embeddings of schur functions into types b c d we consider the problem of embedding the semi ring of schur positive symmetric polynomials into its analogue for the classical types b c d if we preserve highest weights and add the additional lie theoretic parity assumption that the weights in images of schur functions lie in a single translate of the root lattice there are exactly two solutions these naturally extend the kirillov reshetikhin decompositions of representations of symplectic and orthogonal quantum affine algebras uq g

irreducible representations of brauer algebras irreducible representations of brauer algebras are constructed by using the induced representation and the linear equation method as examples some matrix representations of brauer algebras df n with f or 5 are presented

subduction coefficients of brauer algebras and racah coefficients of o n and sp 2 m ii racah coefficients irreducible representations of brauer algebras are discussed in the non standard basis a method for evaluating subduction coefficients sdcs i e the transformation coefficients between standard and non standard bases of brauer algebras is outlined non trivial sdcs of for are derived racah coefficients of o n and sp 2m can be derived from subduction coefficients of brauer algebras by using the schur weyl duality relation between and o n or sp 2m

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 resolving the multiplicity of the branching rule gl 2k 1 c down arrow so 2k 1 c we consider the multiplicity problem of the branching rule gl 2k 1 c down arrow so 2k 1 c finite dimensional irreducible representations of gl 2k 1 c are realized as right translations on subspaces of the holomorphic hilbert bargmann spaces of q 2k 1 complex variables maps are exhibited which carry an irreducible representation of so 2k 1 c into these subspaces an algebra of commuting operators is constructed eigenvalues and eigenvectors of certain of these operators can then be used to resolve the multiplicity in the branching rule

a q brauer algebra we define a new q deformation of brauer s centralizer algebra which contains hecke algebras of type a as unital subalgebras we determine its generic structure as well as the structure of certain semisimple quotients this is expected to have applications for constructions of subfactors of type ii1 factors and for module categories of fusion categories of type a corresponding to certain symmetric spaces

kerdock codes determine unitary 2 designs the non linear binary kerdock codes are known to be gray images of certain extended cyclic codes of length n 2 m over mathbb z 4 we show that exponentiating these mathbb z 4 valued codewords by i triangleq sqrt 1 produces stabilizer states that are quantum states obtained using only clifford unitaries these states are also the common eigenvectors of commuting hermitian matrices forming maximal commutative subgroups mcs of the pauli group we use this quantum description to simplify the derivation of the classical weight distribution of kerdock codes next we organize the stabilizer states to form n 1 mutually unbiased bases and prove that automorphisms of the kerdock code permute their corresponding mcs thereby forming a subgroup of the clifford group when represented as symplectic matrices this subgroup is isomorphic to the projective special linear group psl 2 n we show that this automorphism group acts transitively on the pauli matrices which implies that the ensemble is pauli mixing and hence forms a unitary 2 design the kerdock design described here was originally discovered by cleve et al 2016 but the connection to classical codes is new which simplifies its description and translation to circuits significantly sampling from the design is straightforward the translation to circuits uses only clifford gates and the process does not require ancillary qubits finally we also develop algorithms for optimizing the synthesis of unitary 2 designs on encoded qubits i e to construct logical unitary 2 designs software implementations are available at https github com nrenga symplectic arxiv18a which we use to provide empirical gate complexities for up to 16 qubits

the capelli identity the double commutant theorem and multiplicity free actions 0 the capelli identity cal 3 w p 39 is one of the most celebrated and useful formulas of classical invariant theory w d cl z the double commutant theorem w p 91 is likewise a basic result in the general theory of associative algebras both play key roles in weyl s book the classical groups the main purpose of this paper is to demonstrate a close connection between the two in the context of multiplicity free actions ekj of groups on vector spaces the focus of our discussion will be the structure of the differential operators which commute with a multiplicity free action applications include a new derivation of some formulas of shimura shl and rubenthaler and schiffmann rs for b functions associated to hermitian symmetric spaces and a construction of interesting sets of generators for the center of the universal enveloping algebra of gin we also give a detailed discussion of certain aspects of multiplicity free representations here is an overview of the contents of the paper in sect 1 we review the classical capelli identities we observe how their existence is predicted by a double commutant result 1 6 and 1 8 and we show how they can be used to compute b functions we remark that capelli understood that his operators generate the center of the algebra generated by the polarization operators see b1 p 77 also capelli s motivation in introducing his operators was to explain a formula of cayley essentially the computation of a b function capelli s point of view was remarkably modern and structuralist in certain ways more modern even than that of weyl sections 2 through 9 provide a conceptual context for understanding the capelli identities as a feature of multiplicity free actions they contain a general discussion of the structure of g the algebra of polynomial coefficient differential operators which commute with a given group g of linear transformations the results of the discussion are summarized in theorem 9 1 which says in particular that the polynomial coefficient differential operators commuting with a

