commutant structure of u n 1 u transformations in this paper we have found irreducible representations irreps of the algebra of partially transposed permutation operators on the last subsystem we give here the direct method of irreps construction for an arbitrary number of subsystems n and local dimension d our method is inspired by the representation theory of the symmetric group s n the theory of brauer algebras and walled brauer algebras

brauer algebras and centralizer algebras for so 2n c abstract in 1937 richard brauer identified the centralizer algebra of transformations commuting with the action of the complex special orthogonal groups so 2 n corresponding to the centralizer algebra e k 2 n end so 2 n v k for v c 2 n is a set of diagrams to each diagram d brauer associated a linear transformation d in e k 2 n and showed that e k 2 n is spanned by the transformations d in this paper we first define a product on d k 2 n the c linear span of the diagrams under this product d k 2 n becomes an algebra and extends to an algebra epimorphism since d k 2 n is not associative we denote by d k 2n its largest associative quotient we then show that when k 2 n the semisimple quotient of d k 2n is equal to e k 2 n next we prove some facts about the representation theory of e k 2 n we compute the dimensions of the irreducible e k 2 n modules and give some branching rules

on the blocks of the walled brauer algebra we determine the blocks of the walled brauer algebra in characteristic zero these can be described in terms of orbits of the action of a weyl group of type a on a certain set of weights in positive characteristic we give a linkage principle in terms of orbits of the corresponding affine weyl group we also classify the semisimple walled brauer algebras in all characteristics

quantum walled brauer clifford superalgebras abstract we introduce a new family of superalgebras the quantum walled brauer clifford superalgebras b c r s q the superalgebra bc r s q is a quantum deformation of the walled brauer clifford superalgebra bc r s and a super version of the quantum walled brauer algebra we prove that bc r s q is the centralizer superalgebra of the action of u q q n on the mixed tensor space v q r s v q r v q s when n r s where v q c q n n is the natural representation of the quantum enveloping superalgebra u q q n and v q is its dual space we also provide a diagrammatic realization of bc r s q as the r s bead tangle algebra bt r s q finally we define the notion of q schur superalgebras of type q and establish their basic properties

a unified dressing method for c and s integrable hierarchies the particular example of a 3 1 dimensional n wave equation this paper presents a modification of the dressing method for solving new classes of multidimensional nonlinear partial differential equations pde first we combine a hierarchy of matrix linearizable c integrable equations with a hierarchy of matrix equations integrable by inverse spectral transformation s integrable equations the solution manifold of the associated n 1 dimensional nonlinear pde has an arbitrary dependence on n variables this allows us to outline an algorithm for solving known integrable equations with arbitrary forcing terms second we combine two different s integrable hierarchies two n wave systems in our example the available solution manifold has an arbitrary dependence on two variables multiple scale expansion is useful in both cases for the purpose of simplification and revealing physical applications of the derived nonlinear pde we deal with a 3 1 dimensional space of independent variables but there is no formal restriction on the dimension

from integrable nets to integrable lattices inspired by the results of jonas einsenhart demoulin and bianchi on the permutability property of classical geometrical transformations of conjugate nets and its reductions of pseudo orthogonal pseudo symmetric and pseudo egorov types dressing transformations of the n component kp hierarchy described within the grassmannian are used to generate quadrilateral lattices and its corresponding reductions as a byproduct we get the corresponding discrete dressing transformations in particular we characterize the vectorial fundamental discrete transformations preserving the symmetric lattice

dressing methods for geometric nets ii orthogonal and egorov nets the grassmannian formalism of kp hierarchies is used to study geometric nets of orthogonal type and their subclass of egorov nets efficient dressing methods for cauchy propagators are provided which lead to wide families of explicit nets frobenius manifolds and solutions to the witten dijkgraff verlinde verlinde associativity equations are also constructed

