a characterization of markov equivalence classes for acyclic digraphs undirected graphs and acyclic digraphs adgs as well as their mutual extension to chain graphs are widely used to describe dependencies among variables in multivariate distributions in particular the likelihood functions of adg models admit convenient recursive factorizations that often allow explicit maximum likelihood estimates and that are well suited to building bayesian networks for expert systems whereas the undirected graph associated with a dependence model is uniquely determined there may however be many adgs that determine the same dependence markov model thus the family of all adgs with a given set of vertices is naturally partitioned into markov equivalence classes each class being associated with a unique statistical model statistical procedures such as model selection or model averaging that fail to take into account these equivalence classes may incur substantial computational or other inefficiencies here it is shown that each markov equivalence class is uniquely determined by a single chain graph the essential graph that is itself simultaneously markov equivalent to all adgs in the equivalence class essential graphs are characterized a polynomial time algorithm for their construction is given and their applications to model selection and other statistical questions are described

decomposition matrices of birman murakami wenzl algebras abstract in this paper we calculate decomposition matrices of the birman murakami wenzl algebras over c

the induced representations of brauer algebras and the clebsch gordan coefficients of induced representations of from with are discussed the induction coefficients idcs or the outer product reduction coefficients of with up to a normalization factor are derived by using the linear equation method weyl tableaux for the corresponding gel fand basis of so n are defined the assimilation method for obtaining clebsch gordan coefficients of so n in the gel fand basis for no modification rule involved couplings from idcs of brauer algebras is proposed some isoscalar factors of for the resulting irrep with are tabulated

discriminants of brauer algebras in this paper we compute gram determinants associated to all cell modules of brauer algebras fancyscript b n partial theoretically we know when a cell module of fancyscript b n partial is equal to its simple head this gives a solution of this long standing problem

on a theorem of benson and curtis iwahori conjectured 5 that the hecke algebra over q of a chevalley group g over a finite field f with respect to a bore1 subgroup is isomorphic to the group algebra over q of the weyl group the analogous statement with q replaced by its algebraic closure was proved by tits 3 ex 27 p 561 using a deformation argument in 11 benson and curtis proved using a case by case analysis that iwahori s conjecture is true over q whenever g is simple of type ze e with the help of springer they proved that when g is of type e or e there exists irreducible representations of the hecke algebra whose character involve 4 and hence iwahori s conjecture over q is false in general the main result of this paper is an explicit construction of an isomorphism of the hecke algebra over q q with the group algebra over q q of the weyl group the construction makes essential use of the polynomials p w of 8 at some point in the proof i have to appeal to the interpretation of p l in terms of multiplicities in verma modules 8 i 51 which has recently been established by brylinski and kashiwara 4 and independently by belinson and bernstein

characterizing markov equivalence classes for amp chain graph models chain graphs cg adicyclic graphs use undirected and directed edges to represent both structural and associative dependences like acyclic directed graphs adgs the cg associated with a statistical markov model may not be unique so cgs fall into markov equivalence classes which may be superexponentially large leading to unidentifiability and computational inefficiency in model search and selection it is shown here that under the andersson madigan perlman amp interpretation of a cg each markov equivalence class can be uniquely represented by a single distinguished cg the amp essential graph that is itself simultaneously markov equivalent to all cgs in the amp markov equivalence class a complete characterization of amp essential graphs is obtained like the essential graph previously introduced for adgs the amp essential graph will play a fundamental role for inference and model search and selection for amp cg models

minimal length elements in twisted conjugacy classes of finite coxeter groups let w be a finite coxeter group and let f be an automorphism of w that leaves the set of generators of w invariant we establish certain properties of elements of minimal length in the f conjugacy classes of w that allow us to define character tables for the corresponding twisted iwahori hecke algebras these results are extensions of results obtained by geck and pfeiffer in the case where f is trivial

a graphical representation of equivalence classes of amp chain graphs this paper deals with chain graph models under alternative amp interpretation a new representative of an amp markov equivalence class called the largest deflagged graph is proposed the representative is based on revealed internal structure of the amp markov equivalence class more specifically the amp markov equivalence class decomposes into finer strong equivalence classes and there exists a distinguished strong equivalence class among those forming the amp markov equivalence class the largest deflagged graph is the largest chain graph in that distinguished strong equivalence class a composed graphical procedure to get the largest deflagged graph on the basis of any amp markov equivalent chain graph is presented in general the largest deflagged graph differs from the amp essential graph which is another representative of the amp markov equivalence class

