higher order congruence relations on affine moment graphs the subgeneric case abstract we study the structure algebra z of the stable moment graph for the case of the affine root system a 1 the structure algebra z is an algebra over a symmetric algebra and in particular it is a module over a symmetric algebra we study this module structure on z and we construct a basis by setting c equal to zero in z we obtain the module z c 0 this module can be described in terms of the finite root system a 1 and we show that it is determined by a set of certain divisibility relations these relations can be regarded as a generalization of ordinary moment graph relations that define sections of sheaves on moment graphs and because of this we call them higher order congruence relations

on decomposition of bott samelson sheaves abstract we give an exact algorithm to calculate under some gkm restriction the matrix describing the embedding b s x b s x where the first module is the costalk and the second one is the stalk at x of a bott samelson module sheaf b s this allows us to calculate the first few terms of the decomposition of b s into a sum of indecomposable modules sheaves and to calculate the characters of braden macpherson sheaves in some previously unknown cases

representations of semisimple lie algebras in prime characteristic and the noncommutative springer resolution we prove most of lusztig s conjectures on the canonical basis in homology of a springer ber the conjectures predict that this basis controls numerics of representations of the lie algebra of a semisimple algebraic group over an algebraically closed eld of positive characteristic we check this for almost all characteristics to this end we construct a noncommutative resolution of the nilpotent cone which is derived equivalent to the springer resolution on the one hand this noncommutative resolution is closely related to the positive characteristic derived localization equivalences obtained earlier by the present authors and rumynin on the other hand it is compatible with the t structure arising from an equivalence with the derived category of perverse sheaves on the ane ag variety of the langlands dual group this equivalence established by arkhipov and the rst author ts the framework of local geometric langlands duality the latter compatibility allows one to apply frobenius purity theorem to deduce the desired properties of the basis we expect the noncommutative counterpart of the springer resolution to be of independent interest from the perspectives of algebraic geometry and geometric langlands duality

perturbative analysis of the method of particular solutions for improved inclusion of high lying dirichlet eigenvalues the dirichlet eigenvalue or drum problem in a domain omega subset mathbb r 2 becomes numerically challenging at high eigenvalue frequency e in this regime the method of particular solutions mps gives spectral accuracy for many domain shapes it requires a number of degrees of freedom scaling as sqrt e the number of wavelengths on the boundary in contrast to direct discretization for which this scaling is e our main result is an inclusion bound on eigenvalues that is a factor o sqrt e tighter than the classical bound of moler payne and that is optimal in that it reflects the true slopes of curves appearing in the mps we also present an mps variant that cures a normalization problem in the original method while evaluating basis functions only on the boundary this method is efficient at high frequencies where we show that in practice our inclusion bound can give three extra digits of eigenvalue accuracy with no extra effort

wigner dyson statistics for a class of integrable models we construct an ensemble of second quantized hamiltonians with two bosonic degrees of freedom whose members display with probability one gaussian orthogonal ensemble goe or gaussian unitary ensemble gue statistics nevertheless these hamiltonians have a second integral of motion namely the boson number and thus are integrable to construct this ensemble we use some reverse engineering starting from the fact that n bosons in a two level system with random interactions have an integrable classical limit by the old heisenberg association of boson operators to actions and angles by choosing an n body random interaction and degenerate levels we end up with goe or gue hamiltonians ergodicity of these ensembles completes the example

semiclassical level spacings when regular and chaotic orbits coexist the authors calculate semiclassical limiting level spacing distributions p s for systems whose classical energy surface is divided into a number of separate region in which motion is regular or chaotic in the calculation it is assumed that the spectrum is the superposition of statistically independent sequences of levels from each of the classical phase space regions sequences from regular regions having poisson distributions and those from irregular regions having wigner distributions the formulae for p s depend on the sum of the liouville measures of all the classical regular regions and on the separate liouville measures of the significant chaotic regions

the combinatorics of coxeter categories we present an alternative construction of soergel s category of bimodules associated to a reflection faithful representation of a coxeter system we show that its objects can be viewed as sheaves on the associated moment graph we introduce an exact structure and show that soergel s special bimodules are the projective objects then we construct the indecomposable projectives by both a global and a local method discuss a version of the kazhdan lusztig conjecture and prove it for universal coxeter systems

spacing distributions for rhombus billiards we show that the spacing distributions of rational rhombus billiards fall in a family of universality classes distinctly different from the wigner dyson family of random matrix theory and the poisson distribution some of the distributions find explanation in a recent work of bogomolny gerland and schmit for the irrational billiards despite ergodicity we get the same distribution for the examples considered once again distinct from the wigner dyson distributions all the results are obtained numerically by a method that allows us to reach very high energies

