d i when restricting the conjecture to the case of abelian defect groups it can be reduced to only considering p blocks of a class of groups that are close to being direct products of simple groups ii when restricting the conjecture to the case of principal blocks of finite groups having an abelian sylow p subgroup it can be reduced to only considering p blocks of simple groups a weaker version of donovan s conjecture would suppose that there are only finitely many cartan matrices of p blocks of finite groups with defect group isomorphic to a given one iii this conjecture can be reduced to only considering p blocks of quasisimple groups with centers having orders prime to p

optimal overtaking control for linear time varying systems the problem of tracking given bounded signals is considered for time varying linear systems the cost functional is quadratic and optimal overtaking controls are sought both deterministic and stochastic cases are considered under stabilizability and detectability unique optimal controls are obtained

h optimization and optimal rejection of persistent disturbances abstract in this note it is shown that in a certain well known normed space of persistent signals a natural optimal disturbance rejection problem is equivalent to h optimization thus clarifying certain misconceptions on h optimization in the literature these conclusions are further strengthened using techniques of wiener s generalized harmonic analysis

optimal control of linear systems with almost periodic imputs in this paper we consider linear time invariant and periodic systems with periodic forcing terms we propose new quadratic control problems both deterministic and stochastic we also consider stochastic control with partial observation and show that the separation principle holds our mathematical models cover both finite and infinite dimensional systems

converse problems of fourier expansion and their applications abstract let f c r h have a countable frequency set freq f and satisfy parseval s equality we show that if f satisfies one of the following conditions a uniformly continuous and freq f has a unique limit point at infinity b indefinite integral is lipschitz freq f converges fast in some sense c in the case of euclidean space h all the coefficients are positive then f is pseudo almost periodic example is given to show that the conclusion cannot be improved the results are applied to the theory of riesz fischer and the optimal control theory

modeling of linear fading memory systems motivated by questions of approximate modeling and identification we consider various classes of linear time varying bounded input bounded output bibo stable fading memory systems and the characterizations are proved these include fading memory systems in general almost periodic systems and asymptotically periodic systems we also show that the norm and strong convergence coincide for bibo stable causal fading memory systems

robust performance against time varying structured perturbations in this paper we consider the problem of robust performance analysis for some nominal system m z against bounded linear time varying structured feedback perturbations we introduce a natural input output notion of rate of variation for a linear time varying operator we then exhibit upper and lower bounds on the maximum rate of variation of these perturbations against which robust performance is achievable using these bounds we show that the existence of frequency dependent d scales that render the norm of m z less than one is necessary and sufficient for robust performance against arbitrarily slowly varying structured linear perturbations of norm less than one this result suggests that it is natural and well justified to deal with the frequency dependent upper bound for the spl mu norm rather than spl mu itself this is reassuring given that the upper bound is easily and reliably computable while computation of complex spl mu appears difficult

an equivariant bijection between irreducible brauer characters and weights for sp 2n q abstract the longstanding alperin weight conjecture and its blockwise version have been reduced to simple groups recently by navarro tiep spath and koshitani thus to prove this conjecture it suffices to verify the corresponding inductive condition for all finite simple groups the first is to establish an equivariant bijection between irreducible brauer characters and weights for the universal covering groups of simple groups assume q is a power of some odd prime p we first prove the blockwise alperin weight conjecture for sp 2 n q and odd non defining characteristics if the decomposition matrix of sp 2 n q is unitriangular with respect to an aut sp 2 n q stable basic set this assumption holds for linear primes we can establish an equivariant bijection between the irreducible brauer characters and weights

new limit power function spaces to answer an open question we propose two function spaces one is banach and another is hilbert it is shown that the hilbert space is the largest one among those hilbert spaces in limit power function set whose members have associated fourier series in sense of a new basis and satisfy parseval s equality

the weighted fusion category algebra and the q schur algebra for gl 2 q abstract we show that the weighted fusion category algebra of the principal 2 block b 0 of gl 2 q is the quotient of the q schur algebra s 2 q by its socle for q an odd prime power as a consequence we get a canonical bijection between the set of isomorphism classes of simple k gl 2 q b 0 modules and the set of conjugacy classes of b 0 weights in this case

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

hyperoctahedral schur algebras we study the centralising algebra of a natural action of the hyperoctahedral group i e a finite weyl group of typebr on ther th tensor power of a2n dimensional space the centralising algebra of this is shown to have a product rule similar to schur s product rule in typea we deform this action to an action of the hecke algebra of typeband study the associated centralising algebra of typeband its dual we introduce and studyq permutation modules for the algebra

c sup algebra of strong limit power functions motivated by problems in robust control of power signal set h sub 2 e g h sub 2 is not a vector space fourier analysis cannot be carried out on h sub 2 in general we study those functions in h sub 2 which we call strong limit power we show that the set of all such functions is a sufficiently large c sup algebra fourier analysis is carried out on the functions in particular the uniqueness of the fourier expansion of a strong limit power function is established finally we point out how to analyze and reconstruct such functions

combinatorial verification of the elementary divisors of tensor products abstract using combinatorial arguments we determine the determinantal divisors for the tensor product of two matrices thereby obtaining its elementary divisors and jordan canonical form