orbits and invariants associated with a pair of commuting involutions let be commuting involutions of the connected reductive algebraic group g where and g are defined over a usually algebraically closed field k char k 2 we have fixed point groups h g and k g and an action h k g g where h k g hgk 1 h h k k g g let g h k denote spec o g h k the categorical quotient let a be maximal among subtori s of g such that s s s 1 for all s s there is the associated weyl group w wh k a we show the inclusion a g induces an isomorphism a w g h k in particular the closed h k orbits are precisely those which intersect a the fibers of g g h k are the same as those occurring in certain associated symmetric varieties in particular the fibers consist of finitely many orbits we investigate the structure of w and its relation to other naturally occurring weyl groups and to the action of on the a weight spaces of the relation of the orbit type stratifications of a w and g h k along the way we simplify some of richardson s proofs for the symmetric case and at the end we quickly recover results of berger flensted jensen hoogenboom and matsuki ber57 fj78 hoo84 mat97 for the case k

howe duality and combinatorial character formula for orthosymplectic lie superalgebras we study the howe dualities involving the reductive dual pairs o d spo 2m 2n and sp d osp 2m 2n on the super symmetric tensor of cd cm n we obtain complete decompositions of this space with respect to their respective joint actions we also use these dualities to derive a character formula for these irreducible representations of spo 2m 2n and osp 2m 2n that appear in these decompositions

on resolving the multiplicity of tensor products of irreducible representations of symplectic groups the multiplicity problem of r fold tensor products of irreducible representations of sp 2k c are considered the arbitrary r fold tensor product is shown to be isomorphic to a subspace of the holomorphic hilbert bargmann space of n 2k complex variables maps are constructed which carry an irreducible representation of sp 2k c into this subspace an algebra of commuting operators is constructed and it is shown how eigenvalues and eigenvectors of certain of these operators can be used to resolve the multiplicity

on the decomposition of tensor products of the representations of the classical groups by means of the universal characters on definit les caracteres universels d une representation rationnelle de gl n c et on donne les formules de decomposition des produits tensoriels entre eux en termes de coefficients de littlewood richardson on demontre les formules de decomposition du produit tensoriel de deux representations irreductibles de sp 2n c et de so n c

on odd symplectic schur functions abstract proctor defined combinatorially a family of laurent polynomials called odd symplectic schur functions indexed by pairs c where is partition andcis a column length of a conjecture of proctor invent math 92 1988 307 332 includes the statement that the odd symplectic schur functions are actually characters ofsp 2n 1 c the purpose of the present note is to prove this

quantization of branching coefficients for classical lie groups abstract we study natural quantizations k of branching coefficients corresponding to the restrictions of the classical lie groups to the levi subgroups of their standard parabolic subgroups the polynomials obtained can be regarded as generalizations of lusztig q analogues of weight multiplicities for gl n they coincide with poincare polynomials previously studied by shimozono and weyman they also appear in the hilbert series of the euler characteristic of certain graded virtual g modules and by a result of broer admit nonnegative coefficients providing that restrictive conditions are verified when the levi subgroup considered is isomorphic to a direct product of linear groups we prove that these polynomials admit a stable limit k which decomposes as nonnegative integer combination of poincare polynomials for a general levi subgroup it is conjectured that the polynomials k have nonnegative coefficients when they are parametrized by two partitions when g gl n the polynomials k can be interpreted as quantizations of the littlewood richardson coefficients we show that there also exists a duality between tensor product coefficients for types b c d defined as the analogues of the littlewood richardson coefficients and branching coefficients corresponding to the restriction of so 2 n to subgroups defined from orthogonal decompositions of the root system d n which are not levi subgroups these coefficients can also be quantified but the q analogues obtained may have negative coefficients given a tensor product of irreducible gl n modules we then introduce for each classical group g so n or sp n some q analogues d of the multiplicities obtained by decomposing into its g irreducible components we establish a duality between the polynomials d and k according to a conjecture by shimozono the stable one dimensional sums for nonexceptional affine crystals are expected to coincide with the polynomials d associated to a sequence of rectangular partitions of decreasing heights

on the classification of k involutions let g be a connected reductive algebraic group defined over a field k of characteristic not 2 an involution of g defined over k h a k open subgroup of the fixed point group of and gk resp hk the set of k rational points of g resp h the variety gk hk is called a symmetric k variety these varieties occur in many problems in representation theory geometry and singularity theory over the last few decades the representation theory of these varieties has been extensively studied for k r and c as most of the work in these two cases was completed the study of the representation theory over other fields like local fields and finite fields began the representations of a homogeneous space usually depend heavily on the fine structure of the homogeneous space like the restricted root systems with weyl groups etc thus it is essential to study first this structure and the related geometry in this paper we give a characterization of the isomorphy classes of these symmetric k varieties together with their fine structure of restricted root systems and also a classification of this fine structure for the real numbers p adic numbers finite fields and number fields

