quantum trigonometric calogero sutherland model irreducible characters and clebsch gordan series for the exceptional algebra e7 we reexpress the quantum calogero sutherland model for the lie algebra e7 and the particular value of the coupling constant 1 by using the fundamental irreducible characters of the algebra as dynamical variables for that we need to develop a systematic procedure to obtain all the clebsch gordan series required to perform the change of variables we describe how the resulting quantum hamiltonian operator can be used to compute more characters and clebsch gordan series for this exceptional algebra

raising and lowering operators factorization and differential difference operators of hypergeometric type starting from the rodrigues formula we present a general construction of raising and lowering operators for orthogonal polynomials of continuous and discrete variables on a uniform lattice in order to have these operators mutually adjoint we introduce orthonormal functions with respect to the scalar product of unit weight using the infeld hull factorization method we generate from the raising and lowering operators the second order self adjoint differential difference operator of hypergeometric type

velocity analysis based on minimum entropy seismic resolution is determined by the sparsity of reflection events together with the dispersion of the wavelets representing those events in this paper minimum entropy me norms are introduced as a measure of spatial resolving power it is shown that the lateral dispersion of inverted diffractor responses inverted spatial wavelets increases with increasing velocity error using this property minimum entropy velocity analysis meva is proposed to extract velocity information from diffraction energy meva can be successfully applied to zero offset including poststack data and common offset data with a sufficient amount of diffraction energy in addition meva can be used as an alternative to existing cmp velocity estimation techniques

quantum integrable systems and clebsch gordan series ii the class of quantum integrable systems associated with root systems was introduced as a generalization of the calogero sutherland systems in this letter a new property of such systems is proved to be valid namely in the case of the potential the series for the product of two wavefunctions coincides with the clebsch gordan series this gives the recursive relations for the wavefunctions of such systems and for generalized spherical functions related to them on symmetric spaces one conjectures that the clebsch gordan series is also unchanged under more general two parametric deformation deformation

seismic sparse spike deconvolution via toeplitz sparse matrix factorization abstractwe have developed a new sparse spike deconvolution ssd method based on toeplitz sparse matrix factorization tsmf a bilinear decomposition of a matrix into the product of a toeplitz matrix and a sparse matrix to address the problems of lateral continuity effects of noise and wavelet estimation error in ssd assuming the convolution model a constant source wavelet and the sparse reflectivity a seismic profile can be considered as a matrix that is the product of a toeplitz wavelet matrix and a sparse reflectivity matrix thus we have developed an algorithm of tsmf to simultaneously deconvolve the seismic matrix into a wavelet matrix and a reflectivity matrix by alternatively solving two inversion subproblems related to the toeplitz wavelet matrix and sparse reflectivity matrix respectively because the seismic wavelet is usually compact and smooth the fused lasso was used to constrain the elements in the toeplitz wavelet matrix moreover due to the limitations of computer memory large

simple fusion systems and the solomon 2 local groups we introduce a notion of simple fusion systems which imitates the corresponding notion for finite groups and show that the fusion system on the sylow 2 subgroup of a 7 dimensional spinor group over a field of characteristic 3 considered by ron solomon 18 and by ran levi and bob oliver 11 is simple in this sense

broue s abelian defect group conjecture holds for the harada norton sporadic simple group hn abstract in representation theory of finite groups there is a well known and important conjecture due to m broue he conjectures that for any prime p if a p block a of a finite group g has an abelian defect group p then a and its brauer corresponding block b of the normaliser n g p of p in g are derived equivalent rickard equivalent this conjecture is called broue s abelian defect group conjecture we prove in this paper that broue s abelian defect group conjecture is true for a non principal 3 block a with an elementary abelian defect group p of order 9 of the harada norton simple group hn it then turns out that broue s abelian defect group conjecture holds for all primes p and for all p blocks of the harada norton simple group hn

seismic sparse layer reflectivity inversion using basis pursuit decomposition a basis pursuit inversion of seismic reflection data for reflection coefficients is introduced as an alternative method of incorporating a priori information in the seismic inversion process the inversion is accomplished by building a dictionary of functions representing reflectivity patterns and constituting the seismic trace as a superposition of these patterns basis pursuit decomposition finds a sparse number of reflection responses that sum to form the seismic trace when the dictionary of functions is chosen to be a wedge model of reflection coefficient pairs convolved with the seismic wavelet the resulting reflectivity inversion is a sparse layer inversion rather than a sparse spike inversion synthetic tests suggest that a sparse layer inversion using basis pursuit can better resolve thin beds than a comparable sparse spike inversion application to field data indicates that sparse layer inversion results in the potentially improved detectability and resolution of some thin layers and reveals a