bouncing ball orbits and symmetry breaking effects in a three dimensional chaotic billiard we study the classical and quantum mechanics of a three dimensional stadium billiard it consists of two quarter cylinders that are rotated with respect to each other by 90 ifmmode circ else textdegree fi and it is classically chaotic the billiard exhibits only a few families of nongeneric periodic orbits we introduce an analytic method for their treatment the length spectrum can be understood in terms of the nongeneric and unstable periodic orbits for unequal radii of the quarter cylinders the level statistics agree well with predictions from random matrix theory for equal radii the billiard exhibits an additional symmetry we investigated the effects of symmetry breaking on spectral properties moreover for equal radii we observe a small deviation of the level statistics from random matrix theory this led to the discovery of stable and marginally stable orbits which are absent for unequal radii

vector bundles of rank 2 and linear systems on algebraic surfaces let s be a smooth complex algebraic surface and let l be a divisor on s it is well known that the linear system il ksi where ks is the canonical divisor plays an important role in understanding the geometry of s several authors see for example 8 11 studied various properties of this system our paper treats the same subject but the point of view is different from the works mentioned above the philosophy underlying our approach is almost classical points on s more generally effective 0 cycles in special position with respect to il ksi see definition below contain information about the geometry of s this point of view was recently revived in 5 also 10 in fact this paper contains the technique of going from the effective 0 cycle z in special position with respect to il ksi to the geometry of s for the sake of completeness we give a definition and state a theorem which can be found in 5 10

classical dynamics of a family of billiards with analytic boundaries the classical dynamics of a billiard which is a quadratic conformal image of the unit disc is investigated the author gives the stability analysis of major periodic orbits present the poincare maps demonstrate the mixing properties by following the evolution of a small element in phase space show the existence of homoclinic points and calculate the lyapunov exponent and the kolmogorov entropy h it turns out that the system becomes strongly chaotic positive h for sufficiently large deformations of the unit disc the system shows a generic stochastic transition the computations suggest that the system is mixing if the boundary is not convex

non generic spectral statistics in the quantized stadium billiard we consider the effect of a continuous family of neutral bouncing ball orbits on the energy spectrum of the quantized stadium billiard using a semiclassical approximation we derive analytic expressions for standard two point spectral measures the corrections due to the bouncing ball orbits account for some of the non generic features observed in the analysis of the spectrum of a stadium cavity which was recently measured once the bouncing ball contributions are subtracted the spectrum is shown to be well reproduced by the semi classical trace formula based on unstable periodic orbits we also study special patterns in the spectrum which are due to other non generic features such as edge effects and whispering gallery trajectories

on the kazhdan lusztig polynomials for affine weyl groups in 191 kazhdan and lusztig associated to a coxeter group w certain polynomials p y w e w in the case where w is a weyl group these polynomials or their values at 1 p l give multiplicities of irreducible constituents in the jordan holder series of verma modules for a complex semisimple lie algebra corresponding to w see 3 4 also in the case where w is an affine weyl group it has been conjectured by lusztig lo that these polynomials or their values p l enter the character formula of irreducible rational representations for a semisimple algebraic group corresponding to w over an algebraically closed field of prime characteristic the purpose of this paper is to give some formula 4 2 see also 4 10 concerning the kazhdan lusztig polynomials p for affine weyl groups which appear in this lusztig conjecture namely p for y w dominant this is a generalization of the q analogue of kostant s weight multiplicity formula in 8 as a corollary of the formula we shall show that the lusztig conjecture above is consistent with the steinberg tensor product theorem 15 1 to be more precise let p be the weight lattice of a root system r ii r a positive root system the weyl group w of r acts on p hence we can define the semidirect product of w by p p w d p the element of corresponding to a e p is denoted by t the group contains an afline weyl group w w d q as a normal subgroup where q is the root lattice although is not a coxeter group in general we can define the bruhat order the length function i p la and the kazhdan lusztig polynomials p y w e i v as natural extensions of those for w the main result of this paper under the specialization q 1 is stated as follows corollary 4 10

consolidating boundary methods for finding the eigenstates of billiards the plane wave decomposition method a widely used means of numerically finding eigenstates of the helmholtz equation in billiard systems is described as a variant of the mathematically well established boundary integral method bim a new unified framework encompassing the two methods is discussed furthermore a third numerical method which we call the gauge freedom method is derived from the bim equations this opens the way to further improvements in eigenstate search techniques