resolutions of orthogonal and symplectic analogues of determinantal ideals abstract we consider orthogonal and symplectic analogues of determinantal varieties o r 1 r 2 such varieties simultaneously generalize usual determinantal varieties and rank varieties of symmetric or anti symmetric matrices we find non minimal resolutions of the coordinate rings of the varieties o r 1 r 2 we determine that nearly all such varieties are cohen macaulay and for those that are cohen macaulay we calculate the type furthermore we provide a simple characterization for which varieties o r 1 r 2 are gorenstein as an application we present a class of ideals in k hom e f that are gorenstein of codimension 4

vertex operators arising from jacobi trudi identities we give an interpretation of the boson fermion correspondence as a direct consequence of the jacobi trudi identity this viewpoint enables us to construct from a generalized version of the jacobi trudi identity the action of a clifford algebra on the polynomial algebras that arrive as analogues of the algebra of symmetric functions a generalized giambelli identity is also proved to follow from that identity as applications we obtain explicit formulas for vertex operators corresponding to characters of the classical lie algebras shifted schur functions and generalized schur symmetric functions associated to linear recurrence relations

an approach to the branching rule from sl2n c to sp2n c via littelmann s path model abstract we propose a conjecture describing the branching rule in terms of littelmann s path model from the special linear lie algebra sl 2 n c of type a 2 n 1 to the symplectic lie algebra of type c n embedded as the fixed point subalgebra of the diagram automorphism of sl 2 n c moreover we prove the conjecture in certain cases and also provide some supporting examples in addition we show that the branching coefficients can be obtained explicitly by using the inverse kostka matrix and path models for tensor products of symmetric powers of the defining or natural representation c 2 n of sl 2 n c

young diagrammatic methods for the representation theory of the classical groups of type bn cn dn since the appearance of the celebrated article the classical groups of weyl 7 numerous influences have occurred in the representation theory of the classical groups especially it has become folklore that the facts in this field should be expressed only in terms of young diagrams the aim of this paper is one of the realizations of this principle a classical model of our approach is the results for gl n which are known as littlewood richardson coefficients and the kostka coefficients these two coefficients can be stated in the combinatorial manner using only young diagrams independently of the rank n of gl n we generalize these results to the other classical groups over c namely we give explicit formulas to decompose the tensor product of two irreducible representations into irreducible constituents also we have obtained an interesting relationship linking the irreducible characters of sp and so which we call duality relation between the representations of sp and so as for explicit formulas for the multiplicities of weights in the form of polynomials in the rank n we refer to 3

symmetric polynomials generalized jacobi trudi identities and τ functions an element grnh f of the grassmannian of n dimensional subspaces of the hardy space h h2 extended over the field f c x1 xn may be associated to any polynomial basis 0 1 for c x the plucker coordinates s n x1 xn of labeled by partitions provide an analog of jacobi s bi alternant formula defining a generalization of schur polynomials applying the recursion relations satisfied by the polynomial system to the analog hi 0 of the complete symmetric functions generates a doubly infinite matrix hi j of symmetric polynomials that determine an element h grn h f this is shown to coincide with implying a set of generalized jacobi identities extending a result obtained by sergeev and veselov moscow math j 14 161 168 2014 for the case of orthogonal polynomials the symmetric polynomials s n x1 xn are shown to be kp kadomtsev petviashvili functions in terms of the power sums x of the xa s viewed as kp flow variables a fermionic operator representation is derived for these as well as for the infinite sums s n x s n t associated to any pair of polynomial bases which are shown to be 2d toda lattice functions a number of applications are given including classical group character expansions matrix model partition functions and generators for random processes an element grnh f of the grassmannian of n dimensional subspaces of the hardy space h h2 extended over the field f c x1 xn may be associated to any polynomial basis 0 1 for c x the plucker coordinates s n x1 xn of labeled by partitions provide an analog of jacobi s bi alternant formula defining a generalization of schur polynomials applying the recursion relations satisfied by the polynomial system to the analog hi 0 of the complete symmetric functions generates a doubly infinite matrix hi j of symmetric polynomials that determine an element h grn h f this is shown to coincide with implying a set of generalized jacobi identities extending a result obtained by sergeev and veselov moscow math j 14 161 168 2014 for the case of orthogonal polynomials the symmetric polynomials s n x1 xn are shown to be kp kadomtsev petviashvili functions in terms of the power sums x of the xa s viewed as kp flow variables a fermionic operator re

a branching law for spin 7 c g2 and its applications to unipotent representations abstract we derive a branching law which says how irreducible finite dimensional spin 7 c modules decompose over g2 using the bott borel weil theorem we then use this to calculate the k types on each level of a certain filtered algebra which is also a u lie g2 bimodule explicitly

inhomogeneous discrete time exclusion processes we study discrete time markov processes with periodic or open boundaryconditions and with inhomogeneous rates in the bulk the markov matrices aregiven by the inhomogeneous transfer matrices introduced previously to prove theintegrability of quantum spin chains we show that these processes have asimple graphical interpretation and correspond to a sequential update wecompute their stationary state using a matrix ansatz and express theirnormalization factors as schur polynomials a connection between bethe rootsand lee yang zeros is also pointed out