mixed schur weyl sergeev duality for queer lie superalgebras abstract we introduce a new family of superalgebras b r s for r s 0 such that r s 0 which we call the walled brauer superalgebras and prove the mixed schur weyl sergeev duality for queer lie superalgebras more precisely let q n be the queer lie superalgebra v c n n the natural representation of q n and w the dual of v we prove that if n r s the superalgebra b r s is isomorphic to the supercentralizer algebra end q n v r w s op of the q n action on the mixed tensor space v r w s as an ingredient for the proof of our main result we construct a new diagrammatic realization d k of the sergeev superalgebra ser k finally we give a presentation of b r s in terms of generators and relations

structure and properties of the algebra of partially transposed permutation operators we consider the structure of algebra of operators acting in n fold tensor product space which are partially transposed on the last term using purely algebraical methods we show that this algebra is semi simple and then considering its regular representation we derive basic properties of the algebra in particular we describe all irreducible representations of the algebra of partially transposed operators and derive expressions for matrix elements of the representations it appears that there are two kinds of irreducible representations of the algebra the first one is strictly connected with the representations of the group s n 1 induced by irreducible representations of the group s n 2 the second kind is structurally connected with irreducible representations of the group s n 1

dressing methods for geometric nets i conjugate nets the formalism of multicomponent kp hierarchies is applied to deriving efficient dressing methods for conjugate nets the notion of the cauchy propagator is used for characterizing these nets in terms of spectral data explicit examples in dimensions n 2 and 3 are given in particular periodic nets and cartesian nets with a gaussian localized deformation are exhibited

approximation of fir by iir digital filters an algorithm based on balanced model reduction an algorithm for the approximation of finite impulse response fir filters by infinite impulse response iir filters is presented the algorithm is based on a concept of balanced model reduction the matrix inversions normally associated with this procedure are notoriously error prone due to ill conditioning of the special matrix forms required this difficulty is circumvented here by directly formulating a reduced state space system description which is input output equivalent to the system that would more conventionally be obtained following the explicit step of constructing an interim balanced realization examples of fir by iir filter approximations are included

affine walled brauer algebras and super schur weyl duality abstract a new class of associative algebras referred to as affine walled brauer algebras are introduced these algebras are free with infinite rank over a commutative ring containing 1 then level two walled brauer algebras over c are defined which are some cyclotomic quotients of affine walled brauer algebras we establish a super schur weyl duality between affine walled brauer algebras and general linear lie superalgebras and realize level two walled brauer algebras as endomorphism algebras of tensor modules of kac modules with mixed tensor products of the natural module and its dual over general linear lie superalgebras under some conditions we also prove the weakly cellularity of level two walled brauer algebras and give a classification of their simple modules over c this in turn enables us to classify the indecomposable direct summands of the said tensor modules

the first fundamental theorem of invariant theory for the orthosymplectic super group 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

reciprocal transformations and flat metrics on hurwitz spaces we consider hydrodynamic systems which possess a local hamiltonian structure to such a system there are also associated an infinite number of nonlocal hamiltonian structures we give the necessary and sufficient conditions so that after a nonlinear transformation of the independent variables the reciprocal system still possesses a local hamiltonian structure we show that under our hypotheses bi hamiltonicity is preserved by the reciprocal transformation finally we apply such results to the reciprocal systems of genus g whitham kdv modulation equations

design of linear phase iir filters from fir specifications a simple method is presented for the characterization of a stable iir filter matching a finite portion of the impulse response and autocorrelation coefficients of a given fir filter it is shown that the problem is reduced to the solution of a set of linear equations derived using the impulse response and autocorrelation data the method is characterized by its computational simplicity and is illustrated by some examples to show its superior performance when compared to the existing methods

thick ideals in deligne s category re_p oδ abstract we describe indecomposable objects in deligne s category re p o and explain how to decompose their tensor products we then classify thick ideals in re p o as an application we classify the indecomposable summands of tensor powers of the standard representation of the orthosymplectic supergroup up to isomorphism

