integration of quantum many body problems by affine knizhnik zamolodchikov equations abstract an equivalence of the affine analogues of knizhnik zamolodchikov and dunkl equations is established for arbitrary root systems as a corollary an isomorphism is obtained from the solutions of the above kz equations in the induced representations of hecke algebras to the eigenfunctions for the corresponding matrix quantum calogero sutherland operators which generalizes the matsuo map

static modules stable clifford theory and morita similarities of rings abstract in this paper static modules and stable clifford theory are used to obtain a necessary and sufficient condition for rings to be morita similar simultaneously some equivalences of module categories are also derived

homomorphisms representations and characteristic polynomials of digraphs the existence of a homomorphism between two digraphs often implies many structural properties we collect in this paper some characterizations of various digraph homomorphisms using matrix equations and fiber partitions we also survey the relationship among the characteristic polynomials of a digraph and its divisors this includes an introduction of the concept of branched coverings of digraphs their voltage assignment representations and a decomposition formula for the characteristic polynomial of a branched cover with branch index 1 some open problems are included

exact operator solution of the calogero sutherland model the wave functions of the calogero sutherland model are known to be expressible in terms of jack polynomials a formula which allows to obtain the wave functions of the excited states by acting with a string of creation operators on the wave function of the ground state is presented and derived the creation operators that enter in this formula of rodrigues type for the jack polynomials involve dunkl operators

orthogonality of jack polynomials in superspace abstract jack polynomials in superspace orthogonal with respect to a combinatorial scalar product are constructed they are shown to coincide with the jack polynomials in superspace orthogonal with respect to an analytical scalar product introduced in p desrosiers l lapointe p mathieu jack polynomials in superspace comm math phys 242 2003 331 360 as eigenfunctions of a supersymmetric quantum mechanical many body problem the results of this article rely on generalizing to include an extra parameter the theory of classical symmetric functions in superspace developed recently in p desrosiers l lapointe p mathieu classical symmetric functions in superspace j algebraic combin 24 2006 209 238

static modules and stable clifford theory the object of this paper is to express an important result of dade 2 theorem 7 41 on equivalence of categories from classical stable clifford theory to one in pure ring theory in fact in 2 utilising the properties of fully group graded rings and modules dade described an extended version of cline s stable clifford theory i in the sequel we will work in the category of right d modules where d is an arbitrary ring in our work no group action is involved and the rings and modules are free from any group grading our work is based on a type of module called a static module related to a fixed d module jy or d static d module definition 2 1 if d and s are rings and p 02 3 is an identity preserving ring homomorphism then we will be interested in those b modules which are a static as modules under the restriction of p a source of interest is the following example let y be a group x a normal subgroup of 9 and f any field let j be a g stable s s module then the 9 3 modules of interest here are exactly those which are x static as f x modules our first main theorem theorem 4 9 below shows that the category of all such modules is equivalent to a certain category of d modules where in many cases the latter category is much easier to deal with our second main theorem theorem 5 5 below shows that the category whose objects are those f zj modules which weakly divide as b x modules is equivalent to the category whose objects are those modules which are projective of finite type as modules where 9 end c x m

a unification of knizhnik zamolodchikov and dunkl operators via affine hecke algebras some generalizations of the lusztig lascoux schutzenberger operators for affine hecke algebras are considered as corollaries we obtain lusztig s isomorphisms from affine hecke algebras to their degenerate versions a natural interpretation of the dunkl operators and a new class of differential difference operators generalizing dunkl s ones and the knizhnik zamolodchikov operators from the two dimensional conformal field theory

n symmetric superpolynomials the theory of symmetric functions has been extended to the case where each variable is paired with an anticommuting one the resulting expressions dubbed superpolynomials provide the natural n 1 supersymmetric version of the classical bases of symmetric functions here we consider the case where more than one independent anticommuting variable is attached to each ordinary variable first the n 2 super version of the monomial elementary homogeneous symmetric functions as well as the power sums is constructed systematically using an exterior differential formalism for the multiplicative bases these functions being now indexed by a novel type of superpartitions moreover the scalar product of power sums turns out to have a natural n 2 generalization which preserves the duality between the monomial and homogeneous bases all these results are then generalized to an arbitrary value of n finally for n 2 the scalar product and the homogeneous functions are shown to have a one parameter deformation

regular graphs with four eigenvalues we study the connected regular graphs with four distinct eigenvalues properties and feasibility conditions of the eigenvalues are found several examples constructions and characterizations are given as well as some uniqueness and nonexistence results