pieri resolutions for classical groups abstract we generalize the constructions of eisenbud floystad and weyman for equivariant minimal free resolutions over the general linear group and we construct equivariant resolutions over the orthogonal and symplectic groups we also conjecture and provide some partial results for the existence of an equivariant analogue of boij soderberg decompositions for betti tables which were proven to exist in the non equivariant setting by eisenbud and schreyer many examples are given

weight multiplicity formulas for bivariate representations of classical lie algebras a bivariate representation of a complex simple lie algebra is an irreducible representation whose highest weight is given by a combination of the first two fundamental weights for a complex classical lie algebra we establish an expression for the weight multiplicities of bivariate representations a bivariate representation of a complex simple lie algebra is an irreducible representation whose highest weight is given by a combination of the first two fundamental weights for a complex classical lie algebra we establish an expression for the weight multiplicities of bivariate representations

young modules and filtration multiplicities for brauer algebras we define permutation modules and young modules for the brauer algebra bk r and show that if the characteristic of the field k is neither 2 nor 3 then every permutation module is a sum of young modules respecting an ordering condition similar to that for symmetric groups moreover we determine precisely in which cases cell module filtration multiplicities are well defined as done by hemmer and nakano for symmetric groups

schur weyl duality for orthogonal groups we prove schur weyl duality between the brauer algebra bn m and the orthogonal group om k over an arbitrary infinite field k of odd characteristic if m is even we show that each connected component of the orthogonal monoid is a normal variety this implies that the orthogonal schur algebra associated to the identity component is a generalized schur algebra as an application of the main result an explicit and characteristic free description of the annihilator of n tensor space v n in the brauer algebra bn m is also given

on the semisimplicity of the cyclotomic brauer algebras ii abstract in this paper we give a necessary and sufficient condition for the semisimplicity of cyclotomic brauer algebras b m n of types g m 1 n with m 2 this generalizes h rui a criterion on the semisimple brauer algebras j combin theory ser a 111 2005 78 88 1 2 1 3 and h rui m si a criterion on the semisimple brauer algebras ii j combin theory ser a 113 2006 1199 1203 2 5 on brauer algebras

representation stability for filtrations of torelli groups we show finitely generated rational mathsf vic mathbb q modules and mathsf si mathbb q modules are uniformly representation stable and all their submodules are finitely generated we use this to prove two conjectures of church and farb which state that the quotients of the lower central series of the torelli subgroups of mathrm aut f n and mathrm mod sigma g 1 are uniformly representation stable as sequences of representations of the general linear groups and the symplectic groups respectively furthermore we prove an analogous statement for their johnson filtrations

a reciprocity law and the skew pieri rule for the symplectic group we use the theory of skew duality to show that decomposing the tensor product of k irreducible representations of the symplectic group sp2m sp2m is equivalent to branching from sp2n to sp2n1 sp2nk where n n1 nk are positive integers such that n n1 nk and the njs depend on m as well as the representations in the tensor product using this result and a work of lepowsky we obtain a skew pieri rule for sp2m i e a description of the irreducible decomposition of the tensor product of an irreducible representation of the symplectic group sp2m with a fundamental representation

a brauer algebra theoretic proof of littlewood s restriction rules let u be a complex vector space endowed with an orthogonal or symplectic form and let g be the subgroup of gl u of all the symmetries of this form resp o u or sp u if m is an irreducible gl u module the littlewood s restriction rule describes m as a g module via restriction in this paper we give a new representation theoretic proof of this formula realizing m in a tensor power u otimes f and using schur s duality we reduce to the problem of describing the restriction to an irreducible s f module of an irreducible module for the centralizer algebra of the action of g on u otimes f the latter is a quotient of the brauer algebra and we know the kernel of the natural epimorphism whence we deduce the littlewood s restriction rule

vertex operator realization of symplectic and orthogonal s functions it is shown that symplectic and orthogonal schur functions can be realized by the action of the modes of certain vertex operators on the function the properties of these vertex operators allow one to reproduce the basic propreties of these types of symmetric functions the vertex operator modes are shown to obey free fermionic relations which provide applications such as calculating products and plethysms as well as generating hirota type partial differential equations pdes which have symplectic s functions as functions

real double coset spaces and their invariants abstract let g be a real form of a complex reductive group suppose that we are given involutions and of g let h g denote the fixed group of and let k g denote the fixed group of we are interested in calculating the double coset space h g k we use moment map and invariant theoretic techniques to calculate the double cosets especially the ones that are closed one salient point of our results is a stratification of a quotient of a compact torus over which the closed double cosets fiber as a collection of trivial bundles

a stable finite element method for the axisymmetric three field stokes system abstract a rectangular based velocity pressure extra stress tensor mixed finite element method for solving the three field stokes system in the axisymmetric case is studied the method is to be used in connection with the standard galerkin formulation which makes it particularly suitable for the numerical simulation of viscoelastic flow it is proven to be second order convergent in the natural weighted sobolev norms for the system under consideration

odd symplectic groups on definit des groupes symplectiques sur des espaces vectoriels de dimension impaire ces groupes de lie ne sont ni simples ni reductifs on obtient une formule de caracteres pour certaines representations tensorielles de ces groupes

