degree and valuation of the schur elements of cyclotomic hecke algebras following the generalization of the notion of families of characters defined by lusztig for weyl groups to the case of complex reflection groups thanks to the definition given by rouquier we show that the degree and the valuation of the schur elements functions a and a remain constant on the families of the cyclotomic hecke algebras of the exceptional complex reflection groups the same result has already been obtained for the groups of the infinite series and for some special cases of exceptional groups

on involutions in symmetric groups and a conjecture of lusztig abstract let w s be a coxeter system equipped with a fixed automorphism of order 2 which preserves s lusztig and with vogan in some special cases have shown that the space spanned by set of twisted involutions i e elements w w with w w 1 was naturally endowed with a module structure of the hecke algebra of w s with two distinguished bases which can be viewed as twisted analogues of the well known standard basis and kazhdan lusztig basis the transition matrix between these bases defines a family of polynomials p y w which can be viewed as twisted analogues of the well known kazhdan lusztig polynomials of w s lusztig has conjectured that this module is isomorphic to the right ideal of the hecke algebra with hecke parameter u 2 associated to w s generated by the element x w w u l w t w in this paper we prove this conjecture in the case when id and w s n the symmetric group on n letters our methods are expected to be generalised to all the other finite crystallographic coxeter groups

on the decomposition numbers of the hecke algebra of type dn when n is even abstract let n 4 be an even integer let k be a field with char k 2 and q an invertible element in k such that i 1 n 1 1 q i 0 in this paper we study the decomposition numbers over k of the iwahori hecke algebra h q d n of type d n we obtain some equalities which relate its decomposition numbers with certain schur elements and the decomposition numbers of various iwahori hecke algebras of type a with the same parameter q when char k 0 this completely determine all of its decomposition numbers the main tools we used are the morita equivalence theorem established in j hu a morita equivalence theorem for hecke algebra h q d n when n is even manuscripta math 108 2002 409 430 and certain twining character formulae of weyl modules over a tensor product of two q schur algebras

discriminants of brauer algebras in this paper we compute gram determinants associated to all cell modules of brauer algebras fancyscript b n partial theoretically we know when a cell module of fancyscript b n partial is equal to its simple head this gives a solution of this long standing problem

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

tilting modules for cyclotomic schur algebras this paper investigates the tilting modules of the cycloto mic q schur alge bras the young modules of the ariki koike algebras and the interconnections between them the main tools used to understand the tilting modules are contragredient duality and the specht filtrations and dual specht filtrations of cert ain permutation modules sur prisingly weyl filtrations which are in general more power ful than specht filtrations play only a secondary role

a class of hamiltonian symplectic methods for solving the algebraic riccati equation abstract the algebraic riccati equation can be solved by finding a certain invariant subspace of a related hamiltonian matrix g ammar and v mehrmann devised a method for calculating this invariant subspace that exploits the hamiltonian structure of the matrix by using unitary symplectic similarity transformations this paper discusses a class of methods that find the invariant subspace by using symplectic similarity transformations that are not necessarily unitary the method of ammar and mehrmann is a special case

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

isomorphisms between simple modules of degenerate cyclotomic hecke algebras abstract we give explicit isomorphisms between simple modules of degenerate cyclotomic hecke algebras defined via various cellular bases a special case gives a generalized mullineux involution in the degenerate case

a symplectic qr like algorithm for the solution of the real algebraic riccati equation a method is presented to solve the real algebraic riccati equation xnx xa a t x k 0 where k k t and n n t the solution for the corresponding eigenvalue problem mx lambda x where m is a hamiltonian matrix is computed by an algorithm similar to the qr algorithm special symplectic matrices are used for the transformation of m such that the hamiltonian form is preserved during the computations

stability of homogeneous nonlinear systems with sampled data inputs abstract the main goal of this article is to use properties of homogeneous systems for addressing the problem of stability for a class of nonlinear systems with sampled data inputs this nonlinear strategy leads to several kinds of stability i e local asymptotic stability global asymptotic stability or global asymptotic set stability depending on the sign of the degree of homogeneity the results are illustrated with the case of the double integrator

on hamiltonian and symplectic hessenberg forms abstract characterizations are given for the hamiltonian matrices that can be reduced to hamiltonian hessenberg form and the symplectic matrices that can be reduced to symplectic hessenberg form by orthogonal symplectic similarity transformations the reduction to these special hessenberg forms is the missing link in the solution of the open problem of constructing a stable structure preserving qr like method of complexity o n 3 for the computation of invariant subspaces of hamiltonian and symplectic matrices our considerations lead us to propose an approach to the computation of lagrangian invariant subspaces of a hamiltonian or symplectic matrix

a schur decomposition for hamiltonian matrices abstract a schur type decomposition for hamiltonian matrices is given that relies on unitary symplectic similarity transformations these transformations preserve the hamiltonian structure and are numerically stable making them ideal for analysis and computation using this decomposition and a special singular value decomposition for unitary symplectic matrices a canonical reduction of the algebraic riccati equation is obtained which sheds light on the sensitivity of the nonnegative definite solution after presenting some real decompositions for real hamiltonian matrices we look into the possibility of an orthogonal symplectic version of the qr algorithm suitable for hamiltonian matrices a finite step initial reduction to a hessenberg type canonical form is presented however no extension of the francis implicit shift technique was found and reasons for the difficulty are given

matrix units and generic degrees for the ariki koike algebras this paper uses the murphy basis of the ariki koike algebras to explicitly construct a complete set of primitive idempotents when these algebras are semisimple and q 1 as a consequence we obtain an explicit wedderburn basis for the ariki koike algebras finally we use these idempotents to compute the generic degrees of the ariki koike algebras

the jantzen sum formula for cyclotomic schur algebras the cyclotomic q schur algebra was introduced by dipper james and mathas in order to provide a new tool for studying the ariki koike algebra we here prove an analogue of jantzen s sum formula for the cyclotomic q schur algebra among the applications is a criterion for certain specht modules of the ariki koike algebras to be irreducible

symmetric cyclotomic hecke algebras abstract in this paper we prove that the generic cyclotomic hecke algebras for imprimitive complex reflection groups are symmetric over any ring containing inverses of the parameters for this we show that the determinant of the gram matrix of a certain canonical symmetrizing form introduced by k bremke and g malle indag math 8 1997 453 469 is a unit in any such ring on the way we show that the ariki koike bases of these algebras are also quasi symmetric