a cell filtration of mixed tensor space we construct a cellular basis of the walled brauer algebra which has similar properties as the murphy basis of the group algebra of the symmetric group in particular the restriction of a cell module to a certain subalgebra can be easily described via this basis furthermore the mixed tensor space possesses a filtration by cell modules although not by cell modules of the walled brauer algebra itself but by cell modules of its image in the algebra of endomorphisms of mixed tensor space

pi algebras and their cocharacters abstract all pi algebras r satisfy identities of the form f x y x 1 y 1 x 2 y 2 y n 1 x n theorem a the existence of these identities imply also that cocharacters x n r lie in a hook shaped strip of width depending on the degree of the minimal identity of r theorem c this extends a characterization of rings satisfying a capelli identity theorem b

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

tensor product representations of general linear groups and their connections with brauer algebras abstract for the complex general linear group g gl r c we investigate the tensor product module t p v q v of p copies of its natural representation v c r and q copies of the dual spare v of v we describe the maximal vectors of t and from that obtain an explicit decomposition of t into its irreducible g summands knowledge of the maximal vectors allows us to determine the centralizer algebra c of all transformations on t commuting with the action of g to construct the irreducible c representations and to identify c with a certain subalgebra b r p q of the brauer algebra b r p q

reduced order iir approximation to fir digital filters an algorithm for designing an infinite impulse response iir stable filter using a finite impulse response fir given filter with the objective of reducing the delay and order is described the design is in the time domain using the least squares inverse algorithm which is briefly described in this method the numerator of the approximated filter is part of the fir filter itself and no calculations and minimization are needed to find the numerator coefficients except finding the fir roots an error analysis between the given fir and approximated iir filters is provided this error analysis enables the designer to fix a design parameter often unnoted keeping the energies of the approximated and original filters equal results and two illustrative examples are presented

the second fundamental theorem of invariant theory for the orthogonal group let v c n be endowed with an orthogonal form and g o v be the corresponding orthogonal group brauer showed in 1937 that there is a surjective homomorphism br n endg v r where br n is the r string brauer algebra with parameter n however the kernel of has remained elusive in this paper we show that in analogy with the case of gl v for r n 1 has a kernel which is generated by a single idempotent element e and we give a simple explicit formula for e using the theory of cellular algebras we show how e may be used to determine the multiplicities of the irreducible representations of o v in v r we also show how our results extend to the case where c is replaced by an appropriate eld of positive characteristic and we comment on quantum analogues of our results

diagrammatic kazhdan lusztig theory for the walled brauer algebra we determine the decomposition numbers for the brauer and walled brauer algebras in characteristic zero in terms of certain polynomials associated to cap and curl diagrams recovering a result of martin in the brauer case we consider a second family of polynomials associated to such diagrams and use these to determine projective resolutions of the standard modules we then relate these two families of polynomials to kazhdan lusztig theory via the work of lascoux schutzenberger and boe inspired by work of brundan and stroppel in the cap diagram case

compatible poisson brackets of hydrodynamic type some general properties of compatible poisson brackets of hydrodynamic type are discussed in particular a an invariant differential geometric criterion of the compatibility based on the nijenhuis tensor which is slightly different from those existing in the literature b the lax pair with a spectral parameter governing compatible poisson brackets in the diagonalizable case c the connection of this problem with the class of surfaces in euclidean space which possess non trivial deformations preserving the weingarten operator

transformations of quadrilateral lattices motivated by the classical studies on transformations of conjugate nets we develop the general geometric theory of transformations of their discrete analogs the multidimensional quadrilateral lattices i e lattices x zn rm n m whose elementary quadrilaterals are planar our investigation is based on the discrete analog of the theory of the rectilinear congruences which we also present in detail we study in particular the discrete analogs of the laplace combescure levy radial and fundamental transformations and their interrelations the composition of these transformations and their permutability is also investigated from a geometric point of view the deep connections between transformations and discretizations is also investigated for quadrilateral lattices we finally interpret these results within the formalism