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

multiplicity formulas for fundamental strings of representations of classical lie algebras we call the p fundamental string of a complex simple lie algebra to the sequence of irreducible representations having highest weights of the form k 1 p for k 0 where j denotes the jth fundamental weight of the associated root system for a classical complex lie algebra we establish a closed explicit formula for the weight multiplicities of any representation in any p fundamental string

structure of certain weyl modules for the spin groups abstract let k be an algebraically closed field of characteristic p 0 and let w be a finite dimensional k vector space of dimension greater than or equal to 5 in this paper we give the structure of certain weyl modules for g spin w in the case where p 2 as well as the dimension of the corresponding irreducible finite dimensional rational kg modules in addition we determine the composition factors of the restriction of certain irreducible finite dimensional rational k sl w modules to so w

dirac cohomology k characters and branching laws inspired by work of enright and willenbring ew we prove a generalized littlewood s restriction formula in terms of dirac cohomology our approach is to use a character formula of irreducible unitary lowest weight modules instead of the bernstein gelfand gelfand resolution and the proof is much simpler we also show that our branching formula is equivalent to the formula of enright and willenbring in terms of nilpotent lie algebra cohomology this follows from the close relationship between the dirac cohomology and the corresponding nilpotent lie algebra cohomology for unitary representations of semisimple lie groups of hermitian type which was established in hpr

a generalization of the littlewood richardson rule for the general linear group gz the littlewood richardson rule see 18 193 gives a method to calculate the decomposition of a tensor product of two irreducible g representations the aim of this article is to give a generalization of this rule for all simple simply connected algebraic groups of type a b c d g2 and e we obtain also partial results for g of type f4 e and es the restrictions in the last three cases come from the fact that for the formulation of the decomposition rules we need the notion of a standard young tableau such a notion has been developed by seshadri lakshmibai musili and rajeswari in a series of articles see ll 13 14 16 but not yet for all representations of the last three exceptional groups the advantage of the notion of a young tableau developed by seshadri et al is that it is independent of the type of the group for the convenience of the reader not used to this notion we give first a seperate proof for g sz using the classical notion of a standard young tableau of course for applications the classical notion is much more appropriate in the appendix we give a translation of the notion of a standard young tableau in the sense of seshadri et al into the classical notion of a young tableau for g sp2 and spin in 3 8 we give also such a translation for g g for the other exceptional groups such a translation is not known to the author let p be a dominant weight for g and denote by i the corresponding simple g module we assume that the character of vp is given by ct eycr where the sum has to be taken over all standard young tableaux f of shape p p and v 5 is the weight of the tableau y now by the

on invariant ideals associated to classical groups abstract we study the g 1 g 2 invariant ideals of the coordinate ring of the variety of nullforms associated to a pair of classical groups over a field of characteristic zero

d component rotators as the classical limit of quantum so d vector models the author considers a class of spin systems whose single site configuration space is an orbit of a representation of a compact lie group g for these models the author gets upper and lower bounds to the quantum partition function in terms of two classical partition functions if a certain group theoretic condition is satisfied then these bounds allow one to prove the convergence of a suitable sequence of quantum partition functions to the corresponding classical one this condition is shown to be satisfied in particular for the d component rotators when d is odd the result could be useful for the extension of the lee yang theorem to such models

nonintersecting paths pfaffians and plane partitions gessel and viennot have developed a powerful technique for enumerating various classes of plane partitions the purpose of this article is to show by similar means that one may use pfaffiants to enumerate configurations of nonintersecting paths in which the initial and or terminal vertices of the paths are allowed to vary over specified regions of the digraph this leads to the possibility of enumerating classes of plane partitions in which the shape is allowed to vary whereas the previous applications of gessel and viennot were largely confined to plane partitions of a given shape

a stabilized finite volume element formulation for sedimentation consolidation processes a model of sedimentation consolidation processes in so called clarifier thickener units is given by a parabolic equation describing the evolution of the local solids concentration coupled with a version of the stokes system for an incompressible fluid describing the motion of the mixture in cylindrical coordinates and if an axially symmetric solution is assumed the original problem reduces to two space dimensions this poses the difficulty that the subspaces for the construction of a numerical scheme involve weighted sobolev spaces a novel finite volume element method is introduced for the spatial discretization where the velocity field and the solids concentration are discretized on two different dual meshes the method is based on a stabilized discontinuous galerkin formulation for the concentration field and a multiscale stabilized pair of mathbb p 1 mathbb p 1 elements for velocity and pressure respectively numerical experiments illustrate properties of the model and the satisfactory p

