unitary rotation of square pixellated images the unitary rotation of square pixellated images is based on the finite su 2 oscillator model which describes systems whose values for position momentum and energy are discrete and finite in a two dimensional position space this allows the construction of angular momentum states orthonormal and complete for which rotations are defined as multiplication by phases that carry the rotation angle the decomposition of a digital square images in terms of these angular momentum states determines a unitary hence invertible rotation of the image whose kernel can be computed as a four dimensional array of real numbers

a p q oscillator realization of two parameter quantum algebras it is noted that the study of a quantum algebra sup q 2 with two independent deformation parameters p q leads to a p q oscillator realization for it the analysis is extended to the p q analogues of su 1 1 osp 2 1 and the centreless virasoro algebra the standard single parameter q deformations are obtained in the limit p q

model categories of diagram spectra working in the category mathcal t of based spaces we give the basic theory of diagram spaces and diagram spectra these are functors mathcal d longrightarrow mathcal t for a suitable small topological category mathcal d when mathcal d is symmetric monoidal there is a smash product that gives the category of mathcal d spaces a symmetric monoidal structure examples include begin enumerate item prespectra as defined classically item symmetric spectra as defined by jeff smith item orthogonal spectra a coordinate free analogue of symmetric spectra with symmetric groups replaced by orthogonal groups in the domain category item gamma spaces as defined by graeme segal item mathcal w spaces an analogue of gamma spaces with finite sets replaced by finite cw complexes in the domain category end enumerate we construct and compare model structures on these categories with the caveat that gamma spaces are always connective these categories and their simplicial analogues are quillen equivalent and their associated homotopy categories are equivalent to the classical stable homotopy category monoids in these categories are strict ring spectra often the subcategories of ring spectra module spectra over a ring spectrum and commutative ring spectra are also model categories when this holds the respective categories of ring and module spectra are quillen equivalent and thus have equivalent homotopy categories this allows interchangeable use of these categories in applications 2000 mathematics subject classification primary 55p42 secondary 18a25 18e30 55u35

finite two dimensional oscillator ii the radial model a finite two dimensional radial oscillator of n 1 2 points is proposed with the dynamical lie algebra so 4 su 2 x su 2 y examined in part i of this work but reduced by a subalgebra chain so 4 so 3 so 2 as before there are a finite number of energies and angular momenta the casimir spectrum of the new chain provides the integer radii 0 n and the 2 1 discrete angles on each circle are obtained from the finite fourier transform of angular momenta the wavefunctions of the finite radial oscillator are so 3 clebsch gordan coefficients we define here the hankel hahn transforms with dual hahn polynomials as finite n unitary approximations to hankel integral transforms with bessel functions obtained in the contraction limit n

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

quantum algebraic structures compatible with the harmonic oscillator newton equation we study some of the algebraic structures that are compatible with the quantization of the harmonic oscillator through its newton equation examples of such structures are given they include undeformed and q deformed oscillators as well as the su 2 and the deformed suq 2 lie algebras which appear in a variety of physical models

the development version of the chevie package of gap3 abstract i describe the current state of the development version of the chevie package which allows to transform the theories of coxeter groups reductive algebraic groups complex reflection groups hecke algebras braid monoids etc into actual computations examples are given showing the code to check some results of lusztig

minimal length elements in twisted conjugacy classes of finite coxeter groups let w be a finite coxeter group and let f be an automorphism of w that leaves the set of generators of w invariant we establish certain properties of elements of minimal length in the f conjugacy classes of w that allow us to define character tables for the corresponding twisted iwahori hecke algebras these results are extensions of results obtained by geck and pfeiffer in the case where f is trivial