separation of variables and the computation of fourier transforms on finite groups ii we present a general diagrammatic approach to the construction of efficient algorithms for computing the fourier transform of a function on a finite group by extending work which connects bratteli diagrams to the construction of fast fourier transform algorithms we make explicit use of the path algebra connection to the construction of gel fand tsetlin bases and work in the setting of quivers we relate this framework to the construction of a configuration space derived from a bratteli diagram in this setting the complexity of an algorithm for computing a fourier transform reduces to the calculation of the dimension of the associated configuration space our methods give improved upper bounds for computing the fourier transform for the general linear groups over finite fields the classical weyl groups and homogeneous spaces of finite groups while also recovering the best known algorithms for the symmetric group

murnaghan nakayama rules for characters of iwahori hecke algebras of classical type in this paper we give murnaghan nakayama type formulas for computing the irreducible characters of the iwahori hecke algebras of types an 1 bn and dn our method is a generalization of a derivation of the murnaghan nakayama formula for the irreducible characters of the symmetric group given by curtis greene greene s approach is to sum up the diagonal entries of the matrices of certain cycle permutations in young s seminormal representations the analogues of the young seminormal representations for the iwahori hecke algebras of types an 1 bn and dn were given by hoefsmit

length scale of interaction in spatiotemporal chaos extensive systems have no long scale correlations and behave as a sum of their parts various techniques are introduced to determine a characteristic length scale of interaction beyond which spatiotemporal chaos is extensive in reaction diffusion networks information about network size boundary condition or abnormalities in network topology gets scrambled in spatiotemporal chaos and the attenuation of information provides such characteristic length scales space time information flow associated with the recovery of spatiotemporal chaos from finite perturbations a concept somewhat opposite to the paradigm of lyapunov exponents defines another characteristic length scale high precision computational studies of asymptotic spatiotemporal chaos in the complex ginzburg landau system and transient spatiotemporal chaos in the gray scott network show that these different length scales are comparable and thus suitable to define a length scale of interaction preliminary studies demonstrate the relevance of these length scales for stable chaos

complex ginzburg landau equation in the presence of walls and corners we investigate the influence of walls and corners with dirichlet and neumann boundary conditions in the evolution of two dimensional autooscillating fields described by the complex ginzburg landau equation analytical solutions are found and arguments provided to show that dirichlet walls introduce strong selection mechanisms for the wave pattern corners between walls provide additional synchronization mechanisms and associated selection criteria the numerical results fit well with the theoretical predictions in the parameter range studied

on tensor categories of lie type en n 9 let v be the minuscule representation of the lie algebra g en n 6 7 we show that the centralizer of the action of the quantum group uqg en on v n is generated by the r matrices and one additional element appearing in the n 1 st tensor power a similar description can be given for the classical limit an analogous statement is true for a certain direct summand of v n for uqg en with n 5 n 9 here v is the module whose highest weight is the fundamental weight labeled by the vertex furthest from the triple point of the graph en moreover we obtain a 2 parameter family of braid representations generalizing a q version of the brauer centralizer algebras

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

combinatorics of transformations from standard to non standard bases in brauer algebras transformation coefficients between standard bases for irreducible representations of the brauer centralizer algebra and split bases adapted to the subalgebra f1 f2 f are considered after providing the suitable combinatorial background based on the definition of the i coupling relation on nodes of the subduction grid we introduce a generalized version of the subduction graph which extends the one given in chilla 2006 j phys a math gen 39 7657 for symmetric groups thus we can describe the structure of the subduction system arising from the linear method and give an outline of the form of the solution space an ordering relation on the grid is also given and then as in the case of symmetric groups the choices of the phases and of the free factors governing the multiplicity separations are discussed

secondary structures in a one dimensional complex ginzburg landau equation with homogeneous boundary conditions experiments in extended systems such as the counter rotating couette taylor flow or the taylor dean flow system have shown that patterns with vanishing amplitude may exhibit periodic spatio temporal defects for some range of control parameters these observations could not be interpreted by the complex ginzburg landau equation cgle with periodic boundary conditions we have investigated the one dimensional cgle with homogeneous boundary conditions we found that in the benjamin feir stable region the basic wave train bifurcates to state with periodic spatio temporal defects the numerical results match the observations quite well we have built a new state diagram in the parameter plane spanned by the criticality or equivalently the linear group velocity and the nonlinear frequency detuning

subduction coefficients of birman wenzl algebras and racah coefficients of the quantum groups oq n and spq 2m i subduction coefficients irreducible representations of birman wenzl algebras cf r q in the non standard basis are discussed a procedure for evaluating subduction coefficients sdcs or the transformation coefficients between standard and non standard basis of birman wenzl algebras is formulated based on the linear equation method sdcs of cf r q with f 4 are derived racah coefficients of the quantum groups oq n and spq 2m can be obtained from sdcs of birman wenzl algebras cf r q by using the schur weyl brauer duality relation between birman wenzl algebras cf r q and the quantum groups oq n and spq 2m