certain hypergeometric series related to the root system bc we show that the generalized hypergeometric function 2 f 1 of matrix argument is the series expansion at the origin of a special case of the hypergeometric function associated with the root system of type bc in addition we prove that the jacobi polynomials of matrix argument correspond to the jacobi polynomials associated with the root system of type bc we also give a precise relation between jack polynomials and the jacobi polynomials associated with the root system of type a as a side result one obtains generalized hook length formulas which are related to harish chandra s c function and one can prove a conjecture due to macdonald relating two inner products on a space of symmetric functions

a ghost ring for the left free double burnside ring and an application to fusion systems abstract for a finite group g we define a ghost ring and a mark homomorphism for the double burnside ring b g g of left free g g bisets in analogy to the case of the burnside ring b g the ghost ring has a much simpler ring structure and after tensoring with q one obtains an isomorphism of q algebras as an application of a key lemma we obtain a very general formula for the brauer construction applied to a tensor product of two p permutation bimodules m and n in terms of brauer constructions of the bimodules m and n over a field of characteristic 0 we determine the simple modules of the left free double burnside algebra and prove semisimplicity results for the bifree double burnside algebra these results carry over to results about biset functor categories finally we apply the ghost ring and mark homomorphism to fusion systems on a finite p group we extend a remarkable bijection due to ragnarsson and stancu between saturated fusion systems and certain idempotents of the bifree double burnside algebra over z p to a bijection between all fusion systems and a larger set of idempotents in the bifree double burnside algebra over q

on stable equivalences and blocks with one simple module using a stable equivalence due to rouquier we show that alperin s weight conjecture holds for any p block of a finite group with defect 2 whose brauer correspondent has a unique isomorphism class of simple modules

a generalized brauer construction and linear source modules for a complete discrete valuation ring 0 with residue field f a subgroup h of a finite group g and a homomorphism p h ox we define a functor v 4 v h p from the category of og modules to the category of fng h p modules and investigate its behaviour with respect to linear source modules

some combinatorial properties of jack symmetric functions if 1 l n then write 2 n or 1 n if p is another partition then write p c j if pi 6 for all i i e if the diagram of 1 contains the diagram of p if ipl ii 1 then write p 2 1 reverse lexicographic order if either j i or the first nonvanishing difference e pi is positive for instance writing 2 1 i for ai i it 5 5 41 32 5 311 221 2111 11111 finally write p i if ipl lal and p1 pl2 pli a1 a2 li for all i macdonald m p 63 call the partial ordering the natural ordering but we will call it the dominance ordering let x x x2 be an infinite set of indeterminates as in mi we

on equivalences between blocks of group algebras reduction to the simple components abstract a conjecture of michel broue states that if d is an abelian sylow p subgroup of a finite group g and h n g d then the principal blocks of g and h are rickard equivalent the structure of groups with abelian sylow p subgroups as determined by p fong and m e harris raises the following question assuming that broue s conjecture holds for the simple components of g under what conditions does it hold for g itself due to the structure of g this problem requires mainly the lifting of rickard complexes to p extensions of the simple components and the construction of complexes over wreath products we give here these reduction steps which may be regarded as a clifford theory of tilting complexes

some results on the eigenfunctions of the quantum trigonometric calogero sutherland model related to the lie algebra e6 the quantum trigonometric calogero sutherland models related to lie algebras admit a parametrization in which the dynamical variables are the characters of the fundamental representations of the algebra we develop here this approach for the case of the exceptional lie algebra e6