discrete series representations for mathfrak sl 2 1 meixner polynomials and oscillator models we explore a model for a one dimensional quantum oscillator based on the lie superalgebra for this purpose a class of discrete series representations of is constructed each representation characterized by a real number 0 in this model the position and momentum operators of the oscillator are odd elements of and their expressions involve an arbitrary parameter in each representation the spectrum of the hamiltonian is the same as that of a canonical oscillator the spectrum of a position operator can be continuous or infinite discrete depending on the value of we determine the position wavefunctions both in the continuous and the discrete case and discuss their properties in the discrete case these wavefunctions are given in terms of meixner polynomials from the embedding it can be seen why the case 1 corresponds to a paraboson oscillator consequently taking the values 1 2 1 in the model yields a canonical oscillator

affine deligne lusztig varieties associated to additive affine weyl group elements abstract affine deligne lusztig varieties can be thought of as affine analogs of classical deligne lusztig varieties or frobenius twisted analogs of schubert varieties we provide a method for proving a non emptiness statement for affine deligne lusztig varieties inside the affine flag variety associated to affine weyl group elements satisfying a certain length additivity hypothesis in particular we prove that non emptiness holds whenever it is conjectured to do so for alcoves in the shrunken dominant weyl chamber providing a partial converse to the emptiness results of gortz haines kottwitz and reuman our technique involves the work of geck and pfeiffer on cuspidal conjugacy classes in addition to an analysis of the combinatorics of certain fully commutative elements in the finite weyl group

hz algebra spectra are differential graded algebras we show that the homotopy theory of differential graded algebras coincides with the homotopy theory of h algebra spectra namely we construct quillen equivalences between the quillen model categories of unbounded differential graded algebras and h algebra spectra we also construct quillen equivalences between the differential graded modules and module spectra over these algebras we use these equivalences in turn to produce algebraic models for rational stable model categories we show that basically any rational stable model category is quillen equivalent to modules over a differential graded algebra with many objects

class polynomials for some affine hecke algebras abstract class polynomials attached to affine hecke algebras were first introduced by he in 13 they play an important role in the study of affine deligne lusztig varieties motivated by 14 we compute the class polynomials attached to an affine hecke algebra of type twisted a 2 using these class polynomials we prove a conjecture of gortz haines kottwitz reuman for the general linear group unitary group and division algebra of semisimple rank 2 furthermore we discuss some interesting patterns of affine deligne lusztig varieties

energy spectrum potential and inertia functions of a generalized f oscillator we consider a generalized four parameter q algebra aa q a a q n n a a n a a associating with operators a and a the nonlinear f oscillator operators defined in terms of the usual harmonic oscillator operators as a af n and a f n a where a and a are operators of the weyl heisenberg algebra and n a a the function f n is determined from the commutation relations we write the hamiltonian for the free f oscillator and obtain its energy spectrum besides expressing the hamiltonian in terms of coordinate and momentum we determine the potential and inertia functions coordinate dependent mass and analyse their behaviour by varying the parameters

on bases of centres of iwahori hecke algebras of the symmetric group abstract in 1990 using norms the second author constructed a basis for the centre of the hecke algebra of the symmetric group s n over q trans amer math soc 317 1 1990 361 392 an integral minimal basis was later given by the first author in j algebra 221 1 1999 1 28 following m geck r rouquier centers and simple modules for iwahori hecke algebras in finite reductive groups luminy 1994 birkhauser boston ma 1997 pp 251 272 in principle one can then write elements of the norm basis as integral linear combinations of minimal basis elements in this paper we find an explicit non recursive expression for the coefficients appearing in these linear combinations these coefficients are expressed in terms of certain permutation characters of s n in the process of establishing this main theorem we prove the following items of independent interest a result on the projection of the norms onto parabolic subalgebras the existence of an inner product on the hecke algebra with some interesting properties and the existence of a partial ordering on the norms

