conditional intensity and gibbsianness of determinantal point processes the papangelou intensities of determinantal or fermion point processes are investigated these exhibit a monotonicity property expressing the repulsive nature of the interaction and satisfy a bound implying stochastic domination by a poisson point process we also show that determinantal point processes satisfy the so called condition which is a general form of gibbsianness under a continuity assumption the gibbsian conditional probabilities can be identified explicitly

quantum function algebra at roots of 1 abstract we introduce a form of the quantum function algebra on a drinfeld jimbo quantum group over the ring z q q 1 specializing q to a root of 1 we show that over the cyclotomic field this algebra is a projective module over its central sub algebra which is the usual coordinate algebra of the group we study the induced poisson lie structure of the group a bundle of algebras on a complex simply connected lie group with hamiltonian flows in the bundle is constructed some representations of the quantum function algebra in a root of 1 are constructed as an application an estimate of the dimension of an arbitrary representation is given

reduced standard modules and cohomology first cohomology groups of finite groups with nontrivial irreducible coefficients have been useful in several geometric and arithmetic contexts including wiles s famous paper 1995 internal to group theory 1 cohomology plays a role in the general theory of maximal subgroups of finite groups as developed by aschbacher and scott 1985 one can pass to the case where the group acts faithfully and the underlying module is absolutely irreducible in this case r guralnick 1986 conjectured that there is a universal constant bounding all of the dimensions of these cohomology groups this paper provides the first general positive results on this conjecture proving that the generic 1 cohomology h 1 gen g l lim h 1 g q l see cline parshall scott and van der kallen 1977 of a finite group g q of lie type with absolutely irreducible coefficients l in the defining characteristic of g is bounded by a constant depending only on the root system in all cases we are able to improve this result to a bound on h 1 g q l itself still depending only on the root system the generic h 1 result and related results for ext 1 emerge here as a consequence of a general study of interest in its own right of the homological properties of certain rational modules red red indexed by dominant weights for a reductive group g the modules red and red arise naturally from irreducible representations of the quantum enveloping algebra u of the same type as g at a pth root of unity where p 0 is the characteristic of the defining field for g finally we apply our ext 1 bounds results of bendel nakano and pillen 2006 as well as results of sin 1993 1992 1994 on the ree and suzuki groups to obtain the non generic bounds on h 1 g q l

rigid analytic quantum groups and quantum arens michael envelopes abstract we introduce a rigid analytification of the quantized coordinate algebra of a semisimple algebraic group and a quantized arens michael envelope of the enveloping algebra of the corresponding lie algebra working over a non archimedean field and when q is not a root of unity we show that these analytic quantum groups are topological hopf algebras and frechet stein algebras we then introduce an analogue of the bgg category o for the quantum arens michael envelope and show that it is equivalent to the category o for the corresponding quantized enveloping algebra

cohomology of infinitesimal and discrete groups an infinitesimal algebraic group over a field k is a connected algebraic group scheme whose coordinate ring is a finite dimension al k algebra if one begins with a reductive algebraic group g defined over fp then one has a close relationship between the representation theory of g that of its infinitesimal subgroups g defined as the kernels of the frobenius endomorphisms rr g g and that of its finite chevalley subgroups g fq of fq rational points q pd a basic theorem of cline et al i 9 theorem 6 6 relates the cohomology of g in a rational representation to that of the finite groups g fq related work of cline et al 6 and of the authors 1 16 relates the cohomology of g to that of its infinitesimal subgroups a determination of the cohomology of the first infinitesimal subgroup g 1 of g with trivial coefficients has enabled the authors in 14 to make certain explicit cohomology calculations for the finite groups g fq with certain nontrivial coefficients in this paper we continue our study of infinitesimal subgroups of simple algebraic groups by proving various qualitative properties of their cohomology and by extending our computational knowledge in particular we provide a computation which applies in a range of degrees to chevalley groups over the prime field we also investigate the cohomology of infinitesimal subgroups of nilpotent groups enabling us for example to determine the cohomology of their lie algebras in certain cases we apply this lie algebra cohomology computation to determine the integral cohomology of various torsion free nilpotent groups modulo small prime torsion thereby extending a theorem of dwyer 12 in more detail the paper begins in sect 1 with a spectral sequence converging to the cohomology of the first infinitesimal subgroup g which was previously studied in 14 from this spectral sequence it easily follows that h g1 m is a noetherian h g1 k module for each finite dimensional rational gl module m we then apply the work of andersen and jantzen 1 on the gl cohomology of induced modules to obtain the analogous result for the second infinitesimal subgroup gz our analysis also permits a determination of the generic

an cal h 2 matrix based integral equation solver of reduced complexity and controlled accuracy for solving electrodynamic problems using an h 2 matrix as the mathematical framework we compactly represent a dense system matrix by a reduced set of parameters thus enabling a significant reduction in computational complexity the error bound of the h 2 matrix based representation of an electrodynamic problem was derived we show that exponential convergence with respect to the number of interpolation points can be achieved irrespective of the electric size in addition we show that a direct application of h 2 matrix based techniques to electrodynamic problems would result in a complexity greater than o n with n being the matrix size due to the need of increasing the rank when ascending an inverted tree in order to keep a constant order of accuracy a rank function was hence developed to maintain the same order of accuracy in a wide range of electric sizes without compromising computational complexity with this rank function we demonstrate that given a range of electric sizes which lead to a range of n the dense system of o n 2 parameters can be compactly stored in o n units and the dense matrix vector multiplication can be performed in o n operations moreover the same order of accuracy can be kept across this range the method is kernel independent and hence is suitable for any integral equation based formulation in addition it is applicable to arbitrary structures numerical experiments from small electric sizes to 64 wavelengths have demonstrated the performance of the proposed method

