hub location as the minimization of a supermodular set function this paper highlights how a general class of hub location problems can be modeled as the minimization of a real valued supermodular set function well known problems such as uncapacitated hub location p hub median and hub arc location among others are shown to be particular cases of this class two integer programming formulations are introduced and compared one uses path based variables frequently employed in hub location whereas the other exploits properties of supermodular functions in addition several worst case bounds for a greedy and a local improvement heuristic are obtained for the general class and for some particular cases in which sharper bounds can be devised computational experiments are performed to compare both formulations when used with a general purpose solver computational results obtained on benchmark instances confirm the superiority of the supermodular formulation

su n tensor product multiplicities and virtual berenstein zelevinsky triangles information on su n tensor product multiplicities is neatly encoded in berenstein zelevinsky triangles here we study a generalization of these triangles by allowing negative as well as non negative integer entries for a fixed triple product of weights these generalized berenstein zelevinsky triangles span a lattice in which one may move by adding integer linear combinations of so called virtual triangles inequalities satisfied by the coefficients of the virtual triangles describe a polytope the tensor product multiplicities may be computed as the number of integer points in this convex polytope as our main result we present an explicit formula for this discretised volume as a multiple sum as an application we also address the problem of determining when a tensor product multiplicity is non vanishing the solution is represented by a set of inequalities in the dynkin labels we also allude to the question of when a tensor product multiplicity is greater than a given non negative integer

branching rules for the supergroup su n m from those of su n m the decomposition of representations of supergroups into representations of subgroups is needed in practical applications in this paper we set up and exploit a fruitful one to one correspondence between the lie group branching su n m su n su m u 1 and the supergroup branchings su n m su n su m u 1 and su n1 n2 m1 m2 su n1 m1 su n2 m2 u 1 a simple and useful prescription is discovered for obtaining the su n m branching rules from those of su n m for any representation a large class of examples sufficient for many physical applications we can foresee are explicitly worked out and tabulated

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

generating functions for g2 characters and subgroup branching rules the g2 character generator is given with its help generating functions are derived for branching rules for g2 irreducible representations reduced according to its maximal semisimple subgroups

lie group weight multiplicities from conformal field theory dominant weight multiplicities of simple lie groups are expressed in terms of the modular matrices of wess zumino witten conformal field theories and related objects symmetries of the modular matrices give rise to new relations among multiplicities at least for some lie groups these new relations are strong enough to completely fix all multiplicities

composite young diagrams supercharacters of u m n and modification rules the supercharacter of u m n associated with an arbitrary composite young diagram is defined the distinction is made between standard and non standard supercharacters a modification rule is presented which may be used to express any non standard supercharacter in terms of standard supercharacters in the case n 0 the rule reduces to the rule already known to be appropriate to u m

representations of osp m 2n and young supertableaux young supertableaux are introduced for finite dimensional representations of orthosymplectic superalgebras and used to study their content

quantum game players can have advantage without discord the last two decades have witnessed a rapid development of quantum information processing a new paradigm which studies the power and limit of quantum advantages in various information processing tasks problems such as when quantum advantage exists and if existing how much it could be are at a central position of these studies in a broad class of scenarios there are implicitly or explicitly at least two parties involved who share a state and the correlation in this shared state is the key factor to the efficiency under concern in these scenarios the shared entanglement or discord is usually what accounts for quantum advantage in this paper we examine a fundamental problem of this nature from the perspective of game theory a branch of applied mathematics studying selfish behaviors of two or more players we exhibit a natural zero sum game in which the chance for any player to win the game depends only on the ending correlation we show that in a certain classical equilibrium a situation in which no player can further increase her payoff by any local classical operation whoever first uses a quantum computer has a big advantage over its classical opponent the equilibrium is fair to both players and as a shared correlation it does not contain any discord yet a quantum advantage still exists this indicates that at least in game theory the previous notion of discord as a measure of non classical correlation needs to be reexamined when there are two players with different objectives