investigation of continuous time quantum walk on root lattice an and honeycomb lattice abstract the continuous time quantum walk ctqw on root lattice a n known as hexagonal lattice for n 2 and honeycomb one is investigated by using spectral distribution method to this aim some association schemes are constructed from abelian group z m n and two copies of finite hexagonal lattices such that their underlying graphs tend to root lattice a n and honeycomb one as the size of the underlying graphs grows to infinity the ctqw on these underlying graphs is investigated by using the spectral distribution method and stratification of the graphs based on terwilliger algebra where we get the required results for root lattice a n and honeycomb one from large enough underlying graphs moreover by using the stationary phase method the long time behavior of ctqw on infinite hexagonal lattice is approximated with finite ones for finite distances from the origin while for large distances the scaling behavior of the probability distribution is deduced also it is shown that the probability amplitudes at finite times possess the point group symmetry in the sense that two lattice points have the same probability amplitudes if and only if they belong to the same orbit stratum of the graph of the point group of the lattice apart from physical results it is shown that the bose mesner algebras of our constructed association schemes called n variable p polynomial can be generated by n commuting generators where raising flat and lowering operators as elements of terwilliger algebra are associated with each generator a system of n variable orthogonal polynomials which are special cases of generalized gegenbauer polynomials is constructed where the probability amplitudes are given by integrals over these polynomials or their linear combinations finally the supersymmetric structure of finite honeycomb lattices is revealed

broueʼs abelian defect group conjecture holds for the sporadic simple conway group co3 in the representation theory of finite groups there is a well known and important conjecture due to m broue he conjectures that for any prime p if a p block a of a finite group g has an abelian defect group p then a and its brauer corresponding block an of the normaliser ng p of p in g are derived equivalent rickard equivalent this conjecture is called strong version of broue s abelian defect group conjecture in this paper we prove that the strong version of broue s abelian defect group conjecture is true for the non principal 2 block a with an elementary abelian defect group p of order 8 of the sporadic simple conway group co3 this result completes the verification of the strong version of broue s abelian defect group conjecture for all primes p and for all p blocks of co3

trivial source bimodule rings for blocks and p permutation equivalences we associate with any p block of a finite group a grothendieck ring of certain p permutation bimodules we extend the notion of p permutation equivalences introduced by boltje and xu 4 to source algebras of p blocks of finite groups we show that a p permutation equivalence between two source algebras a b of blocks with a common defect group and same local structure induces an isotypy

recovery of the acoustic impedance from reflection seismograms this paper examines the problem of recovering the acoustic impedance from a band limited normal incidence reflection seismogram the convolutional model for the seismogram is adopted at the outset and it is therefore required that initial processing has removed multiples and recovered true amplitudes as well as possible in the first portion of the paper we investigate the effect of substituting the deconvolved seismic trace that is the band limited version of the reflectivity function into the standard recursion formula for the acoustic impedance the formalism of linear inverse theory is used to show that the logarithm of the normalized acoustic impedance estimated from the deconvolved seismogram is approximately an average of the true logarithm of the impedance moreover the averaging function is identical to that used in deconvolving the initial seismogram the advantage of these averages is that they are unique their disadvantage is that low frequency information which is crucial to making a g

extended trigonometric cherednik algebras and nonstationary schrodinger equations with delta potentials we realize an extended version of the trigonometric cherednik algebra as affine dunkl operators involving heaviside functions we use the quadratic casimir element of the extended trigonometric cherednik algebra to define an explicit nonstationary schrodinger equation with delta potential we use coordinate bethe ansatz methods to construct solutions of the nonstationary schrodinger equation in terms of generalized bethe wave functions it is shown that the generalized bethe wave functions satisfy affine difference knizhnik zamolodchikov equations as functions of the momenta the relation to the vector valued root system analogs of the quantum bose gas on the circle with delta function interactions is indicated

degenerate principal series representations and their holomorphic extensions let x h l be an irreducible real bounded symmetric domain realized as a real form in an hermitian symmetric domain d g k the intersection s of the shilov boundary of d with x defines a distinguished subset of the topological boundary of x and is invariant under h it can be realized as s h p for certain parabolic subgroup p of h we study the spherical representations of h induced from p we find formulas for the spherical functions in terms of the macdonald hypergeometric function this generalizes the earlier result of faraut koranyi for hermitian symmetric spaces d we consider a class of h invariant integral intertwining operators from the representations on l2 s to the holomorphic representations of g restricted to h we construct a new class of complementary series for the groups h so n m su n m with n m 2 and sp n m with n m 1 we realize them as discrete components in the branching rule of the analytic continuation of the holomorphic discrete series of g su n m su n m su n m and su 2n 2m respectively