on the treatment of arbitrary boundary conditions using a fast direct mathcal h matrix solver in mom this communication adapts the formalism of hierarchical mathcal h matrices to the direct solution process lu decomposition of the method of moments mom in the frequency domain a novel clustering approach based on physical constraints is presented and extended to treat dielectric bodies as well as arbitrary boundary conditions comparisons with other numerical methods are provided the analyses show a considerable speedup of the computation times and a significant reduction of required memory compared with the traditional mom solution process especially for highly resonant structures where iterative solvers like the multilevel fast multipole algorithm show poor convergence

fast iterative solution of integral equations with method of moments and matrix decomposition algorithm singular value decomposition the multilevel matrix decomposition algorithm mlmda was originally developed by michielsen and boag for 2d tmz scattering problems and later implemented in 3d by rius et al the 3d mlmda was particularly efficient and accurate for piece wise planar objects such as printed antennas however for arbitrary 3d problems it was not as efficient as the multilevel fast multipole algorithm mlfma and the matrix compression error was too large for practical applications this paper will introduce some improvements in 3d mlmda like new placement of equivalent functions and svd postcompression the first is crucial to have a matrix compression error that converges to zero as the compressed matrix size increases as a result the new mda svd algorithm is comparable with the mlfma and the adaptive cross approximation aca in terms of computation time and memory requirements remarkably in high accuracy computations the mda svd approach obtains a matrix compression error one order of magnitude smaller than aca or mlfma in less computation time like the aca the mda svd algorithm can be implemented on top of an existing mom code with most commonly used green s functions but the mda svd is much more efficient in the analysis of planar or piece wise planar objects like printed antennas

accelerated direct solution of electromagnetic scattering via characteristic basis function method with sherman morrison woodbury formula based algorithm the characteristic basis function method has been proposed to accelerate the direct solution of the method of moments by reducing the number of unknowns however with the increase in the target size the reduced matrix will still become too large to directly solve and store in this communication the algorithm based on the sherman morrison woodbury smw formula is employed to address this problem for the first time it compresses the reduced matrix into a product of block diagonal matrices with the aid of the smw formula and the adaptive cross approximation subsequently the inverse of the reduced matrix can be easily obtained by the inverses of these block diagonal matrices the numerical results about the electromagnetic scattering are shown to demonstrate the performance of the proposed method

on the cohomology of specht modules we investigate the cohomology of the specht module s for the symmetric group d we show if 0 i p 2 then hi d s is isomorphic to hs i b w0 where s d d 1 2 b is the borel subgroup of the algebraic group gld k and 1d is the weight of the determinant representation we obtain similar isomorphisms of ext di s m with b cohomology which in turn yield isomorphisms of cohomology for borel subgroups of gln k for varying n d in the case i 0 and the case i 1 for certain we apply our result and known symmetric group results of james and erdmann to obtain new information about b cohomology finally we show that specht module cohomology is closely related to cohomology for the frobenius kernel b1 for small primes

singular localization for quantum groups at generic q abstract we quantize parabolic flag manifolds and describe categories of quantum d modules on them at a singular central character we compute global sections of generators for these categories for any q c for generic q we prove a singular version of beilinson bernstein localization for a quantized enveloping algebra

second cohomology groups for frobenius kernels and related structures abstract let g be a simple simply connected affine algebraic group over an algebraically closed field k of characteristic p for an odd prime p let b be a borel subgroup of g and u be its unipotent radical in this paper we determine the second cohomology groups of b and its frobenius kernels for all simple b modules we also consider the standard induced modules obtained by inducing a simple b module to g and compute all second cohomology groups of the frobenius kernels of g for these induced modules also included is a calculation of the second ordinary lie algebra cohomology group of lie u with coefficients in k

extensions of modules for algebraic groups let g be a connected semi simple and simply connected algebraic group over an algebraically closed field k of prime characteristic p in this paper we show how one can obtain the extensions of any two simple g modules if one knows the socles of all the g modules extg l l where g1 denotes the kernel of the frobenius endomorphism on g and l as well as l are simple g1 modules alternatively we prove how all such g extensions can be found once one knows the extensions of any two simple g modules belonging to a specified finite set of simple modules with small highest weight such information is available in certain low rank cases it should also be pointed out that it is possible from lusztig s conjecture on irreducible characters 11 to derive the dimensions of ext l1 l2 when l1 and l2 are simple with such small highest weights to obtain our results on exth we need to know that there exists no g1 extension of a simple module by itself this turns out to be somewhat