berry robnik level statistics in a smooth billiard system the berry robnik level spacing distribution is demonstrated clearly in a generic quantized plane billiard for the first time however this ultimate semiclassical distribution is found to only be valid for extremely small semiclassical parameters effective planck s constant where the assumption of statistical independence of regular and irregular levels is achieved for sufficiently larger semiclassical parameters we find fractional power law level repulsion with a phenomenological brody distribution providing an adequate global fit

boundary quasi orthogonality and sharp inclusion bounds for large dirichlet eigenvalues we study eigenfunctions phi j and eigenvalues e j of the dirichlet laplacian on a bounded domain omega subset mathbb r n with piecewise smooth boundary we bound the distance between an arbitrary parameter e 0 and the spectrum e j in terms of the boundary l 2 norm of a normalized trial solution u of the helmholtz equation delta e u 0 we also bound the l 2 norm of the error of this trial solution from an eigenfunction both of these results are sharp up to constants hold for all e greater than a small constant and improve upon the best known bounds of moler payne by a factor of the wavenumber sqrt e one application is to the solution of eigenvalue problems at high frequency via for example the method of particular solutions in the case of planar strictly star shaped domains we give an inclusion bound where the constant is also sharp we give explicit constants in the theorems and show a numerical example where an eigenvalue around the 2500th is computed to 14 digits of relative accuracy the proof makes use of a new quasi orthogonality property of the boundary normal derivatives of the eigenmodes theorem 1 3 of interest in its own right namely the operator norm of the sum of rank 1 operators partial n phi j langle partial n phi j cdot rangle over all e j in a spectral window of width sqrt e a sum with about e n 1 2 terms is at most a constant factor independent of e larger than the operator norm of any one individual term

evanescent wave approach to diffractive phenomena in convex billiards with corners what we are going to call in this paper diffractive phenomena in billiards is far from being deeply understood these are sorts of singularities that for example some kind of corners introduce in the energy eigenfunctions in this paper we use the well known scaling quantization procedure to study them we show how the scaling method can be applied to convex billiards with corners taking into account the strong diffraction at them and the techniques needed to solve their helmholtz equation as an example we study a classically pseudointegrable billiard the truncated triangle then we focus our attention on the spectral behavior a numerical study of the statistical properties of high lying energy levels is carried out it is found that all computed statistical quantities are roughly described by the so called semi poisson statistics but it is not clear whether the semi poisson statistics is the correct one in the semiclassical limit

sheaves on moment graphs and a localization of verma flags abstract to any moment graph g we assign a subcategory v of the category of sheaves on g together with an exact structure we show that in the case that the graph is associated to a non critical block of the equivariant category o over a symmetrizable kac moody algebra v is equivalent as an exact category to the subcategory of modules that admit a verma flag the projective modules correspond under this equivalence to the intersection cohomology sheaves on the graph and hence by a theorem of braden and macpherson to the equivariant intersection cohomologies of schubert varieties associated to kac moody groups

comparable upper and lower bounds for boundary values of neumann eigenfunctions and tight inclusion of eigenvalues for smooth bounded domains in rn we prove upper and lower l2 bounds on the boundary data of neumann eigenfunctions and we prove quasiorthogonality of this boundary data in a spectral window the bounds are tight in the sense that both are independent of the eigenvalues this is achieved by working with an appropriate norm for boundary functions which includes a spectral weight that is a function of the boundary laplacian this spectral weight is chosen to cancel concentration at the boundary that can happen for whispering gallery type eigenfunctions these bounds are closely related to wave equation estimates due to tataru using this we bound the distance from an arbitrary helmholtz parameter e 0 to the nearest neumann eigenvalue in terms of boundary normal derivative data of a trial function u solving the helmholtz equation e u 0 this inclusion bound improves over previously known bounds by a factor of e5 6 analogously to a recently improved inclusion bound in the dirichlet case due to the first two authors finally we apply our theory to present an improved numerical implementation of the method of particular solutions for computation of neumann eigenpairs on smooth planar domains we show that the new inclusion bound improves the relative accuracy in a computed neumann eigenvalue around the 42000th from nine to fourteen digits with negligible extra numerical effort