mixed supertableaux of the superunitary groups ii su n m the young supertableaux of the superunitary groups are studied we give the conditions on the size of legal supertableaux the type of atypicity for tensor representations which are irreducible making the link between the kac dynkin parameters and those of the supertableau a discussion is given on nonfully reducible representations and generalized atypical supertableaux

young diagrams supercharacters of osp m n and modification rules the supercharacter of osp m n associated with an arbitrary young diagram is defined the distinction is made between osp m n standard and non standard supercharacters the corresponding modification rule which may be used to express a non standard supercharacter in terms of standard supercharacters is presented exemplified and proved this rule involves the removal of n 2 1 boundary strips from the young diagram in the case n 0 the rule reduces to the well known rule appropriate to o m for a non standard supercharacter corresponding to a typical irreducible representation of osp m n the modification yields a single typical standard supercharacter on the other hand for a non standard supercharacter corresponding to an atypical irreducible representation the rule yields a linear combination of atypical standard supercharacters

generating functions and elementary young tableaux when constructing polynomial bases for group representations one is led naturally to consider a stretched product of states elementary multiplets are states that cannot be expressed as the stretched product of any other two states and they can be identified with terms occurring in certain generating functions in this paper the authors define an analogous stretched product of young tableaux together with elementary tableaux these may be used in a manner similar to elementary multiplets to construct generating functions they consider several examples where this method can be applied and in particular give new large n rank free branching rule generating functions for the group subgroup pairs su n su m contained in implied by su n m first four labels non zero so n contained in implied by su n first three labels non zero and sp 2n contained in implied by su 2n first five labels non zero this method of finding these generating functions uses the theory of grobner bases which allows the authors to make precise the notion of compatibility relations or forbidden products of elementary tableaux

gromov witten invariants on grassmannians we prove that any three point genus zero gromov witten invariant on a type a grassmannian is equal to a classical intersection number on a two step flag variety we also give symplectic and orthogonal analogues of this result in these cases the two step flag variety is replaced by a sub maximal isotropic grassmannian our theorems are applied in type a to formulate a conjectural quantum littlewood richardson rule and in the other classical lie types to obtain new proofs of the main structure theorems for the quantum cohomology of lagrangian and orthogonal grassmannians

extended poincare supersymmetry rotation groups and branching rules the decomposition of the basic spin irreps of the special orthogonal group so2k into irreps of sod 2 k where d is the spacetime dimension of the extended d dimensional poincare supersymmetry and k is the appropriate automorphism group is determined general results for d 10 several new branching rules for subgroups of so2k are developed a number of specific results are tabulated

representations of supergroups an explicit construction of representations of supergroups is given in terms of direct products of covariant and contravariant fundamental representations the rules of supersymmetrization are characterized by extended young supertableaux this constructive approach leads to explicit transformation properties of higher representations as well as to closed explicit formulas for characters from which other invariants such as dimensions and eigenvalues of all casimir operators can be calculated we have applied this approach so far to the supergroups su n m osp n 2m p n for which we have obtained all the representations constructible as direct products of the fundamental defining representations an argument is presented toward the irreducibility of all these representations

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

cylindric skew schur functions cylindric skew schur functions which are a generalisation of skew schur functions arise naturally in the study of p partitions also recent work of a postnikov shows they have a strong connection with a problem of considerable current interest that of finding a combinatorial proof of the non negativity of the 3 point gromov witten invariants after explaining these motivations we study cylindric skew schur functions from the point of view of schur positivity using a result of i gessel and c krattenthaler we generalise a formula of a bertram i ciocan fontanine and w fulton thus giving an expansion of an arbitrary cylindric skew schur function in terms of skew schur functions while we show that no non trivial cylindric skew schur functions are schur positive we conjecture that this can be reconciled using the new concept of cylindric schur positivity

