truncation diffraction phenomena of floquet waves radiated from semi infinite phased array antenna in a general focus problem this paper presents uniform formulations of radiation mechanisms analogous to the ray decompositions of diffraction theory from a semi infinite and periodic array of antennas the antenna array is excited to radiate the electromagnetic em fields focused at a relatively arbitrarily selected location the ray decomposition results from a closed form formulation by asymptotically evaluating the radiation integral of floquet modes that are obtained by applying the poisson sum formula to the summation of elemental radiations the analysis is relatively general and will reduce to the case of a conventional phased array antenna that is excited to radiate directive beams focused in the far zone and exhibits consistent phenomena theoretical investigations as well as numerical examinations are presented to demonstrate the radiation mechanisms

non integral representation theory of even multiplicity finite w algebras abstract we complete the classification of the finite dimensional irreducible representations of finite w algebras associated to even multiplicity nilpotent elements in classical lie algebras this extends earlier work in brown and goodwin 2013 bg1 where this classification is determined for such representations of integral central character

semisimple orbits of lie algebras and card shuffling measures on coxeter groups abstract random walk on the chambers of hyperplane arrangements is used to define a family of card shuffling measures h w x for a finite coxeter group w and real x 0 by algebraic group theory there is a map from the semisimple orbits of the adjoint action of a finite group of lie type on its lie algebra to the conjugacy classes of the weyl group choosing such a semisimple orbit uniformly at random thereby induces a probability measure on the conjugacy classes of the weyl group for types a b and the identity conjugacy class of w for all types it is proved that for q very good this measure on conjugacy classes is equal to the measure arising from h w q the possibility of refining to a map to elements of the weyl group is discussed

affine shuffles shuffles with cuts the whitehouse module and patience sorting abstract type a affine shuffles are compared with riffle shuffles followed by a cut although these probability measures on the symmetric group s n are different they both satisfy a convolution property strong evidence is given that when the underlying parameter q satisfies gcd n q 1 1 the induced measures on conjugacy classes of the symmetric group coincide this gives rise to interesting combinatorics concerning the modular equidistribution by major index of permutations in a given conjugacy class and with a given number of cyclic descents using representation theoretic work on the whitehouse module a formula is obtained for the cycle structure of a riffle shuffle followed by a cut it is proved that the use of cuts does not speed up the convergence rate of riffle shuffles to randomness generating functions for the first pile size in patience sorting from decks with repeated values are derived this relates to random matrices

asymptotic floquet mode investigation over two dimensional infinite array antennas phased to radiate near zone focused fields this paper extends the previous work in to consider the radiation from an infinite 2 d antenna array for the near field focus applications it employs the poisson sum formula to expand the radiation into floquet modes which may exhibit a class of wave phenomena in terms of geometrical ray fields this analysis automatically reduces to the conventional case of a linearly phased antenna array and results in consistent radiation phenomena it can be considered as a generalization of phased array analysis theoretical investigations as well as numerical illustrations are presented

a one sided version of the poisson sum formula for semi infinite array green s functions the poisson sum formula provides an efficient method for transforming many slowly converging infinite summations into equivalent but more rapidly converging infinite summations electromagnetic applications for this result come in the analysis of infinite arrays of periodic scatterers such as frequency selective surfaces however in some applications such as when one desires the radiation of a semi infinite array of periodically spaced currents the original form of the poisson sum formula is inappropriate for such applications we derive a one sided version of the formula and apply it to the radiation from a semi infinite array of line sources with currents dictated by floquet s theorem the one sided poisson sum transformation yields enhanced convergence characteristics for certain regions of application as a result of the inverse bandwidth relationship between fourier transform pairs the application of it to a semi infinite line source array also provides a plane wave representation for the fields which makes for an extension of the solution to geometries with stratified dielectric media

