towers of recollement and bases for diagram algebras planar diagrams and a little beyond abstract the recollement approach to the representation theory of sequences of algebras is extended to pass basis information directly through the globalisation functor the method is hence adapted to treat sequences that are not necessarily towers by inclusion such as symplectic blob algebras diagram algebra quotients of the type c hecke algebras by carefully reviewing the diagram algebra construction we find a new set of functors interrelating module categories of ordinary blob algebras diagram algebra quotients of the type b hecke algebras at different values of the algebra parameters we show that these functors generalise to determine the structure of symplectic blob algebras and hence of certain two boundary temperley lieb algebras arising in statistical mechanics we identify the diagram basis with a cellular basis for each symplectic blob algebra and prove that these algebras are quasihereditary over a field for almost all parameter choices and generically semisimple that is we give bases for all cell and standard modules

complex representations of finite monoids ii highest weight categories and quivers abstract in this paper we continue our study of complex representations of finite monoids we begin by showing that the complex algebra of a finite regular monoid is a quasi hereditary algebra and we identify the standard and costandard modules we define the concept of a monoid quiver and compute it in terms of the group characters of the standard and costandard modules we use our results to determine the blocks of the complex algebra of the full transformation semigroup we show that there are only two blocks when the degree 3 we also show that when the degree 5 the complex algebra of the full transformation semigroup is not of finite representation type answering negatively a conjecture of ponizovskii

a duality approach to the symmetry of bernstein sato polynomials of free divisors abstract in this paper we prove that the bernstein sato polynomial of any free divisor for which the d s module d s h s admits a spencer logarithmic resolution satisfies the symmetry property b s 2 b s this applies in particular to locally quasi homogeneous free divisors for instance to free hyperplane arrangements or more generally to free divisors of linear jacobian type we also prove that the bernstein sato polynomial of an integrable logarithmic connection e and of its dual e with respect to a free divisor of linear jacobian type are related by the equality b e s b e s 2 our results are based on the behaviour of the modules d s h s and d s e s h s under duality

the partition algebra and the potts model transfer matrix spectrum in high dimensions we construct generalizations pmn q of the partition algebra pn q martin p p 1996 j algebra 183 319 facilitating a representation theoretic approach to the n site transfer matrix spectrum of a high dimensional q state potts model with magnetic field and source terms and to corresponding dichromatic polynomials for each q we describe the irreducible representation theory of the sequence of algebras p q pn q pn1 q pn 1 q n 0 1 2 approaching the large n limit for each positive integer q we extend the potts model representation n of pn q to a representation of p1n q we show how these potts representations embed in the representation theory of the partition algebras these results together provide a tool with which to examine the nature of physical correlation functions for large n the irreducible content of the potts representations can be summarized by sq endpn q vq n and sq 1 endpn1 q vq n where sq is the symmetric group and vq is the space of states of a potts spin we show how the partition algebra formalism matches up the correlation functions of the potts model and the corresponding absolute spectrum degeneracies of its transfer matrix

hecke algebraic approach to the reflection equation for spin chains we use the structural similarity of certain coxeter artin systems to the yang baxter and reflection equations to convert representations of these systems into new solutions of the reflection equation we construct certain bethe ansatz states for these solutions using a parametrization suggested by abstract representation theory

quantum weyl reciprocity and tilting modules quantum weyl reciprocity relates the representation theory of hecke algebras of type a with that of q schur algebras this paper establishes that weyl reciprocity holds integrally i e over the ring q q 1 of laurent polynomials and that it behaves well under base change a key ingredient in our approach involves the theory of tilting modules for q schur algebras new results obtained in that direction include an explicit determination of the ringel dual algebra of a q schur algebra in all cases in particular in the most interesting situation the ringel dual identifies with a natural quotient algebra of the hecke algebra

liftable vector fields over corank one multigerms in this paper a systematic method is given to construct all liftable vector fields over an analytic multigerm f mathbb k n s rightarrow mathbb k p 0 of corank at most one admitting a one parameter stable unfolding