dimension and character formulas for lie supergroups a character formula is derived for lie supergroups the basic technique is that of symmetrization and antisymmetrization associated with young tableaux generalized to supergroups we rewrite the characters of the ordinary lie groups u n o n and sp 2n in terms of traces in the fundamental representation it is then shown that by simply replacing traces with supertraces the characters of certain representations for u n m and osp n 2m are obtained dimension formulas are derived by calculating the characters of a special diagonal supergroup element with 1 and 1 eigenvalues formulas for the eigenvalues of the quadratic casimir operators are given as applications the decomposition of a representation into representations of subgroups is discussed examples are given for the lie supergroup su 6 4 which has physical applications as a dynamical supersymmetry in nuclei

design of linear phase iir filters via impulse response gramians a new method for the design of a linear phase infinite impulse response iir filter is presented it involves designing a finite impulse response fir filter satisfying the given frequency response specifications and subsequently obtaining a significantly lower order iir filter using model reduction based on impulse response gramians the general outline of the method and a brief overview of the existing linear phase fir filter design and model reduction techniques are presented the impulse response gramian and the model reduction algorithm used are presented the method is illustrated by design examples and is compared with other methods for the design of linear phase iir filters using equalizers

on hamiltonian perturbations of hyperbolic systems of conservation laws i quasi triviality of bi hamiltonian perturbations we study the general structure of formal perturbative solutions to the hamiltonian perturbations of spatially one dimensional systems of hyperbolic pdes vt v x 0 under certain genericity assumptions it is proved that any bi hamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives the main tool is in constructing the so called quasimiura transformation of jet coordinates eliminating an arbitrary deformation of a semisimple bi hamiltonian structure of hydrodynamic type the quasi triviality theorem we also describe following 35 the invariants ofsuchbi hamiltonian structures with respect to the group of miura type transformations depending polynomially on the derivatives c

least squares approximation of fir by iir digital filters an algorithm is presented for the least squares approximation of fir filters by iir filters the algorithm is an iterative procedure where each iteration requires the solution of an overdetermined set of linear equations and some digital filtering operations all calculations are performed with the numerator and denominator coefficients of the transfer functions a conversion to state space descriptions is not necessary examples show that the approximation error is as small as that of the iir filters obtained with balanced model reduction moreover the effects of numerical errors are negligible thus our algorithm is applicable even in cases where the fir filter length is large

on the second fundamental theorem of invariant theory for the orthosymplectic supergroup abstract let osp v be the orthosymplectic supergroup on an orthosymplectic vector superspace v of superdimension m 2 n lehrer and zhang showed that there is a surjective algebra homomorphism f r r b r m 2 n end osp v v r where b r m 2 n is the brauer algebra of degree r with parameter m 2 n the second fundamental theorem of invariant theory in this setting seeks to describe the kernel ker f r r of f r r as a 2 sided ideal of b r m 2 n in this paper we show that ker f r r 0 if and only if r r c m 1 n 1 and give a basis and a dimension formula for ker f r r we show that ker f r r as a 2 sided ideal of b r m 2 n is generated by ker f r c r c for any r r c and we provide an explicit set of generators for ker f r c r c these generators coincide in the classical case with those obtained in recent papers of lehrer and zhang on the second fundamental theorem of invariant theory for the orthogonal and symplectic groups as an application we obtain the necessary and sufficient conditions for the endomorphism algebra end osp v v r over the orthosymplectic lie superalgebra osp v to be isomorphic to b r m 2 n

solutions of the wdvv equations and integrable hierarchies of kp type we show that reductions of kp hierarchies related to the loop algebra of sl n with homogeneous gradation give solutions of the darboux egoroff system of pde s using explicit dressing matrices of the riemann hilbert problem generalized to include a set of commuting additional symmetries we construct solutions of the witten dijkgraaf e verlinde h verlinde equations