quantum section method for the soft stadium abstract the soft stadium is defined by a monomial potential with exponent as a parameter such that corresponds to the billiard the practical use of the quantum section method depends only on the partial separability of the system on both sides of the section which holds for all s in particular for 1 0 the system becomes globally separable allowing for a general test of the method for various values of the parameter we also tested the use of the asymptotic wkb type approximation in the construction of green s functions and asymptotic overlap integrals to obtain higher energy eigenvalues we show these approximations to be reliable

robust and efficient solution of the drum problem via nystrom approximation of the fredholm determinant the drum problem finding the eigenvalues and eigenfunctions of the laplacian with dirichlet boundary condition has many applications yet remains challenging for general domains when high accuracy or high frequency is needed boundary integral equations are appealing for large scale problems yet certain difficulties have limited their use we introduce the following two ideas to remedy this 1 we solve the resulting nonlinear eigenvalue problem using boyd s method for analytic root finding applied to the fredholm determinant and we show that this is many times faster than the usual iterative minimization of a singular value 2 we fix the problem of spurious exterior resonances via a combined field representation this also provides the first robust boundary integral eigenvalue method for non simply connected domains we implement the new method in two dimensions using spectrally accurate nystrom product quadrature we prove exponential convergence of the determinant at roots for domains with

computation of the helmholtz eigenvalues in a class of chaotic cavities using the multipole expansion technique in this paper we present a numerical computation of the energy levels and the corresponding wave functions in a microwave resonator using the multipole expansion technique the approach permits closed form fast and robust solutions of the helmholtz equation and the schrodinger equation for two dimensional systems in an important class of wave chaos problem in particular wave functions inside the billiard are expressed in terms of a simple expansion of hankel functions the implementation of the approach is described and the classical bowtie cavity is considered as a case study to demonstrate the versatility and efficiency of the method to validate the accuracy the field distribution and the eigenvalues calculated using this approach are compared to the solution obtained by boundary integral method the case when the cavity contains objects perfect electric conductors and or dielectrics is also presented and discussed

quantising a generic family of billiards with analytic boundaries a generic family of plane billiards has been discovered recently the shape of the boundary is given by the quadratic conformal image of the unit circle and is thus real analytic for small deformations of the unit disc the billiard is a typical kam system but becomes ergodic or even mixing when the curvature of the boundary vanishes at some point the kolmogorov entropy has been calculated and it increases with the deformation of the boundary the author studies aspects of the quantum chaos for this billiard he solves numerically the eigenvalue problem for the laplace operator with dirichlet s boundary condition he examines the spectrum and inspects the avoided crossings at which mixing of nearby states occurs the variation of the nodal structure and of the localisation properties of the eigenfunctions is studied in analysing the level spacing distribution he finds a continuous transition from the poisson distribution towards the wigner distribution the exponent in the level repulsion law varies continuously along with a generic perturbation for small perturbations it seems to be proportional to the square root of the perturbation parameter

from moment graphs to intersection cohomology we describe a method of computing equivariant and ordinary intersection cohomology of certain varieties with actions of algebraic tori in terms of structure of the zero and one dimensional orbits the class of varieties to which our formula applies includes schubert varieties in flag varieties and affine flag varieties we also prove a monotonicity result on local intersection cohomology stalks

koszul duality patterns in representation theory the aim of this paper is to work out a concrete example as well as to provide the general pattern of applications of koszul duality to repre sentation theory the paper consists of three parts relatively independent of each other the first part gives a reasonably selfcontained introduction to koszul rings and koszul duality koszul rings are certain z graded rings with particularly nice homological properties which involve a kind of duality thus to a koszul ring one associates naturally the dual koszul ring the second part is devoted to an application to representation theory of semisimple lie algebras we show license or copyright restrictions may apply to redistribution see http www ams org journal terms of use koszul duality patterns 527 that the block of the bernstein gelfand gelfand category o that corresponds to any fixed central character is governed by the koszul ring moreover the dual of that ring governs a certain subcategory of the category o again this generalizes the selfduality theorem conjectured by beilinson and ginsburg in 1986 and proved by soergel in 1990 in the third part we study certain cate gories of mixed perverse sheaves on a variety stratified by affine linear spaces we provide a general criterion for such a category to be governed by a koszul ring in the flag variety case this reduces to the setup of part two in the more general case of affine flag manifolds and affine grassmannians the criterion should yield interesting results about representations of quantum groups and affine lie algebras department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address sasha math mit edu department of mathematics the university of chicago chicago illinois 60637 e mail address ginzburg math uchicago edu max planck institut fur mathematik gottfried claren strase 26 d 53 bonn 3 germany current address mathematisches institut universitat freiburg albertstrase 23b d 79104 freiburg germany e mail address soergel sun1 mathematik uni freiburg de license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