a ribbon hopf algebra approach to the irreducible representations of centralizer algebras the brauer birman wenzl and type a iwahori hecke algebras abstract we show how the ribbon hopf algebra structure on the drinfel d jimbo quantum groups of types a b c and d can be used to derive formulas giving explicit realizations of the irreducible representations of the iwahori hecke algebras of type a and the birman wenzl algebras we use this derivation to give explicit realizations of the irreducible representations of the brauer algebras as well the derivation is accomplished by way of a combination of techniques from operator algebras quantum groups and the theory of 3 manifold invariants although our applications are in the cases of the quantum groups of types a b c and d most of the aspects of our approach apply in the general setting of ribbon hopf algebras

polynomially convex hulls of singular real manifolds we obtain local and global results on polynomially convex hulls of lagrangian and totally real submanifolds of c n with self intersections and open whitney umbrella points

cohomology of the complement of a free divisor we prove that if d is a strongly quasihomogeneous free divisor in the stein manifold x and u is its complement then the de rham cohomology of u can be computed as the cohomology of the complex of meromorphic differential forms on x with logarithmic poles along d with exterior derivative the class of strongly quasihomogeneous free divisors introduced here includes free hyperplane arrangements and the discriminants of stable mappings in mather s nice dimensions and in particular the discriminants of coxeter groups

deformation of singular lagrangian subvarieties we investigate deformations of lagrangian manifolds with singularities we introduce a complex similar to the de rham complex whose cohomology calculates deformation spaces this cohomology turns out to be constructible in many cases examples of singular lagrangian varieties are presented and deformations are calculated explicitly

symplectic and lagrange stabilities of open whitney umbrellas in this paper we formulate and prove the criteria for the lagrange stability of isotropic map germs of corank at most one they include an analogue of mather s characterisation of the stability of differentiable mappings also they give an generalisation of arnol d s criterion for the lagrange stability of lagrange immersions meanwhile there naturally arise some special singularities the open whitney umbrellas which are characterised as symplectically stable isotropic map germs of corank at most one unless otherwise stated any manifolds and mappings are assumed to be of class c

a variational method for learning sparse and overcomplete representations an expectation maximization algorithm for learning sparse and overcomplete data representations is presented the proposed algorithm exploits a variational approximation to a range of heavy tailed distributions whose limit is the laplacian a rigorous lower bound on the sparse prior distribution is derived which enables the analytic marginalization of a lower bound on the data likelihood this lower bound enables the development of an expectation maximization algorithm for learning the overcomplete basis vectors and inferring the most probable basis coefficients

the computation of the logarithmic cohomology for plane curves we will give algorithms of computing bases of logarithmic cohomology groups for square free polynomials in two variables

the decomposition matrices of the brauer algebra over the complex field the brauer algebra was introduced by r brauer in 1937 as a tool in invariant theory the problem of determining the cartan decomposition matrix of the brauer algebra over the complex field has remained open since then here we determine this fundamental invariant

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

a tower construction for the radical in brauer s centralizer algebras abstract in this paper we study the structure of the brauer centralizer algebras in the case that the multiplication constant x is a rational integer several authors have studied the structure of these algebras for generic values of x in particular hans wenzl showed that the brauer algebras can be obtained from jones basic construction and he used that fact to prove that the brauer algebras are semisimple when x is not a rational integer the tower construction is a method to study towers of semisimple algebras hence it does not apply in our case where the algebras in the tower eventually have radicals our first step in this paper is to modify the tower construction so that it does a simultaneous tower construction of the radicals and the semisimple quotients of a tower of algebras the rest of the paper is devoted to describing these constructions explicitly in the brauer algebra case one surprising corollary of this method is a connection between two seemingly distinct criteria for the simplicity of certain subrings of the brauer algebras it is possible to identify explicitly certain subrings of the brauer algebras which are the matrix rings corresponding to irreducible representations in the semisimple case in previous work these authors gave a combinatorial condition for simplicity of these individual matrix rings when x is a rational integer the tower construction gives a second algebraic condition for simplicity it is difficult to see why these two conditions are equivalent however the methods used in this paper make that clear