on laplace darboux type sequences of generalized weingarten surfaces two novel classes of integrable surfaces generalized weingarten surfaces of class 1 and 2 which may be defined in a coordinate independent manner or equivalently via an extension of the classical lelieuvre formulas are introduced the classes 1 and 2 generalize bianchi surfaces and surfaces of harmonic inverse mean curvature respectively and turn out to be governed by analogs of the ernst weyl equation descriptive of gravitational and neutrino fields in general relativity the latter connection is exploited to derive laplace darboux type transformations for class 2 via the associated loewner konopelchenko rogers representation the laplace darboux type transformations are shown to have remarkable geometric properties in terms of sphere congruences connections with the pohlmeyer lund regge vortex model and an inhomogeneous heisenberg spin equation are recorded a backlund transformation for a particular painleve iii equation is derived in a purely geometric manner and a geometric interpretation of a

deformation of surfaces integrable systems and chern simons theory a few years ago some of us devised a method to obtain integrable systems in 2 1 dimensions from the classical non abelian pure chern simons action via the reduction of the gauge connection in hermitian symmetric spaces in this article we show that the methods developed in studying classical non abelian pure chern simons actions can be naturally implemented by means of a geometrical interpretation of such systems the chern simons equation of motion turns out to be related to time evolving two dimensional surfaces in such a way that these deformations are both locally compatible with the gauss mainardi codazzi equations and completely integrable the properties of these relationships are investigated together with the most relevant consequences explicit examples of integrable surface deformations are displayed and discussed

asymptotic values for degrees associated with strips of young diagrams abstract the sums s l n occur in the representations of the symmetric and the general linear groups in combinatorics and in pi algebras we give asymptotic values for these sums

the blocks of the periplectic brauer algebra in positive characteristic we determine the blocks of the periplectic brauer algebra over any field of odd characteristic

diagram and superfield techniques in the classical superalgebras introduces the concept of graded permutation group in the analysis of tensor operators in the classical superalgebras for u m n and su m n irreducible tensor representations correspond to classes of young tableaux with definite graded symmetry type diagram techniques are given for kronecker products dimensions and branching rules the tensor techniques are complemented by the introduction of a superfluid formalism in which u m n and su m n act on polynomial functions over the appropriate superspace such superfluids may admit constraints a general superfield interpolates between the classes of young tableaux which correspond to particular types of constraint the tensor and superfield techniques are illustrated with case studies of su 2 1 and su m 1

walled brauer algebras as idempotent truncations of level 2 cyclotomic quotients abstract we realize via an explicit isomorphism the walled brauer algebra b r t for arbitrary integral parameter as an idempotent truncation of a level two cyclotomic degenerate affine walled brauer algebra the latter arises naturally in lie theory as the endomorphism ring of so called mixed tensor products i e of a parabolic verma module tensored with some copies of the natural representation and its dual the result provides a method to construct central elements in the walled brauer algebra and also implies the koszulity of the walled brauer algebras if 0

gradings on walled brauer algebras and khovanov s arc algebra abstract we introduce some z graded versions of the walled brauer algebra b r s working over a field of characteristic zero this allows us to prove that b r s is morita equivalent to an idempotent truncation of a certain infinite dimensional version of khovanov s arc algebra we deduce that the walled brauer algebra is koszul whenever 0

the degenerate affine walled brauer algebra abstract we define a degenerate affine version of the walled brauer algebra that has the same role played by the degenerate affine hecke algebra for the symmetric group algebra we use it to prove a higher level mixed schur weyl duality for gl n we consider then families of cyclotomic quotients of level two which appear naturally in lie theory and we prove that they inherit from there a natural grading and a graded cellular structure