an algorithm for characters of hecke algebras hn q of type an 1 recently the characters of irreducible representations of the hecke algebra hn q of type an 1 were identified with the transition coefficients relating q generalized power sum symmetric functions to schur functions king and wybourne 1990 using this result one obtains an easy combinatorial algorithm for calculating the characters

a modified brauer algebra as centralizer algebra of the unitary group the centralizer algebra of the action of u n on the real tensor powers r r v of its natural module v c n is described by means of a modification in the multiplication of the signed brauer algebras the relationships of this algebra with the invariants for u n and with the decomposition of r r v into irreducible submodules is considered

derived equivalences and sl2 categorifications for uq gln abstract we give a construction of sl 2 categorifications for u q gl n for generic q and for q a root of unity

generalized characters of the symmetric group normalized irreducible characters of the symmetric group s n can be understood as zonal spherical functions of the gelfand pair s n s n diags n they form an orthogonal basis in the space of the functions on the group s n invariant with respect to conjugations by s n in this paper we consider a different gelfand pair connected with the symmetric group that is an unbalanced gelfand pair s n s n 1 diags n 1 zonal spherical functions of this gelfand pair form an orthogonal basis in a larger space of functions on s n namely in the space of functions invariant with respect to conjugations by s n 1 we refer to these zonal spherical functions as normalized generalized characters of s n the main discovery of the present paper is that these generalized characters can be computed on the same level as the irreducible characters of the symmetric group the paper gives a murnaghan nakayama type rule a frobenius type formula and an analogue of the determinantal formula for the generalized characters of s n

selection and identity rules for subductions of type a quantum iwahori hecke algebras this paper is concerned with the subduction problem of type a quantum iwahori hecke algebras ch sf q2 with a real deformation parameter q i e the problem of decomposing irreducible representations of such algebras as direct sum of irreducible representations of the subalgebras ch sf1 q2 ch sf2 q2 with f1 f2 f after giving a suitable combinatorial description for the subduction issue we provide a selection rule based on the richardson littlewood criterion which allows to determine the vanishing coupling coefficients between standard basis vectors for such representations and we also present an equivariance condition for the subduction coefficients such results extend those ones corresponding to the subduction problem in symmetric group algebras csf csf1 csf2 which are obtained by q approaching the value of 1

on chain graph models for description of conditional independence structures a chain graph cg is a graph admitting both directed and undirected edges with partially directed cycles forbidden it generalizes both the concept of undirected graph ug and the concept of directed acyclic graph dag a chain graph can be used to describe efficiently the conditional independence structure of a multidimensional discrete probability distribution in the form of a graphoid that is in the form of a list of statements x is independent of y given z obeying a set of five properties axioms an input list of independency statements for every cg is defined and it is shown that the classic moralization criterion for cgs embraces exactly the graphoid closure of the input list a new direct separation criterion for reading independency statements from a cg is introduced and shown to be equivalent to the moralization criterion using this new criterion it is proved that for every cg there exists a strictly positive discrete probability distribution that embodies exactly the independency statements displayed by the graph thus both criteria are shown to be complete and the use of cgs as tools for description of conditional independence structures is justified

casimir elements from the brauer schur weyl duality abstract we consider casimir elements for the orthogonal and symplectic lie algebras constructed with the use of the brauer algebra we calculate the images of these elements under the harish chandra isomorphism and thus show that they together with the pfaffian type element in the even orthogonal case are algebraically independent generators of the centers of the corresponding universal enveloping algebras

separation and completeness properties for amp chain graph markov models pearl s well known d separation criterion for an acyclic directed graph adg is a pathwise separation criterion that can be used to efficiently identify all valid conditional independence relations in the markov model determined by the graph this paper introduces p separation a pathwise separation criterion that efficiently identifies all valid conditional independences under the andersson madigan perlman amp alternative markov property for chain graphs adicyclic graphs which include both adgs and undirected graphs as special cases the equivalence of p separation to the augmentation criterion occurring in the amp global markov property is established and p separation is applied to prove completeness of the global markov property for amp chain graph models strong completeness of the amp markov property is established that is the existence of markov perfect distributions that satisfy those and only those conditional independences implied by the amp property equivalently by p separation a linear time algorithm for determining p separation is presented