upper bound on the characters of the symmetric groups let c be a conjugacy class in the symmetric group s n and be a partition of n let f be the degree of the irreducible representation s c the character of s at c and r c the normalized character c f we prove that there exist constants b 0 and 1 q 0 such that for n 4 for every conjugacy class c in s n and every irreducible representation s of s n r c max q 1 n 1 n b supp c where supp c is the number of non fixed digits under the action of a permutation in c 1 is the size of the largest part in and 1 is the number of parts in the proof is obtained by enumeration of rim hook tableaux the hook formula and probabilistic arguments combinatorial algebraic and statistical applications follow this result in particular we estimate the rate of mixing of random walks on the alternating groups with respect to conjugacy classes

strengths and weaknesses of quantum computing recently a great deal of attention has been focused on quantum computation following a sequence of results bernstein and vazirani in proc 25th annual acm symposium theory comput 1993 pp 11 20 siam j comput 26 1997 pp 1277 1339 simon in proc 35th annual ieee symposium foundations comput sci 1994 pp 116 123 siam j comput 26 1997 pp 1340 1349 shor in proc 35th annual ieee symposium foundations comput sci 1994 pp 124 134 suggesting that quantum computers are more powerful than classical probabilistic computers following shor s result that factoring and the extraction of discrete logarithms are both solvable in quantum polynomial time it is natural to ask whether all of np can be efficiently solved in quantum polynomial time in this paper we address this question by proving that relative to an oracle chosen uniformly at random with probability 1 the class np cannot be solved on a quantum turing machine qtm in time o 2 n 2 we also show that relative to a permutation oracle chosen uniformly at random with probability 1 the class np cap conp cannot be solved on a qtm in time o 2 n 3 the former bound is tight since recent work of grover in it proc 28 th annual acm symposium theory comput 1996 shows how to accept the class np relative to any oracle on a quantum computer in time o 2 n 2

standard young tableaux and weight multiplicities of the classical lie groups by examining the branching rules for all irreducible representations of the classical groups u k su k so 2k 1 sp 2k and so 2k on restriction to u 1 u 1 u 1 standard young tableaux are specified for each of these groups it is shown that these tableaux determine the corresponding characters of the irreducible representations the rules for constructing these tableaux are derived and in this way the determination of weight multiplicities is reduced to a simple combinatorial exercise general formula for such weight multiplicities are given encompassing the most difficult case namely that of so 2k illustrative examples are provided including some yielding the explicit k dependence of weight multiplicities

new example of a generating function for wznw fusion rules the authors review the construction of generating functions for wznw wess zumino novikov witten fusion rules and present the explicit form of the generating function for so 5 in addition to providing a compact and manifestly positive expression for the so 5 fusion rules at all levels this result illustrates a general conjecture on the relation between the generating function for wznw fusion rules and that for finite lie algebra tensor products

branching rules and even dimensional rotation groups so2k unambiguous methods are developed for calculating branching rules for the classical subgroups of the even dimensional rotation group so2k complete results are given for the subgroups suk u1 so2k 2 u1 so2p so2q and so2p 1 so2q 1 a number of examples relevant to problems in supergravity and unification theories are given a complete resolution of the antisymmetric powers of the basic spinor irrep of so10 is given and the results extended to so11

representations of orthosymplectic superalgebras ii young diagrams and weight space techniques for pt i see ibid vol 16 p 473 1983 finite dimensional graded tensor representations of osp m n are enumerated via standard young diagrams and their corresponding highest weight and kac dynkin labels are given a uniform set of conditions on the diagram shape necessary and sufficient for atypicality are presented weight space techniques are used to provide a complete analysis of an atypical representations for the low rank cases osp 2 2 osp 3 2 and osp 4 2

higher genus su n fusion multiplicities as polytope volumes we show how higher genus su n fusion multiplicities may be computed as the discretized volumes of certain polytopes the method is illustrated by explicit analyses of some su 3 and su 4 fusions but applies to all higher point and higher genus su n fusions it is based on an extension of the realm of berenstein zelevinsky triangles by including so called gluing and loop gluing diagrams the identification of the loop gluing diagrams is our main new result since they enable us to characterize higher genus fusions in terms of polytopes also the genus 2 0 point su 3 fusion multiplicity is found to be a simple binomial coefficient in the affine level