reconstruction of a sparse spike train from a portion of its spectrum and application to high resolution deconvolution an algorithm is proposed for the reconstruction of a sparse spike train from an incomplete set of its fourier components it is shown that as little as 20 25 percent of the fourier spectrum is sufficient in practice for a high quality reconstruction the method employs linear programming to minimize the l1 norm of the output because minimization of this norm favors solutions with isolated spikes given a wavelet this technique can be used to perform deconvolution of noisy seismograms when the desired output is a sparse spike series relative reliability of the data is assessed in the frequency domain and only the reliable spectral data are included in the calculation of the spike series equations for the unknown spike amplitudes are solved to an accuracy compatible with the uncertainties in the reliable data in examples with 10 percent random noise the output is superior to that obtained using conventional least squares techniques

macdonald s constant term conjectures for exceptional root systems we announce proofs of macdonald s constant term conjectures for the affine root systems s f 4 and s f 4 v we also give an algorithm for deciding the conjectures for the remaining root systems s e 6 s e 7 and s e 8 and prove that the constant term in question can indeed be expressed in closed form combined with previous work of zeilberger bressoud kadell and gustafson our results imply that macdonald s conjectures are true in form for any root system

on p permutation equivalences between rickard equivalences and isotypies broue and rickard defined in their landmark papers from 1990 and 1996 the notions of an isotypy and a splendid equivalence between p blocks of finite groups here we define a notion of equivalence which we call a p permutation equivalence that implies an isotypy and is implied by a splendid equivalence moreover we study properties of p permutation equivalences

shift operators for the quantum calogero sutherland problems via knizhnik zamolodchikov equation we give a natural interpretation of the shift operators for calogero sutherland quantum problem via kz equation using matsuo cherednik mappings the explicit formulas for the inversions of these mappings and versions of shift operators for kz equations are also found as an application we show that the shift operator can be described via a factorization problem for an appropriate quantum integral discriminant of the calogero system

orthogonality relations and cherednik identities for multivariable baker akhiezer functions abstract we establish orthogonality relations for the baker akhiezer ba eigenfunctions of the macdonald difference operators we also obtain a version of cherednik macdonald mehta integral for these functions as a corollary we give a simple derivation of the norm identity and cherednik macdonald mehta integral for macdonald polynomials in the appendix written by the first author we prove a summation formula for ba functions we also consider more general identities of cherednik type which we use to introduce and construct more general twisted ba functions this leads to a construction of new quantum integrable models of macdonald ruijsenaars type

splendid equivalences derived categories and permutation modules we introduce the notion of a splendid equivalence between blocks of finite groups which is a strengthening of the notion of an equivalence of derived categories such an equivalence is always induced by a two sided tilting complex whose terms are summands of permutation modules a feature that is compatible with the belief that these complexes should be related to the adic cohomology of deligne lusztig varities in the case of reductive groups over finite fields we show that a splendid equivalence has good consequences over and above those that follow from a derived equivalence in particular it gives for principal blocks a structural explanation of brou s concept of an isotypy a compatible family of perfect isometries we give several examples including examples of morita equivalences that are splendid in an unexpectedly non trivial way

sparse multichannel blind deconvolution abstractwe developed a sparse multichannel blind deconvolution smbd method the method is a modification of the multichannel blind deconvolution technique often called euclid deconvolution in which the multichannel impulse response of the earth is estimated by solving an homogeneous system of equations classical euclid deconvolution is unstable in the presence of noise and requires the correct estimation of the length of the seismic wavelet the proposed method on the other hand can tolerate moderate levels of noise and does not require a priori knowledge of the length of the wavelet smbd solves the homogeneous system of equations arising in euclid deconvolution by imposing sparsity on the unknown multichannel impulse response trivial solutions to the aforementioned homogeneous system of equations are avoided by seeking sparse solutions on the unit sphere we tested smbd with synthetic and real data examples synthetic examples were used to judge the viability of the method in terms of noise we f

stochastic sparse spike deconvolution sparse spike deconvolution can be viewed as an inverse problem where the locations and amplitudes of a number of spikes reflectivity are estimated from noisy data seismic traces the main objective is to find the least number of spikes that when convolved with the available band limited seismicwaveletestimate fitthedatawithinagiventolerance errormisfit thedetectionofthespikes timelagsisahighly nonlinearoptimizationproblemthatcanbesolvedusingvery fastsimulatedannealingsa amplitudesareeasilyestimated using linear least squares at each sa iteration at this stage quadratic regularization is used to stabilize the solution to reduce its nonuniqueness and to provide meaningful reflectivitysequences thusavoidingtheneedtoconstrainthe spikes time lags and or amplitudes to force valid solutions impedanceconstraintsalsocanbeincludedatthisstage providing the low frequencies required to recover the acoustic impedance oneadvantageoftheproposedmethodoverother sparse spike deconvolution techniques is that the uncertaintyoftheobtainedsolutionscanbeestimatedstochastically further errorsinthephaseofthewaveletestimatearetolerated foranoptimumconstant phaseshiftisobtainedtocalibratetheeffectivewaveletthatispresentinthedata results using synthetic data including simulated data for the marmousi2 modelandfield 3d data show that physically meaningful high resolution sparse spike sections can be derived fromband limitednoisydata evenwhentheavailablewaveletestimateisinaccurate