anomalous spectral statistics in a symmetrical billiard it is a common assumption that time reversal invariance or some other anti unitary symmetry for a classically chaotic system implies goe type spectral fluctuations for the corresponding quantum system based on previous work on structural invariance we show that a time reversal invariant system for certain point symmetries displays the gue statistics typical of systems with broken time reversal symmetry specifically a billiard having only three fold symmetry shows this unexpected behaviour

numerical accuracy of bogomolny s semiclassical quantization scheme in quantum billiards we use the semiclassical quantization scheme of bogomolny to calculate eigenvalues of the limacon quantum billiard corresponding to a conformal map of the circle billiard we use the entire billiard boundary as the chosen surface of section and use a finite approximation for the transfer operator in coordinate space computation of the eigenvalues of this matrix combined with a quantization condition determines a set of semiclassical eigenvalues which are compared with those obtained by solving the schrodinger equation the classical dynamics of this billiard system undergoes a smooth transition from integrable circle to completely chaotic motion thus providing a test of bogomolny s semiclassical method in coordinate space in terms of the morphology of the wavefunction we analyse the results for billiards which exhibit both soft and hard chaos

generic spectral properties of right triangle billiards this article presents a new method to calculate eigenvalues of right triangle billiards its efficiency is comparable to the boundary integral method and more recently developed variants its simplicity and explicitness however allow new insight into the statistical properties of the spectra we analyse numerically the correlations in level sequences at high level numbers 10 5 for several examples of right triangle billiards we find that the strength of the correlations is closely related to the genus of the invariant surface of the classical billiard flow surprisingly the genus plays an important role at the quantum level also based on this observation a mechanism is discussed which may explain the particular quantum classical correspondence in right triangle billiards though this class of systems is rather small it contains examples for integrable pseudo integrable and non integrable ergodic weakly mixing dynamics so that the results might be relevant in a more general context

the hodge theory of soergel bimodules we prove soergel s conjecture on the characters of indecomposable soergel bimodules we deduce that kazhdan lusztig polynomials have positive coecients for arbitrary coxeter systems using results of soergel one may deduce an algebraic proof of the kazhdan lusztig conjecture

study of regular and irregular states in generic systems in this work we present the results of a numerical and semiclassical analysis of high lying states in a hamiltonian system whose classical mechanics is of a generic mixed type where the energy surface is split into regions of regular and chaotic motion as predicted by the principle of uniform semiclassical condensation when the effective tends to zero each state can be classified as regular or irregular we were able to semiclassically reproduce individual regular states by the einstein brillouin keller torus quantization for which we devise a new approach while for the irregular ones we found the semiclassical prediction of their autocorrelation function in a good agreement with numerics we also looked at the low lying states to get a better understanding of the onset of semiclassical behaviour

loewy filtration and quantum de rham cohomology over quantum divided power algebra abstract the paper explores the indecomposable submodule structures of quantum divided power algebra a q n defined in 23 and its truncated objects a q n m an intertwinedly lifting method is established to prove the indecomposability of a module when its socle is non simple the loewy filtrations are described for all homogeneous subspaces a q s n or a q s n m the loewy layers and dimensions are determined the rigidity of these indecomposable modules is proved an interesting combinatorial identity is derived from our realization model for a class of indecomposable u q sl n modules meanwhile the quantum grassmann algebra q n over a q n is constructed together with the quantum de rham complex q n d via defining the appropriate q differentials and its subcomplex q n m d for the latter the corresponding quantum de rham cohomology modules are decomposed into the direct sum of some sign trivial u q sl n modules

semiempirical theory of level spacing distribution beyond the berry robnik regime modeling the localization and the tunneling effects in this work we study the level spacing distribution in the classically mixed type quantum systems which are generic exhibiting regular motion on invariant tori for some initial conditions and chaotic motion for the complementary initial conditions in the asymptotic regime of the sufficiently deep semiclassical limit sufficiently small effective planck constant the berry and robnik 1984 j phys a math gen 17 2413 picture applies which is very well established we present a new quasi universal semiempirical theory of the level spacing distribution in a regime away from the berry robnik regime the near semiclassical limit by describing both the dynamical localization effects of chaotic eigenstates and the tunneling effects which couple regular and chaotic eigenstates the theory works extremely well in the 2d mixed type billiard system introduced by robnik 1983 j phys a math gen 16 3971 and is also tested in other systems mushroom billiard and prosen billiard