the efficient determination of the percolation threshold by a frontier generating walk in a gradient the frontier in gradient percolation is generated directly by a type of self avoiding random walk the existence of the gradient permits one to generate an infinite walk on a computer of finite memory from this walk the percolation threshold pc for a two dimensional lattice can be determined with apparently maximum efficiency for a naive monte carlo calculation or n 12 for a square lattice the value pc 0 592745 or 0 000002 is found for a simulation of n 2 6 1011 total steps occupied and blocked perimeter sites the power of the method is verified on the kagome site percolation case

new lattice based cryptographic constructions we introduce the use of fourier analysis on lattices as an integral part of a lattice based construction the tools we develop provide an elegant description of certain gaussian distributions around lattice points our results include two cryptographic constructions that are based on the worst case hardness of the unique shortest vector problem the main result is a new public key cryptosystem whose security guarantee is considerably stronger than previous results o n1 5 instead of o n7 this provides the first alternative to ajtai and dwork s original 1996 cryptosystem our second result is a family of collision resistant hash functions with an improved security guarantee in terms of the unique shortest vector problem surprisingly both results are derived from one theorem that presents two indistinguishable distributions on the segment 0 1 it seems that this theorem can have further applications as an example we use it to solve an open problem in quantum computation related to the dihedral hidden subgroup problem

lagrangean heuristics for location problems abstract in this paper we present a framework for developing lagrangean heuristics heuristics based upon lagrangean relaxation and subgradient optimisation with respect to location problems computational results are given for four different location problems p median uncapacitated warehouse location capacitated warehouse location and capacitated warehouse location with single source constraints these results indicate that the framework presented in this paper is robust i e it gives good quality solutions for each of these different location problems

an exact algorithm for maximum entropy sampling we study the experimental design problem of selecting a most informative subset having prespecified size from a set of correlated random variables the problem arises in many applied domains such as meteorology environmental statistics and statistical geology in these applications observations can be collected at different locations and possibly at different times information is measured by entropy in the gaussian case the problem is recast as that of maximizing the determinant of the covariance matrix of the chosen subset we demonstrate that this problem is np hard we establish an upper bound for the entropy based on the eigenvalue interlacing property and we incorporate this bound in a branch and bound algorithm for the exact solution of the problem we present computational results for estimated covariance matrices that correspond to sets of environmental monitoring stations in the united states

branching rules for e8 so16 branching rules for e8 down arrow so16 are derived from those of e8 down arrow su2 e7 down arrow su8 and so16 down arrow u1 su8 by noting that both group chains share a common u1 su8 subgroup additional branching rules are deduced from kronecker products of e8 and so16 irreps in each case the rules distinguish unambiguously between conjugate irreps of so16 an alternative labelling scheme for the irreps of so2k based on its maximal u1 suk subgroup is outlined

the demazure tits subgroup of a simple lie group the demazure tits subgroup of a simple lie group g is the group of invariance of clebsch gordan coefficients tables assuming an appropriate choice of basis the structure of the demazure tits subgroups of an bn cn dn and g2 is described orbits of the permutation action of the dt group in any irreducible finite dimensional representation space of a2 c2 and g2 are decomposed into the sum of irreducible representations of the dt group

internal labels of degenerate representations an expression for the number of internal labels of degenerate irreducible representations of compact semisimple lie groups is given in the same spirit as racah s formula for the nondegenerate case

a subexponential time quantum algorithm for the dihedral hidden subgroup problem we present a quantum algorithm for the dihedral hidden subgroup problem dhsp with time and query complexity 2 o sqrt log n in this problem an oracle computes a function f on the dihedral group d n which is invariant under a hidden reflection in d n by contrast the classical query complexity of dhsp is o sqrt n the algorithm also applies to the hidden shift problem for an arbitrary finitely generated abelian group the algorithm begins as usual with a quantum character transform which in the case of d n is essentially the abelian quantum fourier transform this yields the name of a group representation of d n which is not by itself useful and a state in the representation which is a valuable but indecipherable qubit the algorithm proceeds by repeatedly pairing two unfavorable qubits to make a new qubit in a more favorable representation of d n once the algorithm obtains certain target representations direct measurements reveal the hidden subgroup