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

on the variety of 1 dimensional representations of finite w algebras in low rank abstract let g be a simple lie algebra over c and let e g be nilpotent we consider the finite w algebra u g e associated to e and the problem of determining the variety e g e of 1 dimensional representations of u g e for g of low rank we report on computer calculations that have been used to determine the structure of e g e and the action of the component group e of the centralizer of e on e g e as a consequence we provide two examples where the nilpotent orbit of e is induced but there is a 1 dimensional e stable u g e module which is not induced via losev s parabolic induction functor in turn this gives examples where there is a non induced multiplicity free primitive ideal of u g

quantized symplectic actions and w algebras with a nilpotent element in a semisimple lie algebra g one associates a finitely generated associative algebra w called a w algebra of finite type this algebra is obtained from the universal enveloping algebra u g by a certain hamiltonian reduction we observe that w is the invariant algebra for an action of a reductive group g with lie algebra g on a quantized symplectic affine variety and use this observation to study w our results include an alternative definition of w a relation between the sets of prime ideals of w and of the corresponding universal enveloping algebra the existence of a one dimensional representation of w in the case of classical g and the separation of elements of w by finite dimensional representations

frequency domain green s function for a planar periodic semi infinite phased array i truncated floquet wave formulation this two part sequence deals with the derivation and physical interpretation of a uniform high frequency solution for the field radiated at finite distance by a planar semi infinite phased array of parallel elementary electric dipoles the field obtained by direct summation over the contributions from the individual radiators is restructured into a double series of wavenumber spectral integrals whose asymptotic reduction yields a series encompassing propagating and evanescent floquet waves fws together with corresponding diffracted rays which arise from scattering of the fw at the edge of the array the formal aspects of the solution are treated in the present paper they involve a sequence of manipulations in the complex spectral wavenumber planes that prepare the integrands for subsequent efficient and physically incisive asymptotics based on the method of steepest descent different species of spectral poles define the various species of propagating and evanescent fw their interception by the steepest descent path sdp determines the variety of shadow boundaries for the edge truncated fw the uniform asymptotic reduction of the sdp integrals performed by the van der waerden 1951 procedure and yielding a variety of edge diffracted fields completes the formal treatment

the number of regular semisimple conjugacy classes in the finite classical groups abstract using generating functions we enumerate regular semisimple conjugacy classes in the finite classical groups for the general linear unitary and symplectic groups this gives a different approach to known results for the special orthogonal groups the results are new

floquet mode phenomena of infinite phased array antennas in near field focus applications this paper presents a theoretical investigation on the radiation of an infinite one dimensional antenna array which is phased to radiate electromagnetic fields focused in the near zone of the array aperture in particular the wave phenomena in each of the floquet modes obtained by applying the poisson sum formula to the array excitations are investigated the analysis reduces to the case of a conventional phased antenna array when the radiation is focused in the far zone and exhibits consistent phenomena theoretical investigations as well as numerical illustrations are presented

on the poisson sum formula for the analysis of wave radiation and scattering from large finite arrays poisson sum formulas have been previously presented and utilized in the literature for converting a finite element by element array field summation into an alternative representation that exhibits improved convergence properties with a view toward more efficiently analyzing wave radiation scattering from electrically large finite periodic arrays however different authors appear to use two different versions of the poisson sum formula one of these explicitly shows the end point discontinuity effects due to array truncation whereas the other contains such effects only implicitly it is shown via the sifting property of the dirac delta function that first of all these two versions of the poisson sum formula are equivalent second the version containing implicit end point contributions has often been applied in an incomplete fashion in the literature to solve finite array problems it is also demonstrated that the latter can lead to some errors in finite array field computations

on degrees of irreducible brauer characters based on a large amount of examples which we have checked so far we conjecture that g p 1 2 where p is a prime and the sum runs through the set of irreducible brauer characters in characteristic p of the finite group g we prove the conjecture simultaneously for p solvable groups and groups of lie type in the defining characteristic in non defining characteristics we give asymptotically an affirmative answer in many cases

