kms states of quasi free dynamics on pimsner algebras abstract a continuous one parameter group of unitary isometries of a right hilbert c bimodule induces a quasi free dynamics on the cuntz pimsner c algebra of the bimodule and on its toeplitz extension the restriction of such a dynamics to the algebra of coefficients of the bimodule is trivial and the corresponding kms states of the toeplitz cuntz pimsner and cuntz pimsner c algebras are characterized in terms of traces on the algebra of coefficients this generalizes and sheds light onto various earlier results about kms states of the gauge actions on cuntz algebras cuntz krieger algebras and crossed products by endomorphisms we also obtain a more general characterization in terms of kms weights for the case in which the inducing isometries are not unitary and accordingly the restriction of the quasi free dynamics to the algebra of coefficients is nontrivial

conjugacy classes of g 2 and spectral rigidity the free group 2 is generated by a 1 2 0 1 and b 10 21 and setting x a exp 2 i x b exp 2 i defines a unitary character on 2 for 0 1 a program is devised to compute tr conj class summed over all primitive conjugacy classes of 2 of trace tr combined with a luo sarnak theorem this yields lower bounds for the spectral variance for a large sampling of characters in the range 0 1 the results indicate that the berry conjecture for spectral rigidity does not hold for this set of classically chaotic systems the program is also used to compute x ln n summed over all primitive conjugacy classes of 2 of norm n x the function 0 x is asymptotic to x and the remainder can be written as x x x the values of x are computed for all traces between 3202 and 4802 here x tr 2 2 the s cluster around 0 6 attaining a maximum of 2 3 finally it is proved that the remainder 0 x x has a negative bias by showing that the mean normalized remainder converges to a negative limit

selberg super trace formula for super riemann surfaces ii elliptic and parabolic conjugacy classes and selberg super zeta functions further contributions developing a super analogue of the classical selberg trace formula the selberg super trace formula are presented this paper deals with the calculation of contributions arising from elliptic and parabolic conjugacy classes to the selberg super trace formula for super riemann surfaces analytic properties and the functional equation for the corresponding selberg super zeta functionr0 r1 andzs respectively are derived and discussed in particular the elliptic contributions to a super fuchsian group only alter the multiplicities of the trivial zeros and poles of the selberg super zeta functionr0 r1 andzs respectively already due to the hyperbolic conjugacy classes the parabolic conjugacy classes introduce new features in the analytical structure

level statistics in arithmetical and pseudo arithmetical chaos we investigate a long standing riddle in quantum chaos posed by certain fully chaotic billiards with constant negative curvature whose periodic orbits are highly degenerate in length depending on the boundary conditions for the quantum wavefunctions the energy spectra either have uncorrelated levels usually associated with classical integrability or conform to the universal wigner dyson type although the classical dynamics in both cases is the same the resolution turns out surprisingly simple the maslov indices of orbits within multiplets of degenerate length either yield equal phases for the respective feynman amplitudes and thus poissonian level statistics or give rise to amplitudes with uncorrelated phases leading to wigner dyson level correlations the recent semiclassical explanation of spectral universality in quantum chaos is thus extended to the latter case of pseudo arithmetical chaos

generators of large subgroups of units of integral group rings of nilpotent groups let g be a finite nilpotent group so that all simple components d n n n 2 of qg satisfy the congruence subgroup theorem suppose that for all odd primes p dividing g the hamiltonian quaternions h split over the pth cyclotomic field q p then new units b3 are introduced so that b1 b2 b 2 b3 is of finite index in u zg

periodic orbits and semiclassical quantization of dispersing billiards periodic orbits in a dispersing billiard system consisting of three circular arcs are studied numerically by using a partial coding rule together with an efficient method for enumerating periodic orbits on the real billiard plane by examining several statistical measures it is shown that the length spectrum and the stability exponents are highly uncorrelated the validity of the semiclassical trace formula is also tested and a remarkable agreement of the semiclassical and quantum density of states is obtained at least for about the lower 15 levels

efficient linearly implicit methods for nonlinear multidimensional parabolic problems in this paper we present and analyze new methods to integrate multidimensional parabolic problems with nonlinear reaction terms we consider a first standard spatial semidiscretization stage obtaining a family of stiff nonlinear initial value problems the totally discrete schemes are obtained by numerical integration in time of such problems using new additive runge kutta schemes we show that the resulting algorithms which are only linearly implicit reach unconditional convergence if the additive methods used have suitable properties of linear absolute stability besides they have a computational cost per time step with the same order as the explicit methods finally three numerical experiences are shown in order to illustrate the behavior of our methods