a geometric description of tensor product decompositions in su 3 the direct sum decomposition of tensor products for su 3 has many applications in physics and the problem has been studied extensively this has resulted in many decomposition methods each with its advantages and disadvantages the description given here is geometric in nature and it describes both the constituents of the direct sum and their multiplicities in addition to providing decompositions of specific tensor products this approach is very well suited to studying tensor products as the parameters vary and helping draw general conclusions after a description and proof of the method several consequences are discussed and proved in particular questions regarding multiplicities are considered

generating function method for fusion rules this is the second of two articles devoted to an exposition of the generating function method for computing fusion rules in affine lie algebras the present paper focuses on fusion rules using the machinery developed for tensor products in the companion article although the kac walton algorithm provides a method for constructing a fusion generating function from the corresponding tensor product generating function we describe a more powerful approach which starts by first defining the set of fusion elementary couplings from a natural extension of the set of tensor product elementary couplings a set of inequalities involving the level are derived from this set using farkas lemma these inequalities taken in conjunction with the inequalities defining the tensor products define what we call the fusion basis given this basis the machinery of our previous paper may be applied to construct the fusion generating function new generating functions for sp 4 and su 4 together with a closed form expres

natural labelling schemes for simple roots and irreducible representations of exceptional lie algebras the simple root system of each exceptional simple lie algebra is explicitly constructed in a variety of forms each construction is based on the natural embedding in the exceptional algebra of a classical semi simple algebra of the same rank the procedure adopted leads to the discovery of a number of new chains of subalgebras illustrated by means of supplemented dynkin diagrams the various sets of simple roots are then used to determine natural labels for the irreducible representations of each of the exceptional algebras for each such labelling scheme the modification rules for dealing with non standard representation labels are tabulated the connection between the natural labels and dynkin labels is given in detail and a comparison is made with the labels of b g wybourne and m j bowick 1977

characters of an 1 hecke algebras at roots of unity a frobenius formula is presented for the characters phi sigma p q of the irreducible representations pi sigma of the an 1 hecke algebras hn q labelled by m k standard partitions sigma where q is a primitive pth root of unity with p m k using this result the characters phi sigma p q may be expressed as linear combinations of the characters chi lambda p q of representations which are irreducible when q is not a root of unity the appropriate linear combinations are found by using fusion modification rules for 1 or n or 5 all the remaining characters of irreducible representations of hn q are then found allowing a complete tabulation to be made for 2 or p or 5 along with the corresponding decomposition matrices

the power of strong fourier sampling quantum algorithms for affine groups and hidden shifts many quantum algorithms including shor s celebrated factoring and discrete log algorithms proceed by reduction to a hidden subgroup problem in which an unknown subgroup h of a group g must be determined from a quantum state psi over g that is uniformly supported on a left coset of h these hidden subgroup problems are typically solved by fourier sampling the quantum fourier transform of psi is computed and measured when the underlying group is nonabelian two important variants of the fourier sampling paradigm have been identified the weak standard method where only representation names are measured and the strong standard method where full measurement i e the row and column of the representation in a suitably chosen basis as well as its name occurs it has remained open whether the strong standard method is indeed stronger that is whether there are hidden subgroups that can be reconstructed via the strong method but not by the weak or any other known method in this article we settle this question in the affirmative we show that hidden subgroups h of the q hedral groups i e semidirect products mathbb z q ltimes mathbb z p where q mid p 1 and in particular the affine groups a p can be information theoretically reconstructed using the strong standard method moreover if h p rm polylog p these subgroups can be fully reconstructed with a polynomial amount of quantum and classical computation we compare our algorithms to two weaker methods that have been discussed in the literature the forgetful abelian method and measurement in a random basis and show that both of these are weaker than the strong standard method thus at least for some families of groups it is crucial to use the full power of representation theory and nonabelian fourier analysis namely to measure the high dimensional representations in an adapted basis that respects the group s subgroup structure we apply our algorithm for the hidden subgroup problem to new families of cryptographically motivated hidden shift problems generalizing the work of van dam hallgren and ip on shifts of multiplicative characters finally we close by proving a simple closure property for the class of groups over which the hidden subgroup problem can be solved efficiently