motivic random variables and representation stability ii hypersurface sections abstract we prove geometric and cohomological stabilization results for the universal smooth degree d hypersurface section of a fixed smooth projective variety as d goes to infinity we show that relative configuration spaces of the universal smooth hypersurface section stabilize in the completed grothendieck ring of varieties and deduce from this the stabilization of the hodge euler characteristic of natural families of local systems constructed from the vanishing cohomology we prove explicit formulas for the stable values using a probabilistic interpretation along with the natural analogs in point counting over finite fields we explain how these results provide new geometric examples of a weak version of representation stability for symmetric symplectic and orthogonal groups this interpretation of representation stability was studied in the prequel 20 for configuration spaces

double coset decompositions of reductive lie groups arising from two involutions g 0 h g and g 0 l g here g g g g g for an automorphism of g and f0 denote the connected component of f containing the identity e for a lie group f in 8 we gave standard representatives of double coset decompositions h g l for some typical triples g h l by only using algebraic method as the most elementary example we studied the case g h l gl n if gl p if gl n p if gl r if gl n r if for an arbitrary field if we also studied in 8 other examples related to quadratic forms on vector spaces over the fields ir c and ih in this paper we assume that g is a reductive lie group and we will describe the structure of the double coset decomposition h g l for an arbitrary g h l first we study the compact case section 3 and next the noncompact case section 4 in section 3 we assume that the semisimple part gs of g is compact gs is the analytic subgroup of g for gs g g when g gs and

a new young diagrammatic method for kronecker products of and a new simple young diagrammatic method for kronecker products of o n and sp 2m is proposed based on the representation theory of brauer algebras a general procedure for the decomposition of the tensor products of representations for o n and sp 2m is outlined which is similar to that for u n known as the littlewood rules together with trace contractions from a brauer algebra and some modification rules given by king

finite type 3 manifold invariants and the structure of the torelli group i using the recently developed theory of finite type invariants of integral homology 3 spheres we study the structure of the torelli group of a closed surface explicitly we construct a natural cocycles of the torelli group with coefficients in a space of trivalent graphs and cohomology classes of the abelianized torelli group b group homomorphisms that detect rationally the nontriviality of the lower central series of the torelli group our results are motivated by the appearance of trivalent graphs in topology and in representation theory and the dual role played by the casson invariant in the theory of finite type invariants of integral homology 3 spheres and in morita s study mo2 mo3 of the structure of the torelli group our results generalize those of s morita mo2 mo3 and complement the recent calculation due to r hain ha2 of the i adic completion of the rational group ring of the torelli group we also give analogous results for two other subgroups of the mapping class group

structure of symplectic invariant lie subalgebras of symplectic derivation lie algebras abstract we study the structure of the symplectic invariant part h g 1 sp of the lie algebra h g 1 consisting of symplectic derivations of the free lie algebra generated by the rational homology group of a closed oriented surface g of genus g first we describe the orthogonal direct sum decomposition of this space which is induced by the canonical metric on it and compute it explicitly up to degree 20 in this framework we give a general constraint which is imposed on the sp invariant component of the bracket of two elements in h g 1 second we clarify the relations among h g 1 and the other two related lie algebras h g and h g which correspond to the cases of a closed surface g with and without base point g in particular based on a theorem of labute we formulate a method of determining these differences and describe them explicitly up to degree 20 third by giving a general method of constructing elements of h g 1 sp we reveal a considerable difference between two particular submodules of it one is the sp invariant part of a certain ideal j g 1 and the other is that of the johnson image finally we combine these results to determine the structure of h g 1 completely up to degree 6 including the unstable cases where the genus 1 case has an independent meaning in particular we see a glimpse of the galois obstructions explicitly from our point of view

deformed universal characters for classical and affine algebras abstract creation operators are given for the three distinguished bases of the type bcd universal character ring of koike and terada yielding an elegant way of treating computations for all three types in a unified manner deformed versions of these operators create symmetric function bases whose expansion in the universal character basis has polynomial coefficients in q with nonnegative integer coefficients we conjecture that these polynomials are one dimensional sums associated with crystal bases of finite dimensional modules over quantized affine algebras for all nonexceptional affine types these polynomials satisfy a macdonald type duality

on new multiplicity formulas of weights of representations for the classical groups this is a continuation of 6 in this paper we give the multiplicity formulas of weights in an irreducible representation of the classical groups so n c and sp 2n c c gl 2n c using only young diagrams roughly speaking for each series of classical groups the weight multiplicities can be expressed as a polynomial function in the ranks in this direction r c king and s p 0 plunkett have presented the formulas of this kind using multinomial coefficients in s also see 4 in this paper we present another explicit expression of these polynomial functions moreover we characterize the set of weights dominant integral weights in each irreducible representation only in terms of young diagrams and also give the degrees of these polynomial functions and their leading coefficients our formulas are based on the known result concerning the so called kostka coefficients in the case of gl n and are a natural generalization from the general linear groups to the other classical groups as in the case of gl n the dominant integral weights of sg 2n 1 and sp 2n c gl 2n can be identified with the young diagrams see sect 1 in case of sg 2n the situation is a little bit delicate and for this case see section 1 for each series of the classical groups starting from these identifications we are able to regard the multiplicities of the dominant integral weight q in the irreducible representation with the highest weight 1 as the function of the rank n for fixed young diagrams 2 and q here each series of the classical groups means one of the following