a realisation of the quantum supergroup u glm n abstract we reconstruct the quantum enveloping superalgebra u gl m n over q via finite dimensional quantum schur superalgebras in particular we obtain a new basis containing the standard generators for u gl m n and explicit multiplication formulas of an arbitrary basis element by a generator

hecke algebras of semidirect products and the finite part of the connes marcolli c algebra abstract we study a c dynamical system arising from the ring inclusion of the 2 2 integer matrices in the rational ones the orientation preserving affine groups of these rings form a hecke pair that is closely related to a recent construction of connes and marcolli our dynamical system consists of the associated reduced hecke c algebra endowed with a canonical dynamics defined in terms of the determinant function we show that the schlichting completion also consists of affine groups of matrices over the finite adeles and we obtain results about the structure and induced representations of the hecke c algebra in a somewhat unexpected parallel with the one dimensional case studied by bost and connes there is a group of symmetries given by an action of the finite integral ideles and the corresponding fixed point algebra decomposes as a tensor product over the primes this decomposition allows us to obtain a complete description of a natural class of equilibrium states which conjecturally includes all kms states for 0 1

energy level statistics of an integrable billiard system in a rectangle in the hyperbolic plane a separable billiard system in the hyperbolic strip a particular realization of the hyperbolic plane is investigated the author considers a rectangle bounded by straight lines in the hyperbolic strip as an approximation of either a regular octagon with area a 4 pi or of a hyperbolic rectangle bounded by geodesics respectively due to the simple geometry of the approximation the schrodinger equation is separable and leads to simple transcendental equations for the eigenvalues in terms of legendre functions the statistical properties of the eigenvalue spectrum are investigated and checked with respect to weyl s 1912 law level spacing number variance and spectral rigidity respectively in particular the numerical results for the spectral rigidity are in close agreement with the semi classical theory of berry 1985

a new class of second order linearly implicit fractional step methods a new family of linearly implicit fractional step methods is proposed and analysed in this paper the combination of one of these time integrators with a suitable spatial discretization permits a very efficient numerical solution of semilinear parabolic problems the main quality of this new family of methods compared to other existing time integrators of this type is that they are stable when the spatial differential operator is decomposed in a number m of simpler operators which do not necessarily commute we prove that these methods satisfy this general stability result as well as they are second order consistent both consistency and stability are proven for an operator splitting in an arbitrary number m of terms m 2 finally a numerical experiment illustrates these theoretical results in the last section of the paper

group algebras of kleinian type and groups of units the algebras of kleinian type are finite dimensional semisimple rational algebras a such that the group of units of an order in a is commensurable with a direct product of kleinian groups we classify the schur algebras of kleinian type and the group algebras of kleinian type as an application we characterize the group rings rg with r an order in a number field and g a finite group such that rgis virtually a direct product of free by free groups the study of kleinian groups goes back to the works of poincar poi and bianchi bia and it has been an active field of research ever since in the last decades it is strongly related to the geometrization program of thurston for the classification of 3 manifolds egm mr mas thu the use of the methods of kleinian groups to the study of the groups of units of group rings was started in prr and led to the notion of algebras of kleinian type and finite groups of kleinian type let k be a number field a a central simple k algebra and r a z order in a let r 1 denote the group of units of r of reduced norm 1 every embedding k c induces an embedding a md c where d is the degree of a furthermore r 1 sld c we say that a is of kleinian type if either a k or a is a quaternion algebra over k and r 1 is a discrete subgroup of sl2 c for some embeddingof k in c more generally an algebra of kleinian type prr is by definition a direct sum of simple algebras of kleinian type a finite group g is of kleinian type if and only if the rational group algebra qg is of kleinian type the finite groups of kleinian type have been classified in jprrz where it has been also proved that a finite group g is of kleinian type if and only if the group of units zg of its integral group ring zg is commensurable with a direct product of free by free groups recently alan reid asked us in a private communication about the consequences of replacing the ring of rational integers by another ring of integers this leads to the following two problems problem 1 classify the group algebras of kleinian type of finite groups over number fields problem 2 given a group algebra of kleinian type kg describe the structure of the group of units of the group ring rg for r an order in k