an iteration free domain decomposition method for the fast finite element analysis of electromagnetic problems an iteration free finite element domain decomposition method ddm is proposed for the simulation of 3 d electromagnetic problems the finite element tearing and interconnecting feti with robin type transmission condition is introduced to construct a domain decomposition framework based on hierarchical mathcal h matrix algorithm the numerical green s function of each subdomain can be represented by a data sparse matrix form subsequently the original 3 d problem is reduced to a 2 d problem on the skeleton derived from the finite element discretization of pseudo differential operator the skeleton matrix can also be efficiently approximated with an mathcal h matrix by assembling all the numerical green s functions of subdomains based on mathcal h lu factorization algorithm the computational complexity and memory requirement for the direct solution of the skeleton equation can be reduced to be logarithmic linear the resulting direct feti d feti method completely depends on direct method without any iterative process hence it does not suffer from the issue of convergence rate and avoids redundant computations for multiple right hand side problems numerical examples demonstrate the effectiveness of the proposed method

central extensions and h1 l l of the graded lie algebras of cartan type let f be an algebraically closed field of characteristic p 0 l i rsl i a graded lie algebra of cartan type over f we give a description of the dual adjoint module l by means of the mixed product and the coinduced module such that the computation of h1 l l can be reduced to that of the first cohomology of l 0 then we determine the central extensions and h2 l f by computing h1 l l

localization for quantum groups at a root of unity let c be the field of complex numbers and fix q g c let g be a semi simple lie algebra over c and let g be the corresponding simply connected algebraic group let uq be the quantized enveloping algebra of q let oq be the algebra of quantized functions on g let oq b be the quotient hopf algebra of oq corresponding to a borel subgroup b of g in the paper bk we defined categories of equivariant quantum o modules and pg modules on the quantum flag variety of g we proved that the beilinson bernstein localization theorem holds at a generic q namely the global section functor gives an equivalence between categories of modules and vq iaod es on the quantum flag variety thus one can translate questions about the represen tation theory of quantum groups to the study of the noncommutative geometry of the quantum flag variety in this paper we prove that a derived version of this theorem holds at the root of unity case using this equivalence we get that one can understand the representation theory of quantum groups at roots of unity through the now commutative geometry of the springer fibers we now recall the main results in bk the constructions given there are crucial for the present paper and a fairly detailed survey of the material there is given in the next section we defined an equivariant sheaf of quasi coherent modules over the quantum flag variety to be a left oq mod e equipped with a right q b comod ile structure satisfying certain compatibility conditions such objects form a category denoted ai q gq it contains certain line bundles oq for in the weight lattice we proved that oq x is ample for 0 holds for every q this implies that the category m bq gq is a proj category in the sense of serre

cohomology of infinitesimal algebraic groups in this paper we study the cohomology h br k of a higher frobenius kernel br r 1 of a borel subgroup b of a reductive algebraic group g over an algebraically closed field k of a positive characteristic p for the first frobenius kernel b 1 the cohomology has been completely determined by h h andersen and j c jantzen 3 in case p h the coxeter number of g in low degrees the cohomology is independent of r 1 and their results allow us to compute the first different case h zp i br k for s lt i 3 6 in higher degrees however we have to resort to a different method although h b 2 k has already been computed for sl z also in 3 it is a considerably harder problem for sl 3 we apply algebraic steenrod operations and the transgression theorem to the lyndon hochschildserre or cartan eilenberg spectral sequence arising from a central series of the unipotent radical u of b to find h k for sl 3 5 1 the method we employ here was originally developed by m tezuka and n yagita 21 in order to investigate h u fq k q a power ofp in fact the results of sections 4 and 5 were first conceived from the corresponding results of h u fq k aside from the interests in relation to the cohomology of finite groups the work was also motivated by a close connection of the bl cohomology to g lusztig s conjectural character formula for the irreducible modules of the reductive group g which is described in section 2 hoping that the representation theory may shed more light on some topological problems we intend to take up the steenrod algebra and the a algebras in a subsequent paper

extensions for finite chevalley groups i let g be a connected semisimple algebraic group defined and split over the field fp with p elements and k be the algebraic closure of fp assume further that g is almost simple and simply connected and let g fq be the finite chevalley group consisting of fq rational points of g where q pr for a non negative integer r in this paper formulas are found relating extensions between simple kg fq modules and extensions over g considered as an algebraic group over k one of these formulas which only holds for primes p 3 h 1 where h is the coxeter number of g is then used to show the vanishing of self extensions between simple kg fq modules except for certain simple modules when r 1 and the underlying root system is of type a1 or cn

efficient scalable parallel higher order direct mom sie method with hierarchically semiseparable structures for 3 d scattering a novel fast scalable parallel algorithm is proposed for the solution of large 3 d scattering problems based on 1 the double geometrical and current approximation higher order dho method of moments mom in the surface integral equation sie formulation and 2 a direct solver for dense linear systems utilizing hierarchically semiseparable hss structures namely an hss matrix representation is used for compression factorization and solution of the system matrix in addition a rank revealing qr decomposition for memory compression is used with a stopping criterion in terms of the relative rank tolerance value a method for geometrical preprocessing of the scatterers based on the cobblestone distance sorting technique is employed in order to enhance the hss algorithm accuracy and parallelization numerical examples show how the accuracy of the dho hss mom sie method is easily controllable by using the relative tolerance for the matrix compression moreover the examples demonstrate low memory consumption as well as much faster simulation time when compared to the direct lu decomposition the method enables dramatically faster monostatic scattering computations than iterative solvers and reduced number of unknowns when compared to low order discretizations finally great scalability of the algorithm is demonstrated on more than one thousand processes