difference equation for the heckman opdam hypergeometric function and its confluent whittaker limit abstract we present an explicit difference equation for the heckman opdam hypergeometric function associated with root systems via a confluent hypergeometric limit an analogous difference equation is obtained for the class one whittaker function diagonalizing the open quantum toda chain

explicit solution of the quantum three body calogero sutherland model the class of quantum integrable systems associated with root systems was introduced by olshanetsky and perelomov as a generalization of the calogero sutherland systems it was recently shown by one of the authors that for such systems with a potential the series in the product of two wavefunctions is the deformation of the clebsch gordan series this yields recursion relations for the wavefunctions of those systems and related to them for generalized zonal spherical functions on symmetric spaces in this letter this approach is used to compute the explicit expressions for the three body calogero sutherland wavefunctions which are the jack polynomials we conjecture that similar results are also valid for the more general two parameters deformation deformation introduced by macdonald

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

viterbi sparse spike detection accurate interpretation of seismic traveltimes and amplitudes in the exploration and global scales is complicated by the band limited nature of seismic data we discovered a stochastic method to reduce a seismic waveform into a most probable constituent spike train model waveforms were constructed from a set of candidate spike trains convolved with a source wavelet estimate for each model waveform a profile hidden markov model hmm was constructed to represent the waveform as a stochastic generative model with a linear topology corresponding to a sequence of samples each match state in the hmm represented a sample in the model waveform in which the amplitude was represented by a gaussian distribution insert and delete states allowed the underlying source wavelet to dilate or contract accounting for nonstationarity in the seismic data and errors in the source wavelet estimate the gaussian distribution characterizing each sample s amplitude accounted for random noise the viterbi algorithm was employed to simultaneously find the optimal nonlinear alignment between a model waveform and the seismic data and to assign a score to each candidate spike train the most probable traveltimes and amplitudes were inferred from the alignments of the highest scoring models the method required no implicit assumptions regarding the distribution of traveltimes and amplitudes however in practice the solution set may be limited to mitigate the nonuniqueness of solutions and to reduce the computational effort our analyses found that the method can resolve closely spaced arrivals below traditional resolution limits and that traveltime estimates are robust in the presence of random noise and source wavelet errors the method was particularly well suited to fine scale interpretation problems such as thin bed interpretation tying seismic images to well logs and the analysis of anomalous waveforms in global seismology

an elementary construction of lowering and raising operators for the trigonometric calogero sutherland model the quantum calogero sutherland model of an type calogero f 1971 j math phys 12 419 36 sutherland b 1972 phys rev a 4 2019 21 is completely integrable olshanetsky m a and perelomov a m 1977 lett math phys 2 7 13 olshanetsky m a and perelomov a m 1978 funct anal appl 12 121 8 olshanetsky m a and perelomov a m 1983 phys rep 94 313 404 using this fact we give an elementary construction of lowering and raising operators for the trigonometric case this is similar to but more complicated due to the fact that the energy spectrum is not equidistant than the construction for the rational case perelomov a m 1976 itep preprint no 27

autoregressive recovery of the acoustic impedance this paper presents a method of recovering the acoustic impedance from reflection seismograms using autoregressive ar modeling an approach originally applied to deconvolution by lines and clayton 1977 the algorithm which we describe is novel both in the manner in which the missing low and high frequency information is predicted and in the fact that the prediction may be constrained if acoustic impedance information is available the prediction of the low frequencies treats the missing data as a gap which extends from the low frequency cut off in the negative frequency band to the corresponding frequency in the positive frequency band the conjugate symmetry which governs the behavior of the spectrum in the band is taken into account in the prediction the missing high frequencies are predicted using a modified minimum entropy norm in the frequency domain both synthetic and field examples are presented and illustrate the robustness of the new ar algorithm under a variety of conditions the field e