periodic orbit sum rules for the hadamard gutzwiller model abstract it is shown how a variety of periodic orbit sum rules can be used to extract information about a quantum mechanical system whose classical counterpart is completely chaotic from knowledge only of the classical system and vice versa the basis is the selberg trace formula an exact analogue for the hadamard gutzwiller model of the semiclassical periodic orbit theory of gutzwiller which relates the quantal energies to the lengths of the periodic orbits of the classical system statistical properties of the quantal energies in the low energy region are studied where we restrict ourselves to the level spacing and spectral rigidity

semiclassical limits for the hyperbolic plane we study the concentration properties of high energy eigenfunctions for the laplace beltrami operator of the hyperbolic plane with special consideration of automorphic eigenfunctions at the center of our investigation is the microlocal lift of an eigenfunction to sl 2 introduced by s zelditch the microlocal lift is based on s helgason s fourier transform and has a straightforward description in terms of the lie algebra sl 2 we begin with an elementary demonstration of zelditch s exact differential equation for the microlocal lift further for a sequence of suitably bounded eigenfunctions with eigenvalues tending to infinity we show that the limit of microlocal lifts is a geodesic flow invariant positive measure our main consideration is the microlocal lifts of the elementary eigenfunctions constructed from the macdonald bessel functions we find that with a scaling of auxiliary parameters the corresponding high energy limit converges to the positive dirac measure for the lift to sl 2 of a single geodesic on the upper half plane in particular at high energy the microlocal lift of a macdonald bessel function is concentrated along the lift of a single geodesic we further find that the convergence is uniform in parameters and thus our considerations can be applied to study automorphic eigenfunctions in particular the maass cusp forms for certain cofinite subgroups of sl 2 a formula is presented for the microlocal lift of a cusp form as an integral of a family of dirac measures and a measure given directly by sums of products of the classical fourier coefficients of the cusp form the formula establishes that questions on quantum chaos for automorphic eigenfunctions are equivalent to classically stated questions on twisted sums of coefficients in particular we find that the uniform distribution for the microlocal limit is equivalent to a uniform distribution for the limit of twisted coefficient sums we extend the considerations to the case of the nonholomorphic eisenstein series for the modular group we find that a result of w luo and p sarnak is equivalent to a limit sum formula involving the riemann zeta function and the elementary divisor function the formula is suggestive of a formula of s ramanujan

applications of a computer implementation of poincare s theorem on fundamental polyhedra poincarcs theorem asserts that a group f of isometries of hyperbolic space h is discrete if its generators act suitably on the boundary of some polyhedron in h and when this happens a presentation of f can be derived from this action we explain methods for deducing the precise hypotheses of the theorem from calculation in f when f is algorithmi cally defined and we describe a file of fortran programs that use these methods for groups f acting on the upper half space model of hyperbolic 3 space h we exhibit one modest example of the application of these programs and we summarize computations of repesenta tions of groups psl 2 0 where 6 is an order in a complex quadratic number field in the early 1880 s h poincare discovered a general theorem allowing one to deduce the discreteness of and a presentation for a group g of isometries of hyperbolic space from its action on a hyperbolic polyhedron under certain condi tions theorems of this sort are part of the foundations of his theories of fuchsian and kleinian groups that have become very popular again and h seifert has recently given us a modern proof of a fairly general version of poincare s theorem in 12 see also 7 this theorem has been little used over the past century perhaps partly because its hypotheses have seemed very difficult to verify for a given group g except in very special circumstances one reason for doubting that poincare s theorem is unreasonably difficult to apply to fairly general discrete groups is that no general alternative method for accomplishing its tasks has been proposed the present paper is devoted to demonstrating that poincare s theorem can indeed be applied to given groups in apparently difficult cases and that much of the work can be done by a computer our experience suggests that the theorem is really very helpful in guiding the user to an understanding of the details of the action of g starting from a state of near ignorance an outline of this paper is as follows in section 1 we begin by stating seifert s version of poincare s theorem and explaining how we would apply it to an algorithmically defined group g of isometries of hyperbolic space h o in section 2 we specialize to the situation 00 for which we wrote our file poincare of fortran