localization of modules for a semisimple lie algebra in prime characteristic with an appendix by r bezrukavnikov and s riche computation for sl 3 we show that on the level of derived categories representations of the lie algebra of a semisimple algebraic group over a field of finite characteristic with a given generalized regular central character are the same as coherent sheaves on the formal neighborhood of the corresponding generalized springer fiber the first step is to observe that the derived functor of global sections provides an equivalence between the derived category of d modules with no divided powers on the flag variety and the appropriate derived category of modules over the corresponding lie algebra thus the derived version of the beilinson bernstein localization theorem holds in sufficiently large positive characteristic next one finds that for any smooth variety this algebra of differential operators is an azumaya algebra on the cotangent bundle in the case of the flag variety it splits on springer fibers and this allows us to pass from d modules to coherent sheaves the argument also generalizes to twisted d modules as an application we prove lusztig s conjecture on the number of irreducible modules with a fixed central character we also give a formula for behavior of dimension of a module under translation functors and reprove the kac weisfeiler conjecture the sequel to this paper bmr2 treats singular infinitesimal characters

dynamical localization of chaotic eigenstates in the mixed type systems spectral statistics in a billiard system after separation of regular and chaotic eigenstates we study the quantum mechanics of a billiard robnik 1983 j phys a math gen 16 3971 in the regime of mixed type classical phase space the shape parameter 0 15 at very high lying eigenstates starting at about 1 000 000th eigenstate and including the consecutive 587654 eigenstates by calculating the normalized poincare husimi functions of the eigenstates and comparing them with the classical phase space structure we introduce the overlap criterion which enables us to separate with great accuracy and reliability the regular and chaotic eigenstates and the corresponding energies the chaotic eigenstates appear all to be dynamically localized meaning that they do not uniformly occupy the entire available chaotic classical phase space component but are localized on a proper subset we find with unprecedented precision and statistical significance that the level spacing distributions of the regular levels obey the poisson statistics and the chaotic ones obey the brody statistics as anticipated in a recent paper by batistic and robnik 2010 j phys a math theor 43 215101 where the entire spectrum was found to obey the berry robnik brody statistics there are no effects of dynamical tunneling in this regime due to the high energies as they decay exponentially with the inverse effective planck constant which is proportional to the square root of the energy

dynamical breaking of time reversal symmetry it is a common assumption that quantum systems with time reversal invariance and classically chaotic dynamics have energy spectra distributed according to goe type of statistics here we present a class of systems which fail to follow this rule we show that for convex billiards of constant width with time reversal symmetry and almost chaotic dynamics the energy level distribution is of gue type the effect is due to the lack of ergodicity in the momentum part of the phase space and as we argue is generic in two dimensions

light leaves and lusztig s conjecture we define a map f with domain a certain subset of the set of light leaves certain morphisms between soergel bimodules introduced by the author in an earlier paper and range the set of prime numbers using results of soergel we prove the following property of f if the image p f l of some light leaf l under f is bigger than the coxeter number of the corresponding weyl group then there is a counterexample to lusztig s conjecture in characteristic p we also introduce the double leaves basis which is an improvement of the light leaves basis that has already found interesting applications in particular it forms a cellular basis of soergel bimodules that allows us to produce an algorithm to find the bad primes for lusztig s conjecture

reductive groups the loop grassmannian and the springer resolution in this paper we prove equivalences of categories relating the derived category of a block of the category of representations of a connected reductive algebraic group over an algebraically closed field of characteristic p bigger than the coxeter number and a derived category of equivariant coherent sheaves on the springer resolution or a parabolic counterpart in the case of the principal block combined with previous results this provides a modular version of celebrated constructions due to arkhipov bezrukavnikov ginzburg for lusztig s quantum groups at a root of unity as an application we prove a graded version of a conjecture of finkelberg mirkovic describing the principal block in terms of mixed perverse sheaves on the dual affine grassmannian and deduce a new proof of lusztig s conjecture in large characteristic