the first fundamental theorem of invariant theory for the orthosymplectic supergroup we give an elementary and explicit proof of the first fundamental theorem of invariant theory for the orthosymplectic supergroup by generalising the geometric method of atiyah bott and patodi to the supergroup context we use methods from super algebraic geometry to convert invariants of the orthosymplectic supergroup into invariants of the corresponding general linear supergroup on a different space in this way super schur weyl brauer duality is established between the orthosymplectic supergroup of superdimension m 2n and the brauer algebra with parameter m 2n the result may be interpreted either in terms of the group scheme osp v over mathbb c where v is a finite dimensional super space or as a statement about the orthosymplectic lie supergroup over the infinite dimensional grassmann algebra lambda we take the latter point of view here and also state a corresponding theorem for the orthosymplectic lie superalgebra which involves an extra invariant generator the super pfaffian

the dieck temperley lieb algebras of type b and c in brauer algebras abstract in this paper we will study the dieck temperley lieb algebras of type b n and c n we compute their ranks and describe a basis for each of them by using some results from corresponding brauer algebras and temperley lieb algebras respectively

stable mappings and logarithmic relative symplectic forms let d be the image of a stable weighted homogeneous map f mathbb c n to mathbb c n 1 with rm dim mathbb c rm ker df 0 1 and which is not a trivial deformation of a lower dimensional map by proving a variant of the buchsbaum eisenbud structure theorem for grade 3 gorenstein quotients we show the existence of a form omega in omega 2 log d which restrictst to a non degenerate holomorphic 2 form on the milnor fibres of d experiments with the computer algebra programme macaulay suggest this restriction is closed and is thus a holomorphic symplectic form

multichannel blind source separation using convolution kernel compensation this paper studies a novel decomposition technique suitable for blind separation of linear mixtures of signals comprising finite length symbols the observed symbols are first modeled as channel responses in a multiple input multiple output mimo model while the channel inputs are conceptually considered sparse positive pulse trains carrying the information about the symbol arising times our decomposition approach compensates channel responses and aims at reconstructing the input pulse trains directly the algorithm is derived first for the overdetermined noiseless mimo case a generalized scheme is then provided for the underdetermined mixtures in noisy environments although blind the proposed technique approaches bayesian optimal linear minimum mean square error estimator and is hence significantly noise resistant the results of simulation tests prove it can be applied to considerably underdetermined convolutive mixtures and even to the mixtures of moderately correlated input pulse trains with their cross correlation up to 10 of its maximum possible value

higher dimensional algebra and topological quantum field theory the study of topological quantum field theories increasingly relies upon concepts from higher dimensional algebra such as n categories and n vector spaces we review progress towards a definition of n category suited for this purpose and outline a program in which n dimensional topological quantum field theories tqfts are to be described as n category representations first we describe a suspension operation on n categories and hypothesize that the k fold suspension of a weak n category stabilizes for k n 2 we give evidence for this hypothesis and describe its relation to stable homotopy theory we then propose a description of n dimensional unitary extended tqfts as weak n functors from the free stable weak n category with duals on one object to the n category of n hilbert spaces we conclude by describing n categorical generalizations of deformation quantization and the quantum double construction

characterizing normal crossing hypersurfaces the objective of this article is to give an effective algebraic characterization of normal crossing hypersurfaces in complex manifolds it is shown that a divisor hypersurface has normal crossings if and only if it is a free divisor has a radical jacobian ideal and a smooth normalization using k saito s theory of free divisors also a characterization in terms of logarithmic differential forms and vector fields is found finally we give another description of a normal crossing divisor in terms of the logarithmic residue using recent results of m granger and m schulze

schur weyl duality for the brauer algebra and the ortho symplectic lie superalgebra we give a proof of a schur weyl duality statement between the brauer algebra and the ortho symplectic lie superalgebra mathfrak osp v