dimensions of irreducible modules over w algebras and goldie ranks the main goal of this paper is to compute two related numerical invariants of a primitive ideal in the universal enveloping algebra of a semisimple lie algebra the first one very classical is the goldie rank of an ideal the second one is the dimension of an irreducible module corresponding to this ideal over an appropriate finite w algebra we concentrate on the integral central character case we prove modulo a conjecture that in this case the two are equal our conjecture asserts that there is a one dimensional module over the w algebra with certain additional properties the conjecture is proved for the classical types also modulo the same conjecture we compute certain scale factors introduced by joseph this allows to compute the goldie ranks of the algebras of locally finite endomorphisms of simples in the bgg category mathcal o this completes a program of computing goldie ranks proposed by joseph in the 80 s for integral central characters and modulo our conjecture we also provide an essentially kazhdan lusztig type formula for computing the characters of the irreducibles in the brundan goodwin kleshchev category mathcal o for a w algebra again under the assumption that the central character is integral in particular this allows to compute the dimensions of the finite dimensional irreducible modules the formula is based on a certain functor from an appropriate parabolic category mathcal o to the w algebra category mathcal o this functor can be regarded as a generalization of functors previously constructed by soergel and by brundan kleshchev we prove a number of properties of this functor including the quotient property and the double centralizer property we develop several side topics related to our generalized soergel functor for example we discuss its analog for the category of harish chandra modules we also discuss generalizations to the case of categories mathcal o over dixmier algebras the most interesting example of this situation comes from the theory of quantum groups we prove that an algebra that is a mild quotient of luszitg s form of a quantum group at a root of unity is a dixmier algebra for this we check that the quantum frobenius epimorphism splits

shifted yangians and finite w algebras we give a presentation for the finite w algebra associated to a nilpotent matrix in the general linear lie algebra over c in the special case that the nilpotent matrix consists of n jordan blocks each of the same size l the presentation is that of the yangian of level l associated to gln as was first observed by ragoucy and sorba in the general case we are lead to introduce some generalizations of the yangian which we call the shifted yangians

a truncated floquet wave diffraction method for the full wave analysis of large phased arrays i basic principles and 2 d cases this two part sequence deals with the formulation of an efficient method for the full wave analysis of large phased array antennas this is based on the method of moments mom solution of a fringe integral equation ie in which the unknown function is the difference between the exact solution of the finite array and that of the associated infinite array the unknown currents can be interpreted as produced by the field diffracted at the array edge which is excited by the floquet waves fws pertinent to the infinite configuration following this physical interpretation the unknown in the ie is efficiently represented by a very small number of basis functions with domain on the entire array aperture in order to illustrate the basic concepts the first part of this sequence deals with the two dimensional example of a linearly phased slit array it is shown that the dominant phenomenon fur describing the current perturbation with respect to the infinite array is accurately represented in most cases by only three diffracted ray shaped unknown functions this also permits a simple interpretation of the element by element current oscillation which was described by other authors

degree graphs of simple orthogonal and symplectic groups abstract let g be a finite group and let cd g be the set of irreducible ordinary character degrees of g the degree graph of g is the graph g whose set of vertices is the set of primes dividing degrees in cd g with an edge between primes p 1 and p 2 if p 1 p 2 divides some degree in cd g we determine the graph g for the finite simple groups of types b l c l d l and d l 2 that is for the simple orthogonal and symplectic groups