a frobenius formula for the characters of the hecke algebras this paper uses the theory of quantum groups and the quantum yang baxter equation as a guide in order to produce a method of computing the irreducible characters of the hecke algebra this approach is motivated by an observation of m jimbo giving a representation of the hecke algebra on tensor space which generates the full centralizer of a tensor power of the standard representation of the quantum group u q mathfrak s l n by rewriting the solutions of the quantum yang baxter equation for u q mathfrak s l n in a different form one can avoid the quantum group completely and produce a frobenius formula for the characters of the hecke algebra by elementary methods using this formula we derive a combinatorial rule for computing the irreducible characters of the hecke algebra this combinatorial rule is aq extension of the murnaghan nakayama for computing the irreducible characters of the symmetric group along the way one finds connections apparently unexplored between the irreducible characters of the hecke algebra and hall littlewood symmetric functions and kronecker products of symmetric groups

on cellular algebras with jucys murphy elements abstract we study analogues of jucys murphy elements in cellular algebras arising from repeated jones basic constructions examples include brauer and bmw algebras and their cyclotomic analogues

on brauer algebra simple modules over the complex field this paper gives two results on the simple modules for the brauer algebra bn over the complex field first we describe the module structure of the restriction of all simple bn modules to bn 1 second we give a new geometrical interpretation of ram and wenzl s construction of bases for permissible simple modules

representations and traces of the hecke algebras hn q of type an 1 the notion of the connectivity class of minimal words in the algebra hn q is introduced and a method of explicitly constructing irreducible representation matrices is described and implemented guided by these results the connection between the ocneanu trace on hn q and schur functions is exploited to derive a very simple prescription for calculating the irreducible characters of hn q they appear as the elements of the transition matrix relating certain generalized power sum symmetric functions to schur functions their evaluation involves the use of the littlewood richardson rule which is proved to apply to hn q just as it does to sn both representation matrices and characters are tabulated

on central idempotents in the brauer algebra abstract we provide a method for constructing central idempotents in the brauer algebra using the splitting of short exact sequences of bimodules from this we determine certain primitive central idempotents by working over a suitable integral ring we hence demonstrate an efficient method of constructing pieces of the representation theory of the brauer algebra over artinian rings from the integral case

feigin frenkel center in types b c and d for each simple lie algebra mathfrak g consider the corresponding affine vertex algebra v mathrm crit mathfrak g at the critical level the center of this vertex algebra is a commutative associative algebra whose structure was described by a remarkable theorem of feigin and frenkel about two decades ago however only recently simple formulas for the generators of the center were found for the lie algebras of type a following talalaev s discovery of explicit higher gaudin hamiltonians we give explicit formulas for generators of the centers of the affine vertex algebras v mathrm crit mathfrak g associated with the simple lie algebras mathfrak g of types b c and d the construction relies on the schur weyl duality involving the brauer algebra and the generators are expressed as weighted traces over tensor spaces and equivalently as traces over the spaces of singular vectors for the action of the lie algebra mathfrak sl 2 in the context of howe duality this leads to explicit constructions of commutative subalgebras of the universal enveloping algebras mathrm u mathfrak g t and mathrm u mathfrak g and to higher order hamiltonians in the gaudin model associated with each lie algebra mathfrak g we also introduce analogues of the bethe subalgebras of the yangians mathrm y mathfrak g and show that their graded images coincide with the respective commutative subalgebras of mathrm u mathfrak g t

a recursive rule for kazhdan lusztig characters abstract a formula for computing kazhdan lusztig characters in hecke algebras of coxeter groups is given this formula holds for products of coxeter elements of parabolic subgroups in an arbitrary coxeter group where the parabolic subgroups are of type a and the kazhdan lusztig representations are of finite dimension the formula yields new combinatorial interpretations of murnaghan nakayama type to these characters and to other related characters

bethe subalgebras in affine birman murakami wenzl algebras and flat connections for q kz equations commutative sets of jucys murphy elements for affine braid groups of types were defined construction of r matrix representations of the affine braid group of type and its distinguished commutative subgroup generated by the type jucys murphy elements are given we describe a general method to produce flat connections for the two boundary quantum knizhnik zamolodchikov equations as necessary conditions for sklyanin s type transfer matrix associated with the two boundary multicomponent zamolodchikov algebra to be invariant under the action of the type jucys murphy elements we specify our general construction to the case of the birman murakami wenzl algebras bmw algebras for short as an application we suggest a baxterization of the dunkl cherednik elements in the double affine hecke algebra of type a