hyperbolic octagons and teichmuller space in genus 2 in this paper we study different models of the teichmuller space of compact hyperbolic surfaces with special emphasis on their construction by geodesic octagons such surfaces have become increasingly interesting due to their applications to the physical behavior of chaotic quantum systems they are on the one hand complex enough to show the relevant features and on the other hand they possess a simple mathematical structure allowing practical implementations we give a new description of teichmuller space and the mapping class group in terms of geometric data of the octagons this provides modellings based on the vertices and also on the generators of the associated isometry group in addition we explicitly connect our approach with other models currently used in the literature helling matrices fenchel nielsen parameters the result of the paper may be considered as a computational tool kit to work with a specific model and to relate it with the others

transition from poissonian to gaussian orthogonal ensemble level statistics in a modified artin s billiard one wall of an artin s billiard on the poincar e half plane is replaced by a one parameter mathit c mathit p family of nongeodetic walls a brief description of the classical phase space of this system is given in the quantum domain the continuous and gradual transition from the poisson like to gaussian orthogonal ensemble goe level statistics due to the small perturbations breaking the symmetry responsible for the arithmetic chaos at mathit c mathit p 1 is studied another goe ensuremath rightarrow poisson transition due to the mixed phase space for large perturbations is also investigated a satisfactory description of the intermediate level statistics by the brody distribution was found in both cases the study supports the existence of a scaling region around mathit c mathit p 1 a finite size scaling relation for the brody parameter as a function of 1 mathit c mathit p and the number of levels considered can be established

from classical periodic orbits to the quantization of chaos we present a quantitative analysis of selberg s trace formula viewed as an exact version of gutzwiller s semiclassical periodic orbit theory for the quantization of classically chaotic systems two main applications of the trace formula are discussed in detail i the periodic orbit sum rules giving a smoothing of the quantal energy level density ii the selberg zeta function as a prototype of a dynamical zeta function defined as an euler product over the classical periodic orbits and its analytic continuation across the entropy barrier by means of a dirichlet series it is shown how the long periodic orbits can be effectively taken into account by a universal remainder term which is explicitly given as an integral over an orbit selection function numerical results are presented for the free motion of a point particle on compact riemann surfaces hadamard gutzwiller model which is the primary testing ground for our ideas relating quantum mechanics and classical mechanics in the case of strong chaos our results demonstrate clearly the crucial role played by the long periodic orbits an exact rule for quantizing chaos is derived for such systems where the dirichlet series representing the selberg zeta function converges on the critical line explicit formulae are given for the computation of the abscissae of absolute and conditional convergence respectively of these dynamical dirichlet series for the two riemann surfaces considered it turns out that one can cross the entropy barrier but that the critical line cannot be reached by a convergent dirichlet series it would seem that this is the main reason why the riemann siegel lookalike formula recently conjectured by m v berry and j p keating fails in generating the lower lying quantal energies for these strongly chaotic systems

construction of units in group rings of monomial and symmetric groups let rg be the group ring of a finite group g over an integral domain rand u rg its group of units in 11 for nilpotent groups g with a few exceptions if the order igi is even generators of a subgroup of finite index in the unit group u u 7lg of the integral group ring tlgare given these generators consist of the bass cyclic units which are of the form 1 m u l a ai i m d fi a 1 d l a a a d ord a

on fundamental domains of arithmetic fuchsian groups let k be a totally real algebraic number field and o an order in a quaternion algebra a over k assume that the group o 1 of units in o with reduced norm equal to 1 is embedded into psl 2 r as an arithmetic fuchsian group it is shown how ford s algorithm can be effectively applied in order to determine a fundamental domain of o 1 as well as a complete system of generators of o 1

periodic orbits on the regular hyperbolic octagon abstract the length spectrum of closed geodesics on a compact riemann surface corresponding to a regular octagon on the poincare disc is investigated the general form of the elements of the octagon group a discrete subgroup of nu su 1 1 1 in terms of 2 2 matrices is derived and the previously conjectured law for the length of periodic orbits is proved analytically an algorithm for the multiplicity of geodesics with a given length is developed which leads to an efficient enumeration of the periodic orbits of this strongly chaotic system

additive runge kutta methods for stiff ordinary differential equations certain pairs of runge kutta methods may be used additively to solve a system of n differential equations x j t x g t x pairs of methods of order p 4 where one method is semiexplicit and a stable and the other method is explicit are obtained these methods require the lu factorization of one n x n matrix and p evaluations of g in each step it is shown that such methods have a stability property which is similar to a stability property of perturbed linear differential equations 1 introduction in a recent article 2 the authors showed that certain pairs of methods may be used in an additive fashion to solve an initial value problem for a system of n differential equations x f t x x a xo a t b