markov traces and knot invariants related to iwahori hecke algebras of type b in classical knot theory we study knots inside the 3 sphere modulo isotopy using the alexander and markov theorem we can translate this into a purely algebraic setting in terms of artin braid groups modulo an equivalence relation generated by markov moves one of which is usual conjugation inside the braid group v f r jones 5 used this fact in 1984 for constructing a new knot invariant through trace functions on the associated iwahori hecke algebras of type a with suitable properties that reflect the above markov moves jones s work led to questions of developing knot theory corresponding to other types of coxeter groups it is proved in 6 that there exist braid structures related to arbitrary 3 manifolds which in addition satisfy appropriate markovisotopy equivalence also that if the 3 manifold is a solid torus then the sets of related braids form groups which are in fact the artin tits braid groups related to the b type coxeter groups these results together with a linear trace that we found in 1991 are used in 7 for constructing a 4 variable analogue of the homfly pt 2 variable jones polynomial for oriented knots inside a solid torus we proved the existence of this trace see 7 by following and adapting to the b type case jones s proof of the existence of ocneanu s trace in 5 theorem 5 1

a uniqueness theorem for stable homotopy theory roughly speaking the stable homotopy category of algebraic topology is obtained from the homotopy category of topological spaces by inverting the suspension functor yielding a linear approximation to the homotopy category of spaces the isomorphism classes of objects in the stable homotopy category represent the generalized cohomology theories defined by the eilenberg steenrod axioms es45 without the dimension axiom which distinguishes ordinary from generalized cohomology theories the first construction of the full stable homotopy category was given by boardman b69 nowadays manymodels for the stable homotopy category are known most of which have the additional structure of a closed model category in the sense of quillen q67 in m83 h r margolis introduced a short list of axioms and conjectured that they characterize the stable homotopy category up to an equivalence of categories in theorem 3 2 we prove that these axioms do uniquely specify the stable homotopy category whenever there is some underlying quillen model category we also prove amore structured version theuniqueness theorembelow which states that the model category of spectra itself is uniquely determined by certain equivalent conditions up to so called quillen equivalence of

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

new generalized coherent states we construct a new family of boson coherent states using a specially designed function which is a solution of a functional equation de q x dx e q qx with 0 q 1 and e q 0 1 we use this function in place of the usual exponential to generate new coherent states q z from the vacuum which are normalized and continuous in their label z these states allow the resolution of unity and a corresponding weight function is furnished by the exact solution of the associated stieltjes moment problem they also permit exact evaluation of matrix elements of an arbitrary polynomial given as a normally ordered function of boson operators we exemplify this by showing that the photon number statistics for these states is sub poissonian for any q 1 the states q z are squeezed we obtain and discuss their signal to quantum noise ratio the function e q x allows a natural generation of multiboson coherent states of arbitrary multiplicity which is impossible for the usual coherent states for q 1 all the above resul

minimal length elements of coxeter groups let w be a coxeter group and wj a finite parabolic subgroup we present here a geometric new proof of x he s result on minimal length elements of an orbit of wj which acts on w in a certain way

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

higher homotopy excision and blakers massey theorems for structured ring spectra abstract working in the context of symmetric spectra we prove higher homotopy excision and higher blakers massey theorems and their duals for algebras and left modules over operads in the category of modules over a commutative ring spectrum e g structured ring spectra

a sum rule for scale factors in the goldie rank polynomials this paper is a corollary to s and we adopt the same notation which is also that of s there we showed how two deep questions concerning the enveloping algebra of a semisimple lie algebra could be reduced to a purely combinatorial analysis of the kazhdan lusztig polynomials the first concerns the computation of sot l l wl l yl apart from its own intrinsic interest this question has deep interconnections with geometry via the component group a of the associated nilpotent orbit the second concerns the coefficients z which are ratios of goldie ranks these are central to the determination of the scale factors in the goldie rank polynomials which in turn relate to complete primeness and hence to unitarity see 7 1 4 16 7 121 it first seemed that the results in s did not give any easily extractable information however in 13 lusztig shows that for some canonical quotient a of a there exists a left cell of the weyl group w such that the character table of a can be read off in a well defined manner from data involving the hecke algebra module structure of the cell which we shall call a lusztig cell combined with s this gives several remarkable results first when rl z2 are involutions in a lusztig cell sot l l r a l 12 is determined by multiplying appropriate irreducible characters of a and identifying coefficients this implies that when a e s which occurs in type e it can happen that l l zl l r l is not multiplicity free providing thereby a counterexample to conjecture of 5 5 111 even in the diagonal case second if r is an involution in a lusztig cell then z is the degree of an irreducible character of a third we prove that z divides ial for every element w of a lusztig cell in partial answer to a conjecture in 9 3 231