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

reviving the method of particular solutions fox henrici and moler made famous a method of particular solutions for computing eigenvalues and eigenmodes of the laplacian in planar regions such as polygons we explain why their formulation of this method breaks down when applied to regions that are insufficiently simple and propose a modification that avoids these difficulties the crucial changes are to introduce points in the interior of the region as well as on the boundary and to minimize a subspace angle rather than just a singular value or a determinant similar methods may be used to improve other mesh free algorithms for a variety of computational problems

on spectral statistics of classically integrable systems this work is an extensive study of the spectral statistics of three representative classically integrable systems namely rectangle torus and circle billiards we analyse the statistics and focus on the related level spacing distribution p s and the delta statistics the agreement with the poissonian model is typically found to be perfect up to the outer unfolded energy scale beyond which the saturation is observed in agreement with berry s dynamical theory of the spectral rigidity where as the untypical systems are those where for example p s is not a smooth distribution but a sum of the delta functions due to the granularity of the energy scale for example in the rectangle with the rational squared sides ratio however even there we find reasonable trend towards poissonian statistics for large ranges l but we describe theoretically and numerically the broadening of the delta spikes when the rectangular billiard is slightly distorted away from a rational to an irrational shape and find excellent agreement also in irrational rectangle billiards we show and explain the existence of large fluctuations by one order of magnitude bigger than the statistical ones whose origin is in the closeness to some rational billiard shape these fluctuations and their amplitude are independent of the energy if the bin size shrinks inversely with energy finally we tested the mode distribution i e distribution of the reduced mode fluctuation number w and found that it was not poissonian in agreement with steiner s conjecture and in fact follows the prediction by bleher et al that its tail behaves as the general reason for non universal behaviour of p w is that at largest energy scales we are always in the saturation regime

asymptotic rate of quantum ergodicity in chaotic euclidean billiards the quantum unique ergodicity que conjecture of rudnick and sarnak is that every eigenfunction n of the laplacian on a manifold with uniformly hyperbolic geodesic flow becomes equidistributed in the semiclassical limit eigenvalue en that is strong scars are absent we study numerically the rate of equidistribution for a uniformly hyperbolic sinai type planar euclidean billiard with dirichlet boundary condition the drum problem at unprecedented high e and statistical accuracy via the matrix elements n m of a piecewise constant test function a by collecting 30 000 diagonal elements up to level n 7 105 we find that their variance decays with eigenvalue as a power 0 48 0 01 close to the semiclassical estimate of feingold and peres this contrasts with the results of existing studies which have been limited to en a factor 102 smaller we find strong evidence for que in this system we also compare off diagonal variance as a function of distance from the diagonal against feingold peres or spectral measure at the highest accuracy 0 7 thus far in any chaotic system we outline the efficient scaling method and boundary integral formulae used to calculate eigenfunctions 2006 wiley periodicals inc

a model for the computation of quantum billiards with arbitrary shapes an expansion method for the stationary schrodinger equation of a three dimensional quantum billiard system whose boundary is defined by an arbitrary analytic function is introduced the method is based on a coordinate transformation and an expansion in spherical harmonics the effectiveness is verified and confirmed by a numerical example which is a billiard system depending on a parameter

autocorrelation function of eigenstates in chaotic and mixed systems we study the autocorrelation function of different types of eigenfunctions in quantum mechanical systems with either chaotic or mixed classical limits we obtain an expansion of the autocorrelation function in terms of the correlation distance for localized states in billiards like bouncing ball modes or states living on tori a simple model using only classical input gives good agreement with the exact result in particular a prediction for irregular eigenfunctions in mixed systems is derived and tested for chaotic systems the expansion of the autocorrelation function can be used to test quantum ergodicity on different length scales

schubert calculus and torsion explosion the author observes that certain numbers occurring in schubert calculus for sl also occur as entries in intersection forms controlling decompositions of soergel bimodules in higher rank these numbers grow exponentially this observation gives many counter examples to the expected bounds in lusztig s conjecture on the characters of simple rational modules for sl over fields of positive characteristic the examples also give counter examples to the james conjecture on decomposition numbers for symmetric groups

on the spannedness of adjunction of vector bundles let x be a smooth projective variety of dimension n defined over the field of complex numbers and e be an ample vector bundle on x the linear system kx c e is called the adjoint bundle for vector bundle e the nefness and the ampleness of the adjoint bundle kx cl e have been investigated by several authors abw f p1 yz in this paper we shall investigate the base point freeness spannedness of the adjoint bundle kx c e i e we ask the following

discrete symmetries and spectral statistics we calculate the 2 point spectral statistics associated with a given irreducible representation i e symmetry class for time reversal invariant systems possessing discrete symmetries using semiclassical periodic orbit theory when the representation in question is real or pseudo real our results conform to those of the gaussian orthogonal ensemble goe of random matrices when it is complex we find instead gaussian unitary ensemble gue behaviour this provides a direct semiclassical explanation for the recent observation by leyvraz et al 1996 of gue correlations in the desymmetrized spectra of certain symmetric billiards in the absence of any time reversal invariance breaking e g magnetic fields