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

extensions for frobenius kernels 1 1 let g be a simple algebraic group over an algebraically closed field k of characteristic p 0 for r 1 let gr be the rth frobenius kernel of g it is well known that the representations for g1 are equivalent to the restricted representations for lie g historically the cohomology for frobenius kernels has been best understood for large primes friedlander and parshall fp first computed the cohomology ring h g1 k for p 3 h 1 where h is the coxeter number of the underlying root system they proved that the cohomology ring can be identified with the coordinate algebra of the nullcone andersen and jantzen aj later verified this fact for p h furthermore they generalized this calculation by looking at h g1 h0 where h0 indb for p h their results had some restrictions on the type of root system involved kumar lauritzen and thomsen klt removed the restrictions on the root systems through the use of frobenius splittings the cohomology ring h g1 k modulo nilpotents can be identified in general with the coordinate algebra of the restricted nullcone n1 x lie g x p 0 for good primes nakano parshall and vella npv proved that this variety is irreducible and can be identified with the closure of some richardson orbit recently carlson lin nakano and parshall clnp have given an explicit description of n1 these recent results provide some indication that one can systematically study extensions of frobenius kernels for small primes by using general formulas which exhibit generic behavior for large primes

support varieties for infinitesimal group schemes the representation theory of a connected smooth affine group scheme over a field k of characteristic p 0 is faithfully captured by that of its family of frobenius kernels such frobenius kernels are examples of infinitesimal group schemes affine group schemes g whose coordinate hopf algebra k g is a finite dimensional local k algebra this paper presents a study of the cohomology algebra h g k of an arbitrary infinitesimal group scheme over k we provide a geometric determination of the cohomological support variety cgl spec h g k analogous to that given by d quillen for the cohomology of finite groups q we further study finite dimensional rational g modules m for arbitrary infinitesimal group schemes g over k in a manner initiated by j alperin and l evens a e and j carlson ci for finite groups we consider the variety igim c igi of the ideal im ker hev g k ext m m and provide a geometric description of this variety which is analogous to that given by g avrunin and l scott for finite dimensional modules for finite groups a s this paper is a continuation of our recent work establishing the finite generation of h g k f s and investigating the infinitesimal 1 parameter subgroups of g s f b earlier work of e friedlander and b parshall fp1 fp2 fp3 fp4 and j jantzen ji concerning the cohomology of restricted lie algebras are forerunners of the results presented here finite dimensional restricted lie algebras are in 1 1 correspondence with infinitesimal group schemes of height 1 our main theorems theorems 5 2 and 6 7 below when restricted to infinitesimal group schemes of height 1 significantly strengthen previously known cohomological information for restricted lie algebras an interesting aspect of our work is the extent to which infinitesimal 1 parameter subgroups v g a r g for infinitesimal group schemes g of height r play the role of elementary abelian p subgroups and their generalizations shifted subgroups for finite groups indeed much of our effort is dedicated to proving that cohomology classes are detected modulo nilpotence by such 1 parameter subgroups this is first done in 2 for unipotent infinitesimal group schemes using an induction argument made possible by a structure theorem presented in 1 this structure theorem is the analogue in our context of a theorem of j p serre characterizing elementary abelian p groups s

fast direct solution of method of moments linear system a novel algorithm the compressed block decomposition cbd is presented for highly accelerated direct noniterative method of moments mom solution of electromagnetic scattering and radiation problems the algorithm is based on a block wise subdivision of the mom impedance matrix impedance matrix subblocks corresponding to distant subregions of the problem geometry are not calculated directly but approximated in a compressed form subsequently the matrix is decomposed preserving the compression examples are presented of typical problems in the range of 5000 to 70000 unknowns the total execution time for the largest problem is about 1 h and 20 min for a single excitation vector the main strength of the method is for problems with multiple excitation vectors monostatic rcs computations due to the negligible extra cost for each new excitation for radiation and scattering problems in free space the numerical complexity of the algorithm is shown to be n 2 and the storage requirements scale with n 3 2

frobenius splitting of cotangent bundles of flag varieties let g be a semisimple and simply connected algebraic group over an algebraically closed field of characteristic p 0 let u be the unipotent radical of a borel subgroup b g and u the lie algebra of u springer 16 has shown for good primes that there is a b equivariant isomorphism u u where b acts through conjugation on u and through the adjoint action on u for g sln one has the well known equivariant isomorphism a 7 a i between unipotent and nilpotent upper triangular matrices let p be a good prime for g then there is an isomorphism of homogeneous bundles x g b u g b u where the latter can be identified with the cotangent bundle t g b of g b motivated in part by 12 we establish a link between the g invariant form on the steinberg module st h0 g b p 1 cf 1 8 and frobenius splittings 15 of the cotangent bundle of g b the representation h0 g b 2 p 1 is a quotient of the space of functions h x ox on x here h0 g b m denotes the g module induced from the b module m and half the sum of the roots r opposite to the roots