vecteurs distributions h invariants pour les series principales generalisees d espaces symetriques reductifs et prolongement meromorphe d integrales d eisenstein

multi field finite element methods with discontinuous pressures for axisymmetric incompressible flow two and three field methods are studied for solving the stokes system in the axisymmetric case as a linearized form of different types of fluid flow equations both are designed for the standard galerkin formulation expressed in terms either of the velocity and the pressure or of these two fields together with the extra stress tensor and use discontinuous pressure spaces the first method is related to the rectangular based q2 p1 element due to fortin the other one is linked to the crouzeix raviart triangle both methods satisfy the uniform stability inf sup condition relating the velocity and pressure representations expressed in terms of the natural weighed sobolev norms for the system under consideration this condition is fundamental to derive second order convergence results for solution methods of viscous or viscoelastic incompressible flow problems based on the corresponding finite element spaces in order to illustrate this some numerical results using a method of this type studied by the authors are presented in connection with the three field formulation of the stokes system related to viscoelastic fluids

on the radical of brauer algebras the radical of the brauer algebra mathcal b f x is known to be non trivial when the parameter x is an integer subject to certain conditions with respect to f in these cases we display a wide family of elements in the radical which are explicitly described by means of the diagrams of the usual basis of mathcal b f x the proof is by direct approach for x 0 and via classical invariant theory in the other cases exploiting then the well known representation of brauer algebras as centralizer algebras of orthogonal or symplectic groups acting on tensor powers of their standard representation this also gives a great part of the radical of the generic indecomposable mathcal b f x modules we conjecture that this part is indeed the whole radical in the case of modules and it is the whole part in a suitable step of the standard filtration in the case of the algebra as an application we find some more precise results for the module of pointed chord diagrams and for the temperley lieb algebra realised inside mathcal b f 1 acting on it ahi quanto a dir che sia e cosa dura lo radical dell algebra di brauer pur se l pensier gia muove a congettura n barbecue scholia

a characteristic free approach to brauer algebras brauer algebras arise in representation theory of orthogonal or symplectic groups these algebras are shown to be iterated inflations of group algebras of symmetric groups in particular they are cellular as had been shown before by graham and lehrer this gives some information about block decomposition of brauer algebras

algebraic groups with a commuting pair of involutions and semisimple symmetric spaces let g be a connected reductive algebraic group defined over an algebraically closed field f of characteristic not 2 denote the lie algebra of g by 9 in this paper we shall classify the isomorphism classes of ordered pairs of commuting involutorial automorphisms of g this is shown to be independent of the characteristic of f and can be applied to describe all semisimple locally symmetric spaces together with their line structure involutorial automorphisms of g occur in several places in the literature cartan has already shown that for f c the isomorphism classes of involutorial automorphisms of g correspond bijectively to the isomorphism classes of real semisimple lie algebras which correspond in their turn to the isomorphism classes of riemannian symmetric spaces see helgason 111 if one lifts this involution to the group g then the present work gives a characteristic free description of these isomorphism classes in a similar manner we can show that semisimple locally symmetric spaces correspond to pairs of commuting involutorial automorphisms of g namely let go a be a semisimple locally symmetric pair i e go is a real semisimple lie algebra and rr e aut g an involution then by a result of berger 2 there exists a cartan involution 8 of go such that 00 ea if we denote the complexilication of go by g then o and 8 induce a pair of commuting involutions of g conversely if c 8 e aut g are commuting involutions then c and 8 determine two locally semisimple symmetric pairs for if u is a o and o stable compact real form with conjugation r then ger 1 ge and g f3 i ger are semisimple locally symmetric pairs where

tori invariant under an involutorial automorphism i abstract the geometry of the orbits of a minimal parabolick subgroup acting on a symmetrick variety is essential in several areas but its main importance is in the study of the representations associated with these symmetrick varieties see for example 5 6 20 and 31 up to an action of the restricted weyl group ofg these orbits can be characterized by thehk conjugacy classes of maximalk split tori which are stable underk involution associated with the symmetrick variety herehis a openk subgroup of the fixed point group of this is the second in a series of papers in which we characterize and classify thehk conjugacy classes of maximalk split tori the first paper in this series dealt with the case of algebraically closed fields in this paper we lay the foundation for a characterization and classification for the case of nonalgebraically closed fields this includes a partial classification in the cases where the base field is the real numbers p adic numbers finite fields and number fields

identities for classical group characters of nearly rectangular shape abstract we derive several identities that feature irreducible characters of the general linear the symplectic the orthogonal and the special orthogonal groups all of the identities feature characters that are indexed by shapes that are nearly rectangular by which we mean that the shapes are rectangles except for one row or column that might be shorter than the others as applications we prove new results in plane partitions and tableaux enumeration including new refinements of the bender knuth and macmahon ex conjectures