invariants of the orthosymplectic lie superalgebra and super pfaffians given a complex orthosymplectic superspace v the orthosymplectic lie superalgebra mathfrak osp v and general linear algebra mathfrak gl n both act naturally on the coordinate super ring mathcal s n of the dual space of v otimes mathbb c n and their actions commute hence the subalgebra mathcal s n mathfrak osp v of mathfrak osp v invariants in mathcal s n has a mathfrak gl n module structure sergeev has indicated how to define a pfaffian in this space and announced that together with the invariants of the group mathrm osp v it generates all invariants of mathfrak osp v we introduce a space of super pfaffians show it is a simple mathfrak gl n submodule of mathcal s n mathfrak osp v give an explicit formula for its highest weight vector and prove that the super pfaffians and the mathrm osp v invariants generate mathcal s n mathfrak osp v as an algebra the decomposition of mathcal s n mathfrak osp v as a direct sum of simple mathfrak gl n submodules is obtained and shown to be multiplicity free using howe s mathfrak gl v mathfrak gl n duality on mathcal s n we further analyse the module structure of that space these results also enable us to determine the mathfrak osp v invariants in the tensor powers v otimes r for all r

the curvature line parametrization from circular nets on a surface we study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by discrete nets circular nets and planar quadrilateral nets respectively with edges of order both smooth and discrete geometries are described by integrable systems it is shown that one can obtain an order 2 approximation globally with points of the discrete nets on the smooth surface a new simple geometric construction of principal directions of smooth surfaces is given we study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by discrete nets circular nets and planar quadrilateral nets respectively with edges of order both smooth and discrete geometries are described by integrable systems it is shown that one can obtain an order 2 approximation globally with points of the discrete nets on the smooth surface a new simple geometric construction of principal directions of smooth surfaces is given

reciprocal transformations and local hamiltonian structures of hydrodynamic type systems we start from a hyperbolic dubrovin and novikov dn hydrodynamic type system of dimension n which possesses riemann invariants and we settle the necessary conditions on the conservation laws in the reciprocal transformation so that after such a transformation of the independent variables one of the metrics associated with the initial system is flat we prove the following statement let n 3 in the case of reciprocal transformations of a single independent variable or n 5 in the case of transformations of both the independent variables then the reciprocal metric may be flat only if the conservation laws in the transformation are linear combinations of the canonical densities of conservation laws i e the casimirs the momentum and the hamiltonian densities associated with the hamiltonian operator for the initial metric then we restrict ourselves to the case in which the initial metric is either flat or of constant curvature and we classify the reciprocal transformations of one or both the independent variables so that the reciprocal metric is flat such characterization has an interesting geometric interpretation the hypersurfaces of two diagonalizable dn systems of dimension n 5 are lie equivalent if and only if the corresponding local hamiltonian structures are related by a canonical reciprocal transformation

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

the stackel systems and algebraic curves we show how the abel jacobi map provides all the principal properties of an ample family of integrable mechanical systems associated to hyperelliptic curves we prove that derivative of the abel jacobi map is just the stackel matrix which determines n orthogonal curvilinear coordinate systems in a flat space the lax pairs r matrix algebras and explicit form of the flat coordinates are constructed an application of the weierstrass reduction theory allows us to construct several flat coordinate systems on a common hyperelliptic curve and to connect among themselves different integrable systems on a single phase space

multi matrix models and noncommutative frobenius algebras obtained from symmetric groups and brauer algebras it has been understood that correlation functions of multi trace operators in mathcal n 4 sym can be neatly computed using the group algebra of symmetric groups or walled brauer algebras on the other hand such algebras have been known to construct 2d topological field theories tfts after reviewing the construction of 2d tfts based on symmetric groups we construct 2d tfts based on walled brauer algebras in the construction the introduction of a dual basis manifests a similarity between the two theories we next construct a class of 2d field theories whose physical operators have the same symmetry as multi trace operators constructed from some matrices such field theories correspond to non commutative frobenius algebras a matrix structure arises as a consequence of the noncommutativity correlation functions of the gaussian complex multi matrix models can be translated into correlation functions of the two dimensional field theories

hook young diagrams with applications to combinatorics and to representations of lie superalgebras definition et etude des tableaux k l semi standards applications a la combinatoire et aux representations des superalgebres de lie des diagrammes de young a crochets