differential operators on quantized flag manifolds at roots of unity abstract the quantized flag manifold which is a q analogue of the ordinary flag manifold is realized as a non commutative scheme and we can define the category of d modules on it using the framework of non commutative algebraic geometry however when the parameter q is a root of unity lusztig s frobenius morphism allows us to handle d modules on the quantized flag manifold through modules over a certain sheaf of rings on the ordinary flag manifold in this paper we will show that this sheaf of rings on the ordinary flag manifold is an azumaya algebra over its center we also show that its restriction to certain subsets are split azumaya algebras these are analogues of some results of bezrukavnikov mirkovic rumynin on d modules on flag manifolds in positive characteristics

gibbsianness of fermion random point fields we consider fermion or determinantal random point fields on euclidean space d given a bounded translation invariant and positive definite integral operator j on l2 d we introduce a determinantal interaction for a system of particles moving on d as follows the n points located at x1 x n d have the potential energy given by where j x y is the integral kernel function of the operator j we show that the gibbsian specification for this interaction is well defined when j is of finite range in addition and for d 2 if the intensity is small enough we show that the fermion random point field corresponding to the operator j i j 1 is a gibbs measure admitted to the specification

quantum flag varieties equivariant quantum d modules and localization of quantum groups abstract let o q g be the algebra of quantized functions on an algebraic group g and o q b its quotient algebra corresponding to a borel subgroup b of g we define the category of sheaves on the quantum flag variety of g to be the o q b equivariant o q g modules and prove that this is a proj category we construct a category of equivariant quantum d modules on this quantized flag variety and prove the beilinson bernstein s localization theorem for this category in the case when q is transcendental

hierarchical matrix techniques based on matrix decomposition algorithm for the fast analysis of planar layered structures the matrix decomposition algorithm mda provides an efficient matrix vector product for the iterative solution of the integral equation ie by a blockwise compression of the impedance matrix the mda with a singular value decomposition svd recompression scheme i e so called mda svd method shows strong ability for the analysis of planar layered structures however iterative solution faces the problem of convergence rate an efficient hierarchical h lu decomposition algorithm based on the h matrix techniques is proposed to handle this problem exploiting the data sparse representation of the mda svd compressed impedance matrix h lu decomposition can be efficiently implemented by h matrix arithmetic h matrix techniques provide a flexible way to control the accuracy of the approximate h lu factors h lu decomposition with low accuracy can be used as an efficient preconditioner for the iterative solver due to its low computational cost while h lu decomposition with high accuracy can be used as a direct solver for dealing with multiple right hand side rhs vector problems particularly numerical examples demonstrate that the proposed method is very robust for the analysis of various planar layered structures

on the cohomology of algebraic and related finite groups in the above theorem h gl m denotes the rational cohomology of the algebraic group gl with coefficients in m the rational gl module obtained from the adjoint module m m through twisting by the r th power of the frobenius endomorphism on gl the algebra structure on the cohomology is induced from the associative algebra structure on m on the other hand we provide the following systematic determination of certain rational cohomology groups h g v r where v r is the rational gmodule obtained from the rational g module v by applying the r th frobenius twist see 1 2 this is a somewhat sharpened version of theorem 4 of 36

a note on the adjoint action of a semisimple group on the coordinate algebra of an infinitesimal subgroup abstract let g be a connected semisimple simply connected algebraic group over an algebraically closed field k of positive characteristic the action of g on itself by conjugation induces on the coordinate algebra k g and hence on the coordinate algebra k g r of the infinitesimal subgroup g r the structure of a rational g module in the recent paper 3 by bendel nakano pillen and sobaje the authors ask whether the tensor product of k g r with the rth steinberg module st r has a good filtration here we show that this is the case provided that the characteristic of k is good for g we do so by relating k g r to a certain quotient of the coordinate algebra k g of the lie algebra g of g and using a koszul resolution we note that the argument may also be applied to certain polynomial modules occurring in the context of doty s conjecture 17 4 2 11

sheaf cohomology on gb and tensor products of weyl modulus let k be an algebraically closed field it is assumed that char k p 0 unless otherwise indicated g is a simply connected semisimple algebraic group over k t is a fixed maximal torus and b a bore1 subgroup of g containing t let qi g t be the root system of g d the set of all simple roots and and the sets of all positive and negative roots relative to d respectively we assume that the set a is chosen such that b corresponds to 0 let x be the weight lattice of sp which may be considered as the character group of t or b therefore if 1 e x by abuse of notation we also denote by l the one dimensional tor b module determined by a let x be the set of all dominant weights and w the fundamental weight corresponding to a simple root a e a we denote by 6 the half sum of all positive roots and we know that 6 is the sum of all fundamental weights let w be the weyl group of 0 the reflection with respect to a root a is denoted by s the longest element of w by w there are two ways of w acting on x i the ordinary action if 1 e x m e w the image of i under w is denoted by 2 ii dot action w a 5 2 6 s in particular let a w a the dual root system of is denoted by and the dual root of a e by a i e u 2a a u where is the inner product invariant under the ordinary action of w of the euclidean space spanned by for 1 e x a e let a a 1 a all modules of an algebraic group we consider in this paper are assumed to be rational in particular for an infinite dimensional module the term rational module means this module is a direct limit of finite dimensional rational modules let h be a closed subgroup of g containing t such as g b or t m an h module we define for l e x