su n and sp 2n wzw fusion rules fusion rules for wzw models based on su n and sp 2n are considered using the results of kac 1989 and walton 1989 1990 it is shown that these fusion rules may be computed using young diagram methods by performing extra modifications these modifications take the form of the removal of boundary strips which bear a striking resemblance to the modifications necessary when performing tensor products for these algebras this fact is exploited to exhibit a duality between the fusion rules of su n at level k and su k at level n and also between sp 2n at level k and so 2k at level n the former duality has been discussed in the context of two dimensional statistical mechanics by kuniba and nakanishi 1990 and also by naculich and schnitzer 1990 in the case of wzw models and goodman and wenzl 1989 for hecke algebras at roots of unity for su 3 a manifestly positive combinatorial procedure for computing fusion rules and a generating function for the fusion coefficients are given

the hopf algebra structure of the character rings of classical groups the character ring of covariant irreducible tensor representations of the general linear group admits a hopf algebra structure isomorphic to the hopf algebra of symmetric functions here we study the character rings and of the orthogonal and symplectic subgroups of the general linear group within the same framework of symmetric functions we show that and also admit natural hopf algebra structures that are isomorphic to that of and hence to the isomorphisms are determined explicitly along with the specification of standard bases for and analogous to those used for a major structural change arising from the adoption of these bases is the introduction of new orthogonal and symplectic schur hall scalar products significantly the adjoint with respect to multiplication no longer coincides as it does in the case with a foulkes derivative or skew operation the adjoint and foulkes derivative now require separate definitions and their properties are explored here in the orthogonal and symplectic cases moreover the hopf algebras and are not self dual the dual hopf algebras and are identified finally the hopf algebra of the universal rational character ring of mixed irreducible tensor representations of the general linear group is introduced and its structure maps identified

orthogonal and symplectic analogues of determinantal ideals abstract we consider subvarieties of determinantal varieties determined by an additional rank equation that defines an orthogonal or symplectic structure such varieties simultaneously generalize usual determinantal varieties and rank varieties of symmetric or anti symmetric matrices in this article we find a non trivial class of such orthogonal or symplectic analogues of determinantal varieties for which we can provide a completely combinatorial description of the terms in a minimal resolution of the coordinate ring the results come as an application of the geometric technique and bott s theorem for the cohomology of vector bundles over the grassmannian

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

decompositions related to symmetric varieties abstract let g be a connected reductive group over an algebraically closed field of characteristic 2 and let be an automorphism of g of order 2 whose fixed point group is k the paper establishes for certain g varieties x a decomposition of x in finitely many k stable locally closed pieces namely a if x g p where p is a parabolic subgroup of g and b g adjoint if x is the wonderful compactification of the symmetric variety g k

orbits invariants and representations associated to involutions of reductive groups by an involution of an algebraic group or lie algebra we mean an automorphism 0 such that 0 2 is the identity automorphism let 0 be an involution of a complex reductive lie algebra q and let p denote the 1 eigenspace of 0 on q get g be the adjoint group of q and let k g glog go kostant and rallis 10 have made a detailed study of the action of k on p in particular they obtain very precise information concerning the orbits of k on p and the representation of k on c p the algebra of polynomial functions on p in this paper we consider a global analogue of the problem studied by kostant and rallis let g be a reductive affine algebraic group over an algebraically closed field k of characteristic 2 let 0 be an involution of g and let k gegi0 g g if g is semisimple and the base field is c then g k is the complexification of a riemannian symmetric space we study the action of k and of groups closely related to k by left translation on the coset space g k for technical reasons it is convenient to study a slightly different but equivalent problem let r g g be defined by r g g0 g i then z g p is a closed subvariety of g furthermore z is constant on left cosets of g modulo k and induces an isomorphism o g k p we have o k g k k o g k for k e k and geg thus p is stable under conjugation by k and a is a k isomorphism of affine k varieties we consider the action of k on p by conjugation let x e g have jordan decomposition x x x then we show that x e p if and only if both x and x are in p thus we have a jordan decomposition for elements of p this will play a key role in our results a torus s of g is o anisotropic if o s s 1 for every ses it is known that any two maximal 0 anisotropic tori of g are k let a be a maximal 0 anisotropic torus of g the following result characterizes semisimple elements of p

the brauer algebra and the symplectic schur algebra let k be an algebraically closed field of characteristic p 0 let m r be integers with m 1 r 0 and m r and let s0 2m r be the symplectic schur algebra over k as introduced by the first author we introduce the symplectic schur functor derive some basic properties of it and relate this to work of hartmann and paget we do the same for the orthogonal schur algebra we give a modified jantzen sum formula and a block result for the symplectic schur algebra under the assumption that r and the residue of 2m mod p are small relative to p from this we deduce a block result for the orthogonal schur algebra under similar assumptions we also show that in general the block relations of the brauer algebra and the symplectic and orthogonal schur algebra are the same finally we deduce from the previous results a new proof of the description of the blocks of the brauer algebra in characteristic 0 as obtained by cox de visscher and martin