stratifying endomorphism algebras associated to hecke algebras abstract let g be a finite group of lie type and let k be a field of characteristic distinct from the defining characteristic of g in studying the non describing representation theory of g the endomorphism algebra s g k end kg j ind g p j k plays an increasingly important role in type a by work of dipper and james s g k identifies with a q schur algebra and so serves as a link between the representation theories of the finite general linear groups and certain quantum groups this paper presents the first systematic study of the structure and homological algebra of these algebras for g of arbitrary type because s g k has a reinterpretation as a hecke endomorphism algebra it may be analyzed using the theory of hecke algebras its structure turns out to involve new applications of kazhdan lusztig cell theory in the course of this work we prove two stratification conjectures about coxeter group representations made by e cline b parshall and l scott mem amer math soc 591 1996 and we formulate a new conjecture about the structure of s g k we verify this conjecture here in all rank 2 examples

exterior and symmetric powers of modules for cyclic 2 groups abstract we prove a recursive formula for the exterior and symmetric powers of modules for a cyclic 2 group this makes computation straightforward previously a complete description was only known for cyclic groups of prime order

morita equivalences of ariki koike algebras we prove that every ariki koike algebra is morita equivalent to a direct sum of tensor products of smaller ariki koike algebras which have q connected parameter sets a similar result is proved for the cyclotomic q schur algebras combining our results with work of ariki and uglov the decomposition numbers for the ariki koike algebras defined over fields of characteristic zero are now known in principle

modular irreducibility of cuspidal unipotent characters we prove a long standing conjecture of geck which predicts that cuspidal unipotent characters remain irreducible after ell reduction to this end we construct a progenerator for the category of representations of a finite reductive group coming from generalised gelfand graev representations this is achieved by showing that cuspidal representations appear in the head of generalised gelfand graev representations attached to cuspidal unipotent classes as defined and studied in geck and malle j lond math soc 2 53 63 78 1996

fading memory feedback systems and robust stability abstract this paper considers fading memory for nonlinear time varying systems and associated problems of robust stability we define two notions of fading memory for stable dynamical systems uniform and pointwise we then provide conditions under which stable linear or nonlinear systems exhibit uniform or pointwise fading memory in particular we show that 1 all stable discrete time linear time varying ltv systems have uniform fading memory 2 all stable continuous time ltv systems have pointwise fading memory and 3 stable finite dimensional continuous time ltv systems have uniform fading memory we then show that a version of the small gain theorem which employs the asymptotic gain of a fading memory system is necessary for the stable invertibility of certain feedback operators these results are presented for both continuous time and discrete time systems using general lp or l p notions of input output stability and generalize existing results for l2 stability we further investigate fading memory in a closed loop context for linear plants we parametrize all nonlinear controllers which lead to closed loop pointwise fading memory

quadratic games and h type problems for time varying systems abstract h type control problems with state feedback are considered for time varying systems first general quadratic differential games are studied and then the relationship between h problems and differential games is clarified a necessary and sufficient condition for the existence of a stabilizing controller such that the norm of the operator from the disturbance to the output is less than a prescribed number is given in terms of a riccati equation earlier results on time invariant systems are found to be special cases of our main results the finite horizon problem is also studied

lethargy results in lti system modelling modelling of linear time invariant lti systems is studied in an approximate modelling framework motivated by modern robust control theory as convolution type lti systems are represented by their unit impulse responses i e through a sequence of real complex numbers this paper provides also system theoretic results on modelling of sequences and their associated transfer functions the modelling results derived in this paper are mostly of lethargy type i e they establish limitations in approximate modelling of lti systems non existence of coprime factorizations is established for a number of different types of lti systems this leads to the somewhat surprising conclusion that many lti systems are not even stabilizable and that an even larger class of lti systems cannot be well approximated by rational transfer function models in addition it is discussed that often the rate of rational approximation of lti systems is not essentially better than the rate of approximation in some basis such as finite impulse response fir models furthermore different definitions of the important finite power signal setup are studied and new results given

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

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

elementary divisors of tensor products and p ranks of binomial matrices abstract let f be a field of characteristic p and let a and b be two jordan blocks with nonzero eigenvalues the reduction formulas of c w norman enable us to compute the elementary divisors of a b we investigate connections between the elementary divisors of a b and binomial matrices in particular we show that norman s reduction formulas can be used to compute the p rank of certain binomial matrices

decomposing modular tensor products and periodicity of jordan partitions let jr denote an r r matrix with minimal and characteristic polynomials t 1 r suppose r 6 s it is not hard to show that the jordan canonical form of jr js is similar to j 1 jr where 1 r 0 and p r 1 i rs the partition r s p 1 r of rs which depends only on r s and the characteristic p char f has many applications including to the study of algebraic groups we prove new periodicity and duality results for r s p that depend on the smallest p power exceeding r this generalizes results of j a green b srinivasan and others which depend on the smallest p power exceeding the potentially large integer s we show that for fixed r we can construct a finite table allowing the computation of r s p for all s r and all primes p this generalizes work of k i iima and r iwamatsu

bounded power signal spaces for robust control and modeling the nonlinear space of signals allowing wiener s generalized harmonic analysis gha the linear bounded power signal spaces of beurling marcinkiewicz and wiener and a new linear bounded power space are studied from a control and systems theory perspective specifically it is shown that the system power gain is given by the h infty norm of the system transfer function in each of these spaces for a large class of power stable finite and infinite dimensional systems the gha setup is shown to possess several limitations for the purpose of robustness analysis which motivates the use of the other more general nonstationary signal spaces the natural double sided time axis versions of bounded power signal spaces are shown to break the symmetry between hardy space h infty methods and bounded power operators e g the system transfer function being in h infty does not imply that a causal linear time invariant lti system is bounded as an operator on any of the double sided versions of the studied bounded power signal spaces