large n expansion of q deformed two dimensional yang mills theory and hecke algebras we derive the q deformation of the chiral gross taylor holomorphic string large n expansion of two dimensional su n yang mills theory delta functions on symmetric group algebras are replaced by the corresponding objects canonical trace functions for hecke algebras the role of the schur weyl duality between unitary groups and symmetric groups is now played by q deformed schur weyl duality of quantum groups the appearance of euler characters of configuration spaces of riemann surfaces in the expansion persists we discuss the geometrical meaning of these formulae

representations of q schur superalgebras in positive characteristics abstract in 9 we obtained a classification of irreducible representations for a q schur superalgebra s k m n r at a root of unity in a field k of characteristic 0 under the assumption m n r in this paper we will generalise this result to fields of positive characteristics p 2

centers of generic hecke algebras let w be a weyl group and let w be a parabolic subgroup of w define a as follows a r 0q u w where x w is the generic algebra of type a over q u u an indeterminate associated with the group w and r is a q u algebra possibly of infinite rank in which u is invertible similarly we define a associated with w let m be an a a bimodule and let b e m define the relative norm 14 nw w b eul t at bat tet where t is the set of distinguished right coset representives for w in w we show that if b e zm a m e mlma a m va e a then nw w b e zm a in addition other properties of the relative norm are given and used to develop a theory of induced modules for generic hecke algebras including a markey decomposition this section of the paper is previously unpublished work of p hoefsmit and l l scott let e kl k2 k be a partition of n and let sc h l1 sk be a left justified parabolic subgroup of sn of shape c define bct nsn s where z 4171 nsk s avu i 1 with aw a k cycle of length k 1 in sk then the main result of this paper is theorem the set b f n is a basis for za sn a sn over q u u remnark the norms b in za sn a sn are analogs of conjugacy class sums in the center of qsn and in fact specialization of these norms at u 1 gives the standard conjugacy class sum basis of the center of qsn up to coefficients from q

free groups in quaternion algebras abstract in juriaans et al 2009 9 we constructed pairs of units u v in z orders of a quaternion algebra over q d d a positive and square free integer with d 7 mod 8 such that u n v n is free for some n n here we extend this result to any imaginary quadratic extension of q thus including matrix algebras more precisely we show that u n v n is a free group for all n 1 and d 2 and for d 2 and all n 2 the units we use arise from pell s and gauss equations

centralizers of iwahori hecke algebras to date integral bases for the centre of the iwahori hecke algebra of a finite coxeter group have relied on character theoretical results and the isomorphism between the iwahori hecke algebra when semisimple and the group algebra of the finite coxeter group in this paper we generalize the minimal basis approach of f1 to provide a way of describing and calculating elements of the minimal basis for the centre of an iwahori hecke algebra which is entirely combinatorial in nature and independent of both the above mentioned theories this opens the door to further generalization of the minimal basis approach to other cases in particular we show that generalizing it to centralizers of parabolic subalgebras requires only certain properties in the coxeter group we show here that these properties hold for groups of type a and b giving us the minimal basis theory for centralizers of any parabolic subalgebra in these types of iwahori hecke algebra

groups of units of integral group rings commensurable with direct products of free by free groups research partially supported by onderzoeksraad of vrije universiteit brussel fonds voor wetenschappelijk onderzoek belgium d g i of spain fundacion seneca of murcia cnpq and fapdf ams classification index primary 16u60 secondary 11r27 16a26

the fundamental invariant of the hecke algebra hn q characterizes the representations of hn q sn suq n and su n the irreducible representations irreps of the hecke algebra hn q are shown to be completely characterized by the fundamental invariant of this algebra cn this fundamental invariant is related to the quadratic casimir operator c2 of suq n and reduces to the transposition class sum 2 n of sn when q 1 the projection operators constructed in terms of cn for the various irreps of hn q are well behaved in the limit q 1 even when approaching degenerate eigenvalues of 2 n in the latter case for which the irreps of sn are not fully characterized by the corresponding eigenvalue of the transposition class sum the limiting form of the projection operator constructed in terms of cn gives rise to factors that depend on higher class sums of sn which effect the desired characterization expanding this limiting form of the projection operator into a linear combination of class sums of sn the coefficients constitute the corresponding row in the character table of sn the properties of the funda