wigner distribution function for euclidean systems euclidean systems include poly and monochromatic wide angle optics acoustics and also infinite discrete data sets we use a recently defined wigner operator and quasiprobability distribution function to set up and study the phase space evolution of these models subject to differential and difference equations respectively infinite data sets and two dimensional monochromatic helmholtz fields are thus shown by their wigner function on a cylinder of direction and location the wigner function for polychromatic wavefields has c number coordinates of two dimensional wavenumber and position

finite two dimensional oscillator i the cartesian model a finite two dimensional oscillator is built as the direct product of two finite one dimensional oscillators using the dynamical lie algebra su 2 x su 2 y the position space in this model is a square grid of points while the ordinary continuous two dimensional quantum oscillator has a symmetry algebra u 2 the symmetry algebra of the finite model is only u 1 x u 1 y because it lacks rotations in the position and momentum plane we show how to import an so 2 group of rotations from the continuum model that transforms unitarily the finite wavefunctions on the fixed square grid we thus propose a finite analogue for fractional u 2 fourier transforms

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

elements with finite coxeter part in an affine weyl group abstract let w a be an affine weyl group and w a w 0 be the natural projection to the corresponding finite weyl group we say that w w a has finite coxeter part if w is conjugate to a coxeter element of w 0 the elements with finite coxeter part are a union of conjugacy classes of w a we show that for each conjugacy class o of w a with finite coxeter part there exists a unique maximal proper parabolic subgroup w j of w a such that the set of minimal length elements in o is exactly the set of coxeter elements in w j similar results hold for twisted conjugacy classes

a frobenius formula for the characters of ariki koike algebras abstract let h n r be the ariki koike algebra associated to the complex reflection group wn r g r 1 n in this paper we give a new presentation of h n r by making use of the schur weyl reciprocity for h n r established by m sakamoto and t shoji 1999 j algebra 221 293 314 this allows us to construct various non parabolic subalgebras of h n r we construct all the irreducible representations of h n r as induced modules from such subalgebras we show the existence of a partition of unity in h n r which is specialized to a partition of unity in the group algebra c wn r then we prove a frobenius formula for the characters of h n r which is an analogy of the frobenius formula proved by a ram 1991 invent math 106 461 488 for the iwahori hecke algebra of type a

deligne lusztig varieties and period domains over finite fields abstract we prove that the drinfeld space is essentially the only deligne lusztig variety which is at the same time a period domain over a finite field we also discuss an affineness criterion for deligne lusztig varieties and its relation to cohomology vanishing theorems

algebras and modules in monoidal model categories in recent years the theory of structured ring spectra formerly known as a infty and e infty ring spectra has been simplified by the discovery of categories of spectra with strictly associative and commutative smash products now a ring spectrum can simply be defined as a monoid with respect to the smash product in one of these new categories of spectra in this paper we provide a general method for constructing model category structures for categories of ring algebra and module spectra this provides the necessary input for obtaining model categories of symmetric ring spectra functors with smash product gamma rings and diagram ring spectra algebraic examples to which our methods apply include the stable module category over the group algebra of a finite group and unbounded chain complexes over a differential graded algebra 1991 mathematics subject classification primary 55u35 secondary 18d10