the data correcting algorithm for the minimization of supermodular functions the data correcting dc algorithm is a recursive branch and bound type algorithm in which the data of a given problem instance are heuristically corrected at each branching in such a way that the new instance will be as close as possible to polynomially solvable and the result satisfies a prescribed accuracy the difference between optimal and current solution in this paper the dc algorithm is applied to determining exact or approximate global minima of supermodular functions the working of the algorithm is illustrated by an instance of the simple plant location spl problem computational results obtained for the quadratic cost partition problem qcp show that the dc algorithm outperforms a branch and cut algorithm not only for sparse graphs but also for nonsparse graphs with density more than 40 often with speeds 100 times faster

relation between the supertableaux of the supergroups osp 2 2 and su 1 2 the classification and the interpretation of the young supertableaux of the orthosymplectic group osp 2 2 are given a comparison is made with the supertableaux of the superunitary group su 1 2 taking advantage of the isomorphism of the corresponding superalgebras

the weyl groups and weight multiplicities of the exceptional lie groups the weyl group wg of each exceptional simple lie group g is described in detail its structure is defined in terms of its coset decomposition with respect to the weyl group wh of a classical semi simple lie group h embedded naturally in g the concepts of g dominance and g equivalence are defined and used to determine from the character formula of weyl the branching rule associated with the restriction of group elements from g to h the weyl group wg is used further to impose constraints on both the branching multiplicities for g to h and the weight multiplicities of g these constraints are used to evaluate the weight multiplicities of f4 e6 e7 and e8 together with the branching multiplicities for e8 to so 16

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

an application of combinatorial optimization to statistical physics and circuit layout design we study the problem of finding ground states of spin glasses with exterior magnetic field and the problem of minimizing the number of vias holes on a printed circuit board or contacts on a chip subject to pin preassignments and layer preferences the former problem comes up in solid state physics and the latter in very large scale integrated vlsi circuit design and in printed circuit board design both problems can be reduced to the max cut problem in graphs based on a partial characterization of the cut polytope we design a cutting plane algorithm and report on computational experience with it our method has been used to solve max cut problems on graphs with up to 1 600 nodes

kronecker products for compact semisimple lie groups a review is given of the application of s function techniques to the evaluation of kronecker products of irreducible representations of compact semisimple lie groups explicit formulae are derived for all irreducible representations of all such groups recent developments involving composite young diagrams are brought to fruition and the vexed problem of so2k is dealt with completely new branching rules for the classical groups are given in an appendix these are exploited in the evaluation of kronecker products by means of a technique which is applied to both classical and exceptional groups a discussion is made of various modification rules which are needed to express the final results in standard form

limitations of quantum coset states for graph isomorphism it has been known for some time that graph isomorphism reduces to the hidden subgroup problem hsp what is more most exponential speedups in quantum computation are obtained by solving instances of the hsp a common feature of the resulting algorithms is the use of quantum coset states which encode the hidden subgroup an open question has been how hard it is to use these states to solve graph isomorphism it was recently shown by moore et al 2005 that only an exponentially small amount of information is available from one or a pair of coset states a potential source of power to exploit are entangled quantum measurements that act jointly on many states at once we show that entangled quantum measurements on at least n log n coset states are necessary to get useful information for the case of graph isomorphism matching an information theoretic upper bound this may be viewed as a negative result because in general it seems hard to implement a given highly entangled measurement our main theorem is very general and also rules out using joint measurements on few coset states for some other groups such as gl n fpm and gn where g is finite and satisfies a suitable property