invariant forms on irreducible modules of simple algebraic groups let g be a simple linear algebraic group over an algebraically closed field k of characteristic p 0p 0 and let v be an irreducible rational g module with highest weight when v is self dual a basic question to ask is whether v has a non degenerate g invariant alternating bilinear form or a non degenerate g invariant quadratic form if p 2p 2 the answer is well known and easily described in terms of in the case where p 2p 2 we know that if v is self dual it always has a non degenerate g invariant alternating bilinear form however determining when v has a non degenerate g invariant quadratic form is a classical problem that still remains open we solve the problem in the case where g is of classical type and is a fundamental highest weight i i and in the case where g is of type alal and r s r s for 1 r s l1 r s l we also give a solution in some specific cases when g is of exceptional type as an application of our results we refine seitz s 1987 description of maximal subgroups of simple algebraic groups of classical type one consequence of this is the following result if x y sl v x y sl v are simple algebraic groups and v xv x is irreducible then one of the following holds 1 v yv y is not self dual 2 both or neither of the modules v yv y and v xv x have a non degenerate invariant quadratic form 3 p 2p 2 x so v x so v and y sp v y sp v

a note on decomposition numbers for reductive algebraic groups in a recent paper 7 james proves two theorems on the decomposition numbers for the general linear groups and working within the context of green s monograph 5 derives properties of the decomposition numbers for the symmetric groups this note grew out of an attempt to formulate for arbitrary reductive groups these theorems of james we prove a natural result on decomposition numbers for reductive groups from which james s theorems are deduced the result the theorem of section 2 is the analogue of jantzen s satz 1 18 of 8 on decomposition numbers for representations of complex semisimple lie algebras also proved independently by deodhar very little direct use of 5 is made though some of the ideas are cribbed from that source as one may see by comparing the properties of the functor fa in section 2 with green s treatment see section 6 2 of s of ese theory s a ring e an idempotent

approximation of boundary element matrices this article considers the problem of approximating a general asymptotically smooth function in two variables typically arising in integral formulations of boundary value problems by a sum of products of two functions in one variable from these results an iterative algorithm for the low rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed this algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance

rational cherednik algebras and hilbert schemes let h c be the rational cherednik algebra of type a n 1 with spherical subalgebra u c e h c e then u c is filtered by order of differential operators with associated graded ring mbox gr u c mathbb c mathfrak h oplus mathfrak h where w is the n th symmetric group we construct a filtered mathbb z algebra b such that under mild conditions on c the category b qgr of graded noetherian b modules modulo torsion is equivalent to u c mod the associated graded mathbb z algebra mbox gr b has mbox gr b lqgr simeq coh hilb n this can be regarded as saying that u c simultaneously gives a non commutative deformation of mathfrak h oplus mathfrak h w and of its resolution of singularities hilb n rightarrow mathfrak h oplus mathfrak h as we show elsewhere this result is a powerful tool for studying the representation theory of h c and its relationship to hilb n

the frobenius morphism in invariant theory abstract let r be the homogeneous coordinate ring of the grassmannian g gr 2 n defined over an algebraically closed field of characteristic p 0 in this paper we give a completely characteristic free description of the decomposition of r considered as a graded r p module into indecomposables frobenius summands as a corollary we obtain a similar decomposition for the frobenius pushforward of the structure sheaf of g and we obtain in particular that this pushforward is almost never a tilting bundle on the other hand we show that r provides a noncommutative resolution for r p when p n 2 generalizing a result known to be true for toric varieties in both the invariant theory and the geometric setting we observe that if the characteristic is not too small the frobenius summands do not depend on the characteristic in a suitable sense in the geometric setting this is an explicit version of a general result by bezrukavnikov and mirkovic on frobenius decompositions for partial flag varieties we are hopeful that it is an instance of a more general p uniformity principle

cohomology of classical algebraic groups from the functorial viewpoint abstract we prove that extension groups in strict polynomial functor categories compute the rational cohomology of classical algebraic groups this result was previously known only for general linear groups we give several applications to the study of classical algebraic groups such as a cohomological stabilization property the injectivity of external cup products and the existence of hopf algebra structures on the stable cohomology of a classical algebraic group with coefficients in a hopf algebra our result also opens the way to new explicit cohomology computations we give an example inspired by recent computations of djament and vespa

iterative analysis of finite sized planar frequency selective surfaces with rectangular patches or perforations frequency selective surfaces fss and other periodic gratings are often analyzed under the assumption that they are infinite in extent most existing methods for analyzing periodic structures are based on the use of a floquet type representation of the fields in a unit cell whose dimensions are typically comparable to the wavelength in this work a finite truncated version of an infinite periodic structure is dealt with directly without the benefit of the assumption that the structure is periodic this in turn requires the handling of a large number of unknowns and makes it difficult to solve the problem using conventional matrix methods two different iteration approaches to solving the finite fss problem are discussed in the paper both of which employ the spectral domain formulation the first of these employs the spectral iteration technique and the second one uses the conjugate gradient cg iteration algorithm convergence characteristics of both of these methods are investigated and the results are reported