energy level statistics of the hadamard gutzwiller ensemble abstract the statistical properties of the quantal energy levels of the hadamard gutzwiller ensemble whose classical members belong to the class of systems with hard chaos are investigated based on a sample of 4500 energy levels it is shown that the short range statistics as nearest neighbour spacing distributions are governed by the goe predictions of random matrix theory which was first surmised by wigner and by landau and smorodinsky for nuclear level statistics this result strengthens the hypothesis that quantum systems with chaotic classical counterpart display level repulsion as predicted by random matrix theory however the level statistics describing correlations over greater level distances deviate from the goe predictions which is explained as a simple consequrnce of the fact that the spectral rigidity 3 l introduced by dyson and mehta saturates non universally at a finite value for l in complete agreement with the semiclassical theory developed by berry

correlations for pairs of closed geodesics in this article we consider natural counting problems for closed geodesics on negatively curved surfaces we present asymptotic estimates for pairs of closed geodesics the differences of whose lengths lie in a prescribed family of shrinking intervals related pair correlation problems have been studied in both quantum chaos and number theory

on the periodic orbits of a strongly chaotic system abstract a point particle sliding freely on a two dimensional surface of constant negative curvature hadamard gutzwiller model exemplifies the simplest chaotic hamiltonian system exploiting the close connection between hyperbolic geometry and the group su 1 1 1 we construct an algorithm symboliv dynamics which generates the periodic orbits of the system for the simplest compact riemann surface having as its fundamental group the octagon group we present an enumeration of more than 206 million periodic orbits for the length of the n th primitive periodic orbit we find a simple expression in terms of algebraic numbers of the form m 2 n m n n are governed by a particular beatty sequence which reveals a strange arithmetical structure of chaos knowledge of the length spectrum is crucial for quantization via the selberg trace formula periodic orbit theory which in turn is expected to unravel the mystery of quantum chaos

presentations of the unit group of an order in a non split quaternion algebra abstract we give an algorithm to determine a finite set of generators of the unit group of an order in a non split classical quaternion algebra h k over an imaginary quadratic extension k of the rationals we then apply this method to obtain a presentation for the unit group of h z 1 7 2 as a consequence a presentation is discovered for the orthogonal group so 3 z 1 7 2 these results provide the first examples of a characterization of the unit group of some group rings that have an epimorphic image that is an order in a non commutative division algebra that is not a totally definite quaternion algebra

crossing the entropy barrier of dynamical zeta functions dynamical zeta functions are an important tool to quantize chaotic dynamical systems the basic quantization rules require the computation of the zeta functions on the real energy axis where their euler product representations running over the classical periodic orbits usually do not converge due to the existence of the so called entropy barrier determined by the topological entropy of the classical system we show that the convergence properties of the dynamical zeta functions rewritten as dirichlet series are governed not only by the well known topological and metric entropy but depend crucially on subtle statistical properties of the maslov indices and of the multiplicities of the periodic orbits that are measured by a new parameter for which we introduce the notion of a third entropy if and only if the third entropy is nonvanishing one can cross the entropy barrier if it exceeds a certain value one can even compute the zeta function in the physical region by means of a convergent dirichlet series a simple statistical model is presented which allows to compute the third entropy four examples of chaotic systems are studied in detail to test the model numerically

the minimal basis for the centre of an iwahori hecke algebra abstract this paper arose out of an attempt to generalize the q q q 1 basis for the centre of an iwahori hecke algebra h q found by jones to z q q 1 and to other types considering the iwahori hecke algebra h over a subring z of z q1 2 q 1 2 where q1 2 q 1 2 we use a new and natural definition of positivity on h to describe the minimal z basis for z h in terms of a partial order on the positive part of h the main result is to show that this minimal basis is the set of primitive minimal elements of the positive part of the centre for any weyl group in addition the primitive minimal positive central elements can be characterized as exactly those positive central elements which specialize on setting 0 the equivalent of setting q 1 in z q1 2 q 1 2 to the sum of elements in a conjugacy class and which apart from the shortest elements from that conjugacy class sum contain no other terms corresponding to shortest elements of any conjugacy class a constructive algorithm is provided for obtaining the minimal basis we use the results of jones to achieve the result in type an without the need for character theory and give the result for all weyl groups using the character theoretical results of geck and rouquier finally we discuss the non crystallographic cases and give some explicit examples