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

maximizing a submodular function by integer programming polyhedral results for the quadratic case abstract we give a new mixed integer programming mip formulation for the quadratic cost partition problem that is derived from a mip formulation for maximizing a submodular function several classes of valid inequalities for the convex hull of the feasible solutions are derived using the valid inequalities for the node packing polyhedron facet defining conditions and separation algorithms are discussed and computational results are reported

generating functions for su 2 plethysms with fixed exchange symmetry a number of fixed plethysm generating are given for su 2 a fixed plethysm generating function gives the content of the component with definite exchange symmetry of the direct product of a given number of copies of an irreducible representation rl with l running through all values new symmetries are found relating antisymmetric to symmetric products and relating plethysms in which the number of factors is interchanged with the factor representation label l expressions for two box plethysm generating function for su 3 and for fixed plethysm generating functions for su 2 based on reducible representations are also given

the hidden subgroup problem and quantum computation using group representations the hidden subgroup problem is the foundation of many quantum algorithms an efficient solution is known for the problem over abelian groups employed by both simon s algorithm and shor s factoring and discrete log algorithms the nonabelian case however remains open an efficient solution would give rise to an efficient quantum algorithm for graph isomorphism we fully analyze a natural generalization of the algorithm for the abelian case to the nonabelian case and show that the algorithm determines the normal core of a hidden subgroup in particular normal subgroups can be determined we show however that this immediate generalization of the abelian algorithm does not efficiently solve graph isomorphism

multi level facility location as the maximization of a submodular set function abstract in this paper we model the multi level uncapacitated facility location problem as two different combinatorial optimization problems the first model is the classical representation of the problem which uses a set of vertices as combinatorial objects to represent solutions whereas in the second model we propose the use of a set of paths an interesting observation is that the real valued set function associated with the first combinatorial problem does not satisfy the submodular property whereas the set function associated with the second problem does satisfy this property this illustrates the fact that submodularity is not a property intrinsic to an optimization problem but rather to its mathematical representation

higher su n tensor products we extend our recent results on ordinary su n tensor product multiplicities to higher su n tensor products particular emphasis is put on four point couplings where the tensor product of four highest weight modules is considered the number of times the singlet occurs in the decomposition is the associated multiplicity in this framework ordinary tensor products correspond to three point couplings as in that case the four point multiplicity may be expressed explicitly as a multiple sum measuring the discretized volume of a convex polytope this description extends to higher point couplings as well we also address the problem of determining when a higher point coupling exists i e when the associated multiplicity is non vanishing the solution is a set of inequalities in the dynkin labels

constrained maximum entropy sampling a fundamental experimental design problem is to select a most informative subset having prespecified size from a set of correlated random variables instances of this problem arise in many applied domains such as meteorology environmental statistics and statistical geology in these applications observations can be collected at different locations and possibly at different times information is measured by entropy practical situations have further restrictions on the design space for example budgetary limits geographical considerations as well as legislative and political considerations may restrict the design space in a complicated manner using techniques of linear algebra combinatorial optimization and convex optimization we develop upper and lower bounds on the optimal value for the gaussian case we describe how these bounds can be integrated into a branch and bound algorithm for the exact solution of these design problems finally we describe how we have implemented this algorithm and we present computational results for estimated covariance matrices corresponding to sets of environmental monitoring stations in the ohio valley of the united states

on character generators for simple lie algebras we study character generating functions character generators of simple lie algebras the expression due to patera and sharp derived from the weyl character formula is first reviewed a new general formula is then found it makes clear the distinct roles of outside and inside elements of the integrity basis and helps determine their quadratic incompatibilities we review analyse and extend the results obtained by gaskell using the demazure character formulae we find that the fundamental generalized poset graphs underlying the character generators can be deduced from such calculations these graphs introduced by baclawski and towber can be simplified for the purposes of constructing the character generator the generating functions can be written easily using the simplified graphs and associated demazure expressions the rank two algebras are treated in detail but we believe our results are indicative of those for general simple lie algebras