quantum localization of chaotic eigenstates and the level spacing distribution the phenomenon of quantum localization in classically chaotic eigenstates is one of the main issues in quantum chaos or wave chaos and thus plays an important role in general quantum mechanics or even in general wave mechanics in this work we propose two different localization measures characterizing the degree of quantum localization and study their relation to another fundamental aspect of quantum chaos namely the energy spectral statistics our approach and method is quite general and we apply it to billiard systems one of the signatures of the localization of chaotic eigenstates is a fractional power law repulsion between the nearest energy levels in the sense that the probability density to find successive levels on a distance s goes like proportionality s for small s where 0 1 and 1 corresponds to completely extended states we show that there is a clear functional relation between the exponent and the two different localization measures one is based on the information entropy and the other one on the correlation properties of the husimi functions we show that the two definitions are surprisingly linearly equivalent the approach is applied in the case of a mixed type billiard system m robnik j phys a math gen 16 3971 1983 in which the separation of regular and chaotic eigenstates is performed

an efficient fredholm method for the calculation of highly excited states of billiards a numerically efficient fredholm formulation of the billiard problem is presented the standard solution in the framework of the boundary integral method in terms of a search for roots of a secular determinant is reviewed first we next reformulate the singularity condition in terms of a flow in the space of an auxiliary one parameter family of eigenproblems and argue that the eigenvalues and eigenfunctions are analytic functions within a certain domain based on this analytic behavior we present a numerical algorithm to compute a range of billiard eigenvalues and associated eigenvectors by only two diagonalizations

comment on energy level statistics in the mixed regime we comment on the recent paper by abul magd 1996 29 1 concerning the energy level statistics in the mixed regime i e such having the mixed classical dynamics where regular and chaotic regions coexist in the phase space we point out that his basic assumption on the additive property of the level repulsion function r s conditional probability density in the sense of dividing it linearly into the regular and chaotic part in proportion to the classical fractional phase space volumes and is not justified since among other things it relies on the type of berry s ergodic assumption which however is only correct in a homogeneous ensemble of ergodic systems not in the neighbourhood of an integrable system thus his resulting distribution cannot be regarded as a theoretically well founded object we point out that the semiclassical limiting energy level spacing distribution must be of berry robnik 1984 type and explain what transitional behaviour of the brody type with fractional power law energy level repulsion we observe in the near semiclassical regime where effective is not yet small enough thus we refer to the derivation arguments and conclusions in our previous paper prosen and robnik 1994 26 8059 and explain again the behaviour in this double transition region

fast computation of high frequency dirichlet eigenmodes via spectral flow of the interior neumann to dirichlet map we present a new algorithm for numerical computation of large eigenvalues and associated eigenfunctions of the dirichlet laplacian in a smooth star shaped domain in d d 2 conventional boundary based methods require a root search in eigenfrequency k hence take o n3 effort per eigenpair found where n o kd 1 is the number of unknowns required to discretize the boundary our method is o n faster achieved by linearizing with respect to k the spectrum of a weighted interior neumann to dirichlet ntd operator for the helmholtz equation approximations to the square roots kj of all o n eigenvalues lying in k k where o 1 are found with o n3 effort we prove an error estimate with c independent of k we present a higher order variant with eigenvalue error scaling empirically as o 5 and eigenfunction error as o 3 the former improving upon the scaling method of vergini and saraceno for planar domains d 2 with an assumption of absence of spectral concentration we also prove rigorous error bounds that are close to those numerically observed for d 2 we compute robustly the spectrum of the ntd operator via potential theory nystrom discretization and the cayley transform at high frequencies 400 wavelengths across with eigenfrequency relative error 10 10 we show that the method is 103 times faster than standard ones based upon a root search 2014 wiley periodicals inc

kazhdan lusztig combinatorics in the moment graph setting abstract motivated by a question on the graded rank of the stalks of the canonical sheaf on a bruhat graph we lift some equalities concerning parabolic kazhdan lusztig polynomials to this moment graph setting our proofs hold also in positive characteristic under some technical assumptions

bogomolny section for the stadium i quantum theory the quarter stadium can be decomposed into a rectangle and a quarter circle in each of these the helmholtz equation is separable we thus construct green s functions for both regions and a fully quantum mechanical bogomolny matrix this results in a very efficient algorithm for calculating eigenvalues and eigenfunctions we discuss the relation of the family of periodic bouncing ball orbits to the contribution of the non oscillating modes