representations of rational cherednik algebras with minimal support and torus knots abstract in this paper we obtain several results about representations of rational cherednik algebras and discuss their applications our first result is the cohen macaulayness property as modules over the polynomial ring of cherednik algebra modules with minimal support our second result is an explicit formula for the character of an irreducible minimal support module in type a n 1 for c m n and an expression of its quasispherical part i e the isotypic part of hooks in terms of the homfly polynomial of a torus knot colored by a young diagram we use this formula and the work of calaque enriquez and etingof to give explicit formulas for the characters of the irreducible equivariant d modules on the nilpotent cone for sl m our third result is the construction of the koszul bgg complex for the rational cherednik algebra which generalizes the construction of the koszul bgg resolution from 3 and 21 and the calculation of its homology in type a we also show in type a that the differentials in the koszul bgg complex are uniquely determined by the condition that they are nonzero homomorphisms of modules over the cherednik algebra finally our fourth result is the symmetry theorem which identifies the quasispherical components in the representations with minimal support over the rational cherednik algebras h m n s n and h n m s m in fact we show that the simple quotients of the corresponding quasispherical subalgebras are isomorphic as filtered algebras this symmetry was essentially established in 8 in the spherical case and in 24 in the case gcd m n 1 and it has a natural interpretation in terms of invariants of torus knots

commutative quotients of finite w algebras abstract in this paper we reduce the problem of 1 dimensional representations for the finite w algebras and humphreys conjecture on small representations of reduced enveloping algebras to the case of rigid nilpotent elements in exceptional lie algebras we use katsylo s results on sections of sheets to determine the krull dimension of the largest commutative quotient of the finite w algebra u g e

special transverse slices and their enveloping algebras let g be a simple simply connected algebraic group over image image lie g image image the nilpotent cone of image and e h f an imageimage2 triple in image let s e ker ad f the special transverse slice to the adjoint orbit of e and s0 s image image the coordinate ring image s0 is naturally graded see slodowy simple singularities and simple algebraic groups lecture notes in mathematics vol 815 springer verlag berlin heidelberg new york 1980 let z image be the centre of the enveloping algebra u image and z image image an algebra homomorphism identify image with image via a killing isomorphism and let denote the linear function on image corresponding to e following kawanaka generalized gelfand graev representations and ennola duality in algebraic groups and related topics advanced studies in pure mathematics vol 6 pp 175 206 north holland amsterdam new york oxford 1985 moeglin c r acad sci paris ser i 303 no 17 1986 845 848 and premet invent math 121 1995 79 117 we attach to a nilpotent subalgebra image subset ofimage of dimension dim 2 and a 1 dimensional image module image let imageh height 14 width 14 denote the algebra opposite to endimage u image multiply sign in circleu image image and imageh height 14 width 14 imageh height 14 width 14 multiply sign in circlez image image it is proved in the paper that the algebra imageh height 14 width 14 has a natural filtration such that gr imageh height 14 width 14 the associated graded algebra is isomorphic to image s0 this construction yields natural noncommutative deformations of all singularities associated with the adjoint quotient map of image

finite dimensional representations of w algebras w algebras of finite type are certain finitely generated associative algebras closely related to the universal enveloping algebras of semisimple lie algebras in this paper we prove a conjecture of premet that gives an almost complete classification of finite dimensional irreducible modules for w algebras also we study a relation between harish chandra bimodules and bimodules over w algebras

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

finite dimensional irreducible representations of finite w algebras associated to even multiplicity nilpotent orbits in classical lie algebras we consider finite w algebras u mathfrak g e associated to even multiplicity nilpotent elements in classical lie algebras we give a classification of finite dimensional irreducible u mathfrak g e modules with integral central character in terms of the highest weight theory from brundan et al int math res notices 15 art id rnn051 2008 as a corollary we obtain a parametrization of primitive ideals of u mathfrak g with associated variety the closure of the adjoint orbit of e and integral central character

character d modules via drinfeld center of harish chandra bimodules the category of character d modules is realized as drinfeld center of the abelian monoidal category of harish chandra bimodules tensor product of harish chandra bimodules is related to convolution of d modules via the long intertwining functor radon transform by a result of beilinson and ginzburg represent theory 3 1 31 1999 exactness property of the long intertwining functor on a cell subquotient of the harish chandra bimodules category shows that the truncated convolution category of lusztig adv math 129 85 98 1997 can be realized as a subquotient of the category of harish chandra bimodules together with the description of the truncated convolution category bezrukavnikov et al in isr j math 170 207 234 2009 this allows us to derive under a mild technical assumption a classification of irreducible character sheaves over obtained by lusztig by a different method