minimum entropy deconvolution with frequency domain constraints a method for reconstructing the reflectivity spectrum using the minimum entropy criterion is presented the algorithm fmed described is compared with the classical minimum entropy deconvolution med as well as with the linear programming lp and autoregressive ar approaches the med is performed by maximizing an entropy norm with respect to the coefficients of a linear operator that deconvolves the seismic trace by comparison the approach presented here maximizes the norm with respect to the missing frequencies of the reflectivity series spectrum this procedure reduces to a nonlinear algorithm that is able to carry out the deconvolution of band limited data avoiding the inherent limitations of linear operators the proposed method is illustrated under a variety of synthetic examples field data are also used to test the algorithm the results show that the proposed method is an effective way to process band limited data the fmed and the lp arise from similar conceptions both methods seek an extremum of a particular norm subjected to frequency constraints in the lp approach the linear programming problem is solved using an adaptation of the simplex method which is a very expensive procedure the fmed uses only two fast fourier transforms ffts per iteration hence the computational cost of the inversion is reduced

bosonic and fermionic eigenstates for generalized sutherland models we construct bosonic and fermionic eigenstates for the generalized sutherland models associated with arbitrary reduced root systems respectively through w symmetrization and w anti symmetrization of heckman opdam s nonsymmetric jacobi polynomials square norms of the nonsymmetric heckman opdam polynomials are evaluated from their rodrigues formulae the w symmetrization and w anti symmetrization of the nonsymmetric polynomials enable us to evaluate square norms of bosonic and fermionic eigenstates for the generalized sutherland models

the hamiltonian of the quantum trigonometric calogero sutherland model in the exceptional algebra e8 we express the hamiltonian of the quantum trigonometric calogero sutherland model for the lie algebra e8 and coupling constant by using the fundamental irreducible characters of the algebra as dynamical independent variables

the homotopy theory of fusion systems the main goal of this paper is to identify and study a certain class of spaces which in many ways behave like p completed classifying spaces of finite groups these spaces occur as the classifying spaces of certain algebraic objects which we call p local finite groups a p local finite group consists roughly speaking of a finite p group s and fusion data on subgroups of s encoded in a way explained below our starting point is our earlier paper blo on p completed classifying spaces of finite groups together with the axiomatic treatment by lluis puig pu pu2 of systems of fusion among subgroups of a given p group the p completion of a space x is a space xp which isolates the properties of x at the prime p and more precisely the properties which determine its mod p cohomology for example a map of spaces x f y induces a homotopy equivalence xp y p if and only if f induces an isomorphism in mod p cohomology and h xp fp h x fp in favorable cases if x is p good when g is a finite group the p completion bgp of its classifying space encodes many of the properties of g at p for example not only the mod p cohomology of bg but also the sylow p subgroup of g together with all fusion among its subgroups are determined up to isomorphism by the homotopy type of bgp our goal here is to give a direct link between p local structures and homotopy types which arise from them this theory tries to make explicit the essence of what it means to be the p completed classifying space of a finite group and at the same time yields new spaces which are not of this type but which still enjoy most of the properties a space of the form bgp would have we hope that the ideas presented here will have further applications and generalizations in algebraic topology but this theory also fits well with certain aspects of modular representation theory in particular it may give a way of constructing classifying spaces for blocks in the group ring of a finite group over an algebraically closed field of characteristic p a saturated fusion system f over a p group s consists of a set homf p q of monomorphisms for each pair of subgroups p q s which form a category under composition include all monomorphisms induced by conjugation in s and satisfy certain other axioms formulated by puig definitions 1 1 and 1 2 below in particular these axioms are satisfied by the conjugacy homomorphisms in a finite group we refer to pu and pu2 for more details of puig s work on saturated

on stable equivalences with endopermutation source abstract we show that a bimodule between block algebras which has a fusion stable endopermutation module as a source and which induces morita equivalences between centralisers of nontrivial subgroups of a defect group induces a stable equivalence of morita type this is the converse to a theorem of puig the special case where the source is trivial has long been known by many authors the earliest instance of a result deducing a stable equivalence of morita type from local morita equivalences with possibly nontrivial endopermutation source is due to puig in the context of blocks with abelian defect groups with a frobenius inertial quotient the present note is motivated by an application due to biland to blocks of finite groups with structural properties known to hold for hypothetical minimal counterexamples to the z p theorem