the beilinson bernstein correspondence for quantized enveloping algebras theory of the quantized flag manifold as a quasi scheme non commutative scheme has been developed by lunts rosenberg 15 they have formulated an analogue of the beilinson bernstein correspondence using the q differential operators introduced in their earlier paper 14 in this paper we shall establish its modified version using a class of q differential operators which is possibly smaller than the one in 14

a multilevel matrix decomposition algorithm for analyzing scattering from large structures a multilevel algorithm is presented for analyzing scattering from electrically large surfaces the algorithm accelerates the iterative solution of integral equations that arise in computational electromagnetics the algorithm permits a fast matrix vector multiplication by decomposing the traditional method of moment matrix into a large number of blocks with each describing the interaction between distant scatterers the multiplication of each block by a trial solution vector is executed using a multilevel scheme that resembles a fast fourier transform fft and that only relies on well known algebraic techniques the computational complexity and the memory requirements of the proposed algorithm are o n log sup 2 n

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

strata hasse invariants hecke algebras and galois representations we construct group theoretical generalizations of the hasse invariant on strata closures of the stacks mathop g hbox mathtt zip nolimits mu restricting to zip data of hodge type we obtain a group theoretical hasse invariant on every ekedahl oort stratum closure of a general hodge type shimura variety a key tool is the construction of a stack of zip flags mathop g hbox mathtt zipflag nolimits mu fibered in flag varieties over mathop g hbox mathtt zip nolimits mu it provides a simultaneous generalization of the classical case homogeneous complex manifolds studied by griffiths schmid and the flag space for siegel varieties studied by ekedahl van der geer four applications are obtained 1 pseudo representations are attached to the coherent cohomology of hodge type shimura varieties modulo a prime power 2 galois representations are associated to many automorphic representations with non degenerate limit of discrete series archimedean component 3 it is shown that all ekedahl oort strata in the minimal compactification of a hodge type shimura variety are affine thereby proving a conjecture of oort 4 part of serre s letter to tate on mod p modular forms is generalized to general hodge type shimura varieties

differentiating the weyl generic dimension formula with applications to support varieties abstract the authors compute the support varieties of all irreducible modules for the small quantum group u g where g is a finite dimensional simple complex lie algebra and is a primitive l th root of unity with l larger than the coxeter number of g the calculation employs the prior calculations and techniques of ostrik and of nakano parshall and vella as well as deep results involving the validity of the lusztig character formula for quantum groups and the positivity of parabolic kazhdan lusztig polynomials for the affine weyl group analogous support variety calculations are provided for the first frobenius kernel g 1 of a reductive algebraic group scheme g defined over the prime field f p

extensions for finite chevalley groups iii rational and generic cohomology abstract let g be a connected reductive algebraic group and b be a borel subgroup defined over an algebraically closed field of characteristic p 0 in this paper the authors study the existence of generic g cohomology and its stability with rational g cohomology groups via the use of methods from the authors earlier work new results on the vanishing of g and b cohomology groups are presented furthermore vanishing ranges for the associated finite group cohomology of g f q are established which generalize earlier work of hiller in addition to stability ranges for generic cohomology which improve on seminal work of cline parshall scott and van der kallen

complexes and vanishing of cohomology for group schemes abstract completely reducible homology complexes were originally introduced in the context of finite groups and used to study the question of vanishing of cohomology in this paper we study these complexes and the vanishing of cohomology for arbitrary finite dimensional cocommutative hopf algebras applications are later provided for infinitesimal group schemes of reductive and solvable algebraic groups

support varieties of line bundle cohomology groups for sl3 k abstract let g sl 3 k where k is a field of characteristic p 0 and let x t be any weight with corresponding line bundle l on g b in this paper we compute the support varieties for all modules of the form h i h i g b l over the first frobenius kernel g 1 the calculation involves certain recursive character formulas given by donkin cf 5 which can be used to compute the characters of the line bundle cohomology groups in the case where is a p regular weight and m h i 0 for some i these formulas are used to show that any pth root of unity is not a root of the generic dimension of m to handle the case where is not p regular we employ techniques similar to those used by drupieski nakano and parshall cf 7 to show that the module h i is not projective over g 1 whenever it is nonzero and lies outside of the steinberg block

on conjugating representations and adjoint representations of semisimple groups where ea is a direct sum of irreducible g modules of highest weight 2 here we consider the conjugating representation in arbitrary characteristic and prove the appropriate version of richardson s theorem under small and almost certainly unnecessary characteristic root system restrictions one can no longer decompose a g into homogeneous components but instead we prove the existence of an ascending

quasi invariance and integration by parts for determinantal and permanental processes abstract determinantal and permanental processes are point processes with a correlation function given by a determinant or a permanent their atoms exhibit mutual attraction of repulsion thus these processes are very far from the uncorrelated situation encountered in poisson models we establish a quasi invariance result we show that if atom locations are perturbed along a vector field the resulting process is still a determinantal respectively permanental process the law of which is absolutely continuous with respect to the original distribution based on this formula following bismut approach of malliavin calculus we then give an integration by parts formula