n point and higher genus osp 1 2 fusion we study affine osp 1 2 fusion the fusion in osp 1 2 conformal field theory for example higher point and higher genus fusion is discussed the fusion multiplicities are characterized as discretized volumes of certain convex polytopes and are written explicitly as multiple sums measuring those volumes we extend recent methods developed to treat affine su 2 fusion they are based on the concept of generalized berenstein zelevinsky triangles and virtual couplings higher point tensor products of finite dimensional irreducible osp 1 2 representations are also considered the associated multiplicities are computed and written as multiple sums

maximization of submodular functions theory and enumeration algorithms abstract submodular functions are powerful tools to model and solve either to optimality or approximately many operational research problems including problems defined on graphs after reviewing some long standing theoretical results about the structure of local and global maxima of submodular functions cherenin s selection rules and his dichotomy algorithm we revise the above mentioned theory and show that our revision is useful for creating new non binary branching algorithms and finding either approximation solutions with guaranteed accuracy or exact ones

fusion bases as facets of polytopes a new way of constructing fusion bases i e the set of inequalities governing fusion rules out of fusion elementary couplings is presented it relies on a polytope reinterpretation of the problem the elementary couplings are associated with the vertices of the polytope while the inequalities defining the fusion basis are the facets the symmetry group of the polytope associated with the lowest rank affine lie algebras is found it has order 24 for su 2 432 for su 3 and quite surprisingly it reduces to 36 for su 4 while it is only of order 4 for sp 4 this drastic reduction in the order of the symmetry group as the algebra gets more complicated is rooted in the presence of many linear relations between the elementary couplings that break most of the potential symmetries for su 2 and su 3 it is shown that the fusion basis defining inequalities can be generated from few one and two respectively elementary ones for su 3 new symmetries of the fusion coefficients are found

quantum computation and lattice problems we present the first explicit connection between quantum computation and lattice problems namely our main result is a solution to the unique shortest vector problem svp under the assumption that there exists an algorithm that solves the hidden subgroup problem on the dihedral group by coset sampling additionally we present an approach to solving the hidden sub group problem on the dihedral group by using an average case subset sum routine

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

quantum cohomology and the k schur basis we prove that structure constants related to hecke algebras at roots of unity are special cases of k littlewood richardson coefficients associated to a product of k schur functions as a consequence both the 3 point gromov witten invariants appearing in the quantum cohomology of the grassmannian and the fusion coefficients for the wzw conformal field theories associated to su l are shown to be k littlewood richardson coefficients from this mark shimozono conjectured that the k schur functions form the schubert basis for the homology of the loop grassmannian whereas k schur coproducts correspond to the integral cohomology of the loop grassmannian we introduce dual k schur functions defined on weights of k tableaux that given shimozono s conjecture form the schubert basis for the cohomology of the loop grassmannian we derive several properties of these functions that extend those of skew schur functions

standard young tableaux and character generators of classical lie groups character generators are derived for each of the classical groups su k so 2k 1 sp 2k and so 2k in order to do this use is made of various generalisations of standard young tableaux one set of these is in one to one correspondence with the contributions to the characters of each irreducible representation and the other involving shifted young diagrams provides a means of enumerating those maximal chains in the fundamental posets of each group which serve to generate via their descent subchains all possible multichains the character generators are then written down in terms of a suitable labelling of the poset elements illustrative examples are given

character generators for unitary and symplectic groups combinatorial algorithms for computing the character generators of u n su n and sp 2n c are described these algorithms produce relatively compact nested expressions for the character generators moreover the terms appearing in these expressions all have positive coefficients this feature is not shared by the expression for the character generator which uses the weyl character formula

affine su 3 and su 4 fusion multiplicities as polytope volumes affine su 3 and su 4 fusion multiplicities are characterized as discretized volumes of certain convex polytopes the volumes are measured explicitly resulting in multiple sum formulae these are the first polytope volume formulae for higher rank fusion multiplicities the associated threshold levels are also discussed for any simple lie algebra we derive an upper bound on the threshold levels using a refined version of the gepner witten depth rule