on a family of non unitarizable ribbon categories we consider several families of categories the first are quotients of h andersen s tilting module categories for quantum groups of lie type b at odd roots of unity the second consists of categories of type bc constructed from idempotents in bmw algebras our main result is to show that these families coincide as braided tensor categories using a recent theorem of tuba and wenzl by appealing to similar results of blanchet and beliakova we obtain another interesting equivalence with these two families of categories and the quantum group categories of lie type c at odd roots of unity the morphism spaces in these categories can be equipped with a hermitian form and we are able to show that these categories are never unitary and no braided tensor category sharing the grothendieck semiring common to these families is unitarizable

the size distribution for markov equivalence classes of acyclic digraph models bayesian networks equivalently graphical markov models determined by acyclic digraphs or adgs also called directed acyclic graphs or dags have proved to be both effective and efficient for representing complex multivariate dependence structures in terms of local relations however model search and selection is potentially complicated by the many to one correspondence between adgs and the statistical models that they represent if the adgs models ratio is large search procedures based on unique graphical representations of equivalence classes of adgs could provide substantial computational efficiency hitherto the value of the adgs models ratio has been calculated only for graphs with n 5 or fewer vertices in the present study a computer program was written to enumerate the equivalence classes of adg models and study the distributions of class sizes and number of edges for graphs up to n 10 vertices the ratio of adgs to numbers of classes appears to approach an asymptote of about 3 7 distributions of the classes according to number of edges and class size were produced which also appear to be approaching asymptotic limits imposing a bound on the maximum number of parents to any vertex causes little change if the bound is sufficiently large with four being a possible minimum the program also includes a new variation of orderly algorithm for generating undirected graphs

on the hecke algebras and the colored homfly polynomial the colored homfly polynomial is the quantum invariant of oriented links in s 3 associated with irreducible representations of the quantum group u q sl n in this paper using an approach to calculate quantum invariants of links via the cabling projection rule we derive a formula for the colored homfly polynomial in terms of the characters of the hecke algebras and schur polynomials the technique leads to a fairly simple formula for the colored homfly polynomial of torus links this formula allows us to test the labastida marino vafa conjecture which reveals a deep relationship between chern simons gauge theory and string theory on torus links

strongly multiplicity free modules for lie algebras and quantum groups abstract let u be either the universal enveloping algebra of a complex semisimple lie algebra g or its drinfel d jimbo quantisation over the field c z of rational functions in the indeterminate z we define the notion of strongly multiplicity free smf for a finite dimensional u module v and prove that for such modules the endomorphism algebras end u v r are generic in the sense that in the classical unquantised case they are quotients of kohno s infinitesimal braid algebra t r while in the quantum case they are quotients of the group ring c z b r of the r string braid group b r in the classical case the generators are generalisations of the quadratic casimir operator c of u while in the quantum case they arise from r matrices which may be thought of as square roots of a quantum analogue of c in a completion of u r this unifies many known results and brings some new cases into their context these include the irreducible 7 dimensional module in type g 2 and arbitrary irreducibles for sl 2 the work leads naturally to questions concerning non semisimple deformations of the relevant endomorphism algebras which arise when the ground rings are varied

a quantum analogue of the first fundamental theorem of classical invariant theory we establish a noncommutative analogue of the first fundamental theorem of classical invariant theory for each quantum group associated with a classical lie algebra we construct a noncommutative associative algebra whose underlying vector space forms a module for the quantum group and whose algebraic structure is preserved by the quantum group action the subspace of invariants is shown to form a subalgebra which is finitely generated we determine generators of this subalgebra of invariants and determine their commutation relations in each case considered the noncommutative modules we construct are flat deformations of their classical commutative analogues our results are therefore noncommutative generalisations of the first fundamental theorem of classical invariant theory which follows from our results by taking the limit as q 1 our method similarly leads to a definition of quantum spheres which is a noncommutative generalisation of the classical case with orthogonal quantum group symmetry

solution of lambda omega systems theta schemes and multigrid methods the numerical solution of a time dependent reaction diffusion lambda omega system discretized by theta schemes and finite differences is considered stability and accuracy of finite difference theta schemes for this system are established to solve the time implicit evolution equations a nonlinear multigrid method is applied the convergence properties of this solver are investigated considering a linearized lambda omega model

frt duals as quantum enveloping algebras the hopf algebra generated by the l functionals on the quantum double cq g cq g is considered where cq g is the coordinate algebra of a standard quantum group and q is not a root of unity it is shown to be isomorphic to cq g op uq g this proves a conjecture by t hodges as an algebra it can be embedded into uq g uq g here it is proven that there is no bialgebra structure on uq g uq g for which this embedding becomes a homomorphism of bialgebras in particular it is not an isomorphism as a preliminary a lemma of hodges concerning the structure of l functionals on cq g is generalized for the classical groups a certain choice of root vectors is expressed in terms of l functionals a formula for their coproduct is derived