cohomologie des varietes de deligne lusztig abstract we study the cohomology of deligne lusztig varieties with aim the construction of actions of hecke algebras on such cohomologies as predicted by the conjectures of broue malle and michel ultimately aimed at providing an explicit version of the abelian defect conjecture we develop the theory for varieties associated to elements of the braid monoid and partial compactifications of them we are able to compute the cohomology of varieties associated to possibly twisted rank 2 groups and powers of the longest element w 0 some indeterminacies remain for g 2 we use this to construct hecke algebra actions on the cohomology of varieties associated to w 0 or its square for groups of arbitrary rank in the subsequent work f digne j michel endomorphisms of deligne lusztig varieties nagoya j math 183 2006 we construct actions associated to more general regular elements and we study their traces on cohomology

on the affineness of deligne lusztig varieties abstract we prove that the deligne lusztig variety associated to minimal length elements in any conjugacy class of the weyl group is affine which was conjectured by orlik and rapoport in s orlik m rapoport deligne lusztig varieties and period domains over finite fields arxiv 0705 1646

a simple difference realization of the heisenberg q algebra a realization of the heisenberg q algebra whose generators are first order difference operators on the full real line is discussed herein the eigenfunctions of the corresponding q oscillator hamiltonian are given explicitly in terms of the q 1 hermite polynomials the nonuniqueness of the measure for these q oscillator states is also studied

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

decomposition numbers for generic iwahori hecke algebras of noncrystallographic type we compute the e modular decomposition matrices for the generic iwahori hecke algebras of typei2 m form n m 2 h3 andh4 for alle n leading to nontrivial decomposition maps the results are obtained by a combined use of different ideas from computational representation theory and by application of the computer algebra systemsgap chevie vectorenumeratorandmeataxe

wigner distribution function for finite systems we construct a wigner distribution function for finite data sets it is based on a finite optical system a linear wave guide where the finite number of discrete sensors is equal to the number of modes which the guide can carry the dynamical group for this model is su 2 and the wave functions are sets of n 2l 1 data points the wigner distribution function assigns classical c numbers to the operators of position momentum and wave guide mode

minimal length elements in some double cosets of coxeter groups abstract we study the minimal length elements in some double cosets of coxeter groups and use them to study lusztig s g stable pieces and the generalization of g stable pieces introduced by lu and yakimov we also use them to study the minimal length elements in a conjugacy class of a finite coxeter group and prove a conjecture in m geck s kim g pfeiffer minimal length elements in twisted conjugacy classes of finite coxeter groups j algebra 229 2 2000 570 600

discrete spherical harmonic oscillator transforms on the cartesian grids using transformation coefficients the analog harmonic oscillators are well studied in quantum physics including their energy states wavefunctions orthogonal properties and eigenfunctions of the fourier transform in addition the continuous solutions in different dimension and coordinate systems are known some discrete equivalents of the 1d wavefunctions were also studied however in the 3d spherical coordinate system the discrete equivalents of the 3d wavefunctions are not established in this paper we focus on the spherical harmonic oscillator wavefunctions shows the spherical harmonic oscillator transforms shots and their discrete implementation the shows can be synthesized by linear combinations of the hermite gaussian functions with proper transformation coefficients we find that computing the coefficients can be speeded up using the fast fourier transforms or some recursive relations these coefficients relate the hermite transforms with the shots some applications of the discrete shows and the discrete shots are introduced first the shows are exactly the eigenfunctions of the 3d dft also the shots can be used to derive the spherical harmonic oscillator descriptor shod which is a rotational invariant descriptor we find that the shod is not only compatible with the existing rotational descriptors for the spherically sampled data but also outperforms the existing rotational descriptors for 3d cartesian sampled bandlimited input data besides the shots can be used to decompose 3d signals into spherical components hence 3d signal reconstruction is done using partially chosen spherical component and 3d data compression for mri data is demonstrated using shots for medical applications