isomorphisms of quantizations via quantization of resolutions abstract in this paper we prove the existence of isomorphisms between certain non commutative algebras that are interesting from the representation theoretic perspective and arise as quantizations of certain poisson algebras we show that quantizations of kleinian singularities obtained by three different constructions are isomorphic to each other the constructions are via symplectic reflection algebras quantum hamiltonian reduction and w algebras next we prove that parabolic w algebras in type a are isomorphic to quantum hamiltonian reductions associated to quivers of type a finally we show that the symplectic reflection algebras for wreath products of the symmetric group and a kleinian group are isomorphic to certain quantum hamiltonian reductions our results involving w algebras are new while for those dealing with symplectic reflection algebras we just find new proofs a key ingredient in our proofs is the study of quantizations of symplectic resolutions of appropriate poisson varieties

counting semisimple orbits of finite lie algebras by genus abstract the adjoint action of a finite group of lie type on its lie algebra is studied a simple formula is conjectured for the number of semisimple orbits of a given split genus this conjecture is proved for type a and partial results are obtained for other types for type a a probabilistic interpretation is given in terms of card shuffling

hammersley s interacting particle process and longest increasing subsequences in a famous paper 8 hammersley investigated the lengthl n of the longest increasing subsequence of a randomn permutation implicit in that paper is a certain one dimensional continuous space interacting particle process by studying a hydrodynamical limit for hammersley s process we show by fairly soft arguments that limn 1 2 el n 2 this is a known result but previous proofs 14 11 relied on hard analysis of combinatorial asymptotics

bernstein inequality and holonomic modules abstract in this paper we study the representation theory of filtered algebras with commutative associated graded whose spectrum has finitely many symplectic leaves examples are provided by the algebras of global sections of quantizations of symplectic resolutions quantum hamiltonian reductions and spherical symplectic reflection algebras we introduce the notion of holonomic modules for such algebras we show that provided the algebraic fundamental groups of all leaves are finite the generalized bernstein inequality holds for the simple modules and turns into equality for holonomic simples under the same finiteness assumption we prove that the associated variety of a simple holonomic module is equi dimensional we also prove that if the regular bimodule has finite length then any holonomic module has finite length this allows one to reduce the bernstein inequality for arbitrary modules to simple ones we prove that the regular bimodule has finite length for the global sections of quantizations of symplectic resolutions for quantum hamiltonian reductions and for rational cherednik algebras the paper contains a joint appendix by the author and etingof that motivates the definition of a holonomic module in the case of global sections of a quantization of a symplectic resolution

super formal darboux weinstein theorem and finite w superalgebras abstract let v v 0 v 1 be a z 2 graded super vector space with an even c action and v 0 be a fixed point of the induced action in this paper we prove an equivariant darboux weinstein theorem for the formal polynomial algebras a s v 0 v 1 we also give a quantum version of it let g g 0 g 1 be a basic lie superalgebra and e g 0 be a nilpotent element we use the equivariant quantum darboux weinstein theorem to give a poisson geometric realization of the finite w superalgebra u g e in the sense of losev an indirect relation between finite w super algebras u g e and u g 0 e is presented finally we use such a realization to study the finite dimensional irreducible modules over u g e

on kostant s theorem for the lie superalgebra q n abstract in this paper we study finite w algebras for basic superalgebras and q n associated with the regular even nilpotent coadjoint orbits we prove that this algebra satisfies the amitsur levitzki identity and therefore all its irreducible representations are finite dimensional in the case of q n we give an explicit description of the w algebra in terms of generators and relations and realize it as a quotient of the super yangian of q 1