infinitesimal 1 parameter subgroups and cohomology this is the first of two papers in which we determine the spectrum of the cohomology algebra of infinitesimal group schemes over a field k of characteristic p 0 whereas sfb is concerned with detection of cohomology classes the present paper introduces the graded algebra k vr g of functions on the scheme of infinitesimal 1 parameter subgroups of height r on an affine group scheme g and demonstrates that this algebra is essentially a retract of h g k provided that g is an infinitesimal group scheme of height kr this work is a continuation of f s in which the existence of certain universal extension classes was established thereby enabling the proof of finite generation of h g k for any finite group scheme g over k the role of the scheme of infinitesimal 1 parameter subgroups of g was foreshadowed in f p where h g 1 k was shown to be isomorphic to the coordinate algebra of the scheme of p nilpotent elements of g lie g for g a smooth reductive group g 1 the first frobenius kernel of g and p char k sufficiently large indeed p nilpotent elements of g correspond precisely to infinitesimal 1 parameter subgroups of g 1 much of our effort in this present paper involves the analysis of the restriction of the universal extension classes to infinitesimal 1 parameter subgroups in 1 we construct the affine scheme vr g of homomorphisms from ga r to g which we call the scheme of infinitesimal 1 parameter subgroups of height r in g in the special case r 1 this is the scheme of p nilpotent elements of the p restricted lie algebra lie g for various classical groups g vr g is the scheme of r tuples of p nilpotent pairwise commuting elements of lie g more generally an embedding g c gln determines a closed embedding of vr g into the scheme of r tuples of p nilpotent pairwise commuting elements of gl1 lie gln the relationship between k vr g the coordinate algebra of vr g and h g k is introduced in theorem 1 14 a natural homomorphism of graded k algebras

equivariant factorization homology of global quotient orbifolds abstract we introduce equivariant factorization homology extending the axiomatic framework of ayala francis to encompass multiplicative invariants of manifolds equipped with finite group actions examples of such equivariant factorization homology theories include bredon equivariant homology and twisted versions of hochschild homology our main result is that equivariant factorization homology satisfies an equivariant version of excision and is uniquely characterised by this property we also discuss applications to representation theory such as constructions of categorical braid group actions

limits of weight spaces lusztig s q analogs and fiberings of adjoint orbits let g be a connected complex semisimple algebraic group and t a maximal torus inside a borel subgroup b with g t and b their lie algebras let v be a representation in the category a for g the t decomposition v met va of v into a direct sum of finite dimensional weight spaces is central in the representation theory of g in this paper we introduce on weight spaces a new structure the principal filtration je v where e is a principal nilpotent in g chosen to be compatible with t for example e can be the sum of the simple root vectors relative to t b this filtration is constructed in a very simple way by taking j v to be the space of vectors annihilated by the p 1 th power of e for p 0 our approach is motivated by kostant s fundamental work ki k2 on actions of the principal tds three dimensional subalgebra and coordinate rings of regular adjoint orbits in theorem 3 4 we give a new description in terms of the dimension jumps of the principal filtration of the weight space of lusztig s l q analog mm q of dominant weight multiplicity in a finite dimensional irreducible g representation v the polynomial m q was defined algebraically as an alternating sum over the weyl group through a q analog of kostant s weight multiplicity formula we prove that m q is equal to the jump polynomial

modular representation theory of lie algebras on etudie les representations non necessairement restreintes de l algebre universelle enveloppante u 7b g pour une algebre de lie restreinte 7b g

support varieties for modules over chevalley groups and classical lie algebras let g be a connected reductive algebraic group over an algebraically closed field of characteristic p 0 g 1 be the first frobenius kernel and g f p be the corresponding finite chevalley group let m be a rational g module in this paper we relate the support variety of m over the first frobenius kernel with the support variety of m over the group algebra kg f p this provides an answer to a question of parshall applications of our new techniques are presented which allow us to extend results of alperin mason and janiszczak jantzen and to calculate the dimensions of support varieties for finite chevalley groups

random point fields associated with certain fredholm determinants ii fermion shifts and their ergodic and gibbs properties we construct and study a family of probability measures on the configuration space over countable discrete space associated with nonnegative definite symmetric operators via determinants under a mild condition they turn out unique gibbs measures also some ergodic properties including the entropy positivity are discussed in the lattice case

localization of quantum biequivariant ɗ modules and q w algebras the author is grateful to y kremnizer for useful discussions and to the referee for careful reading of the manuscript

multiscale compressed block decomposition for fast direct solution of method of moments linear system the multiscale compressed block decomposition algorithm ms cbd is presented for highly accelerated direct non iterative solution of electromagnetic scattering and radiation problems with the method of moments mom the algorithm is demonstrated to exhibit n2 computational complexity and storage requirements scaling with n1 5 for electrically large objects several numerical examples illustrate the efficiency of the method in particular for problems with multiple excitation vectors the largest problem presented in this paper is the monostatic rcs of the nasa almond at 50 ghz for one thousand incidence angles discretized using 442 089 rwg basis functions being entirely algebraic ms cbd is independent of the greens function of the problem

moment formulae for general point processes abstract the goal of this paper is to generalize most of the moment formulae obtained in 12 more precisely we consider a general point process and show that the quantities relevant to our problem are the so called papangelou intensities when the papangelou intensities of are well defined we show some general formulae to recover the moment of order n of the stochastic integral of the point process we will use these extended results to introduce a divergence operator and study a random transformation of the point process