an algebraic model for free rational g spectra for connected compact lie groups g we show that the category of free rational g spectra for a connected compact lie group g is quillen equivalent to the category of torsion differential graded modules over the polynomial ring h bg the ingredients are the enriched morita equivalences of schwede and shipley topology 42 1 103 153 2003 the functors of shipley am j math 129 351 379 2007 making rational spectra algebraic koszul duality and thick subcategory arguments based on the simplicity of the derived category of a polynomial ring

conjugacy relation on coxeter elements abstract let w s be an irreducible finitely presented coxeter system the present paper is mainly concerned with a conjugacy relation on coxeter elements in the case where contains just one circle in particular when is itself a circle in the cases where is either a three multiple circle or a circle with three nodes we show that the ss equivalence relation on coxeter elements of w is the same as the w conjugacy relation an explicit formula is given for the characteristic polynomial of a coxeter element in the natural reflection representation of w when is a circle we also give the answers to some questions raised by coleman and extend some results of geck and pfeiffer concerning conjugacy relation in coxeter groups

generalized deformed oscillator and nonlinear algebras the harmonic oscillator is deformed arbitrarily and the properties of the deformed oscillator algebra are studied the eigenstates and eigenvalues of the deformed oscillator are calculated this generalized deformed oscillator algebra is used to create nonlinear deformation of the classical su 2 algebra the deformed boson realizations of an arbitrary nonlinear su 2 algebra are calculated the algebra of the q deformed oscillator is a special case of the generalized deformed oscillator algebra

geometric and homological properties of affine deligne lusztig varieties this paper studies ane deligne lusztig varieties x w b in the ane ag variety of a quasi split tamely ramied group we describe the geometric structure of x w b for a minimal length element w in the conjugacy class of an extended ane weyl group we then provide a reduction method that relates the structure of x w b for arbitrary elements w in the extended ane weyl group to those associated with minimal length elements based on this reduction we establish a connection between the dimension of ane deligne lusztig varieties and the degree of the class polynomial of ane

s modules and symmetric spectra we study a symmetric monoidal adjoint functor pair between the category of s modules and the category of symmetric spectra the functors induce equivalences between the respective homotopy categories of spectra module spectra and ring spectra then at approximately the same time two categories of spectra with nice smash products were discov ered elmendorf kriz mandell and may constructed the category of s modules ekmm and je smith introduced symmetric spectra hss both categories are quillen model categories and have associated notions of ring and module spectra however these two categories arise in completely dierent ways and even though the homotopy categories are equivalent it is not a priori clear if both frameworks give rise to the same homotopy theory of rings and modules both categories have their merits described in detail in the introductions of ekmm and hss and it is desirable to be able to translate results obtained in one category into conclusions valid in the other the present paper describes an easy mechanism which facilitates such comparisons below we dene a lax symmetric monoidal functor ms sp from the category of s modules to the category of symmetric spectra the functor preserves homotopy groups and has a strong symmetric monoidal left adjoint we show that the two functors induce inverse equivalences of the homotopy categories of spectra ring spectra commutative ring spectra and module spectra main theorem the functor from the category ms of s modules to the category sp of symmetric spectra passes to a symmetric monoidal equivalence of homotopy categories

on the q oscillator and the quantum algebra suq 1 1 the authors discuss some problems of the q deformed harmonic oscillator different q bosonisations of the quantum suq 1 1 algebra are given and the corresponding infinite dimensional representations of discrete series are analysed

polynomial heisenberg algebras polynomial deformations of the heisenberg algebra are studied in detail some of their natural realizations are given by the higher order susy partners and not only by those of first order as is already known of the harmonic oscillator for even order polynomials here it is shown that the susy partners of the radial oscillator play a similar role when the order of the polynomial is odd moreover it will be proved that the general systems ruled by such kinds of algebras in the quadratic and cubic cases involve painleve transcendents of types iv and v respectively

solutions of the compatibility conditions for a wigner quantum oscillator we consider the compatibility conditions for an n particle d dimensional wigner quantum oscillator these conditions can be rewritten as certain triple relations involving anticommutators so it is natural to look for solutions in terms of lie superalgebras in the recent classification of generalized quantum statistics for the basic classical lie superalgebras 1 each such statistics is characterized by a set of creation and annihilation operators plus a set of triple relations in the present paper we investigate which cases of this classification also lead to solutions of the compatibility conditions our analysis yields some known solutions and several classes of new solutions

monoidal uniqueness of stable homotopy theory we show that the monoidal product on the stable homotopy category of spectra is essentially unique this strengthens work of this author with schwede on the uniqueness of models of the stable homotopy theory of spectra also the equivalences constructed here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra s modules w spaces orthogonal spectra simplicial functors with symmetric spectra as an application we show that with an added assumption about underlying model structures margolis axioms uniquely determine the stable homotopy category of spectra up to monoidal equivalence

localization of algebras over coloured operads we give sufficient conditions for homotopical localization functors to preserve algebras over coloured operads in monoidal model categories our approach encompasses a number of previous results about preservation of structures under localizations such as loop spaces or infinite loop spaces and provides new results of the same kind for instance under suitable assumptions homotopical localizations preserve ring spectra in the strict sense not only up to homotopy modules over ring spectra and algebras over commutative ring spectra as well as maps between ring spectra module maps and algebra maps it is principally the treatment of module spectra and their maps that led us to the use of coloured operads also called enriched multicategories in this context

murnaghan nakayama rules for characters of iwahori hecke algebras of classical type in this paper we give murnaghan nakayama type formulas for computing the irreducible characters of the iwahori hecke algebras of types an 1 bn and dn our method is a generalization of a derivation of the murnaghan nakayama formula for the irreducible characters of the symmetric group given by curtis greene greene s approach is to sum up the diagonal entries of the matrices of certain cycle permutations in young s seminormal representations the analogues of the young seminormal representations for the iwahori hecke algebras of types an 1 bn and dn were given by hoefsmit

parabolic deligne lusztig varieties abstract motivated by the broue conjecture on blocks with abelian defect groups for finite reductive groups we study parabolic deligne lusztig varieties and construct on those which occur in the broue conjecture an action of a braid monoid whose action on their l adic cohomology will conjecturally factor through a cyclotomic hecke algebra in order to construct this action we need to enlarge the set of varieties we consider to varieties attached to a ribbon category this category has a garside family which plays an important role in our constructions so we devote the first part of our paper to the necessary background on categories with garside families

quasi fibonacci oscillators we study the properties of the sequences of the energy eigenvalues for some generalizations of q deformed oscillators including the p q oscillator and the three four and five parameter deformed oscillators given in the literature it is shown that most of the considered models belong to the class of so called fibonacci oscillators for which any three consecutive energy levels satisfy the relation en 1 en en 1 with real constants on the other hand for a certain oscillator known since 1993 we prove its non fibonacci nature possible generalizations of the three term fibonacci relation are discussed among which for the oscillator we choose as the most adequate the so called quasi fibonacci or local fibonacci property of the energy levels the property is encoded in the three term quasi fibonacci qf relation with the non constant n dependent coefficients and various aspects of the qf relation are elaborated for the oscillator and some of its extensions

finite q oscillator the finite q oscillator is a model that obeys the dynamics of the harmonic oscillator with the operators of position momentum and hamiltonian being functions of elements of the q algebra suq 2 the spectrum of position in this discrete system in a fixed representation j consists of 2j 1 sensor points and similarly for the momentum observable the spectrum of energies is finite and equally spaced so the system supports coherent states the wavefunctions involve dual q kravchuk polynomials which are solutions to a finite difference schrodinger equation time evolution times a phase defines the fractional fourier q kravchuk transform in the classical limit as q 1 we recover the finite oscillator lie algebra the n 2j limit returns the macfarlane biedenharn q oscillator and both limits contract the generators to the standard quantum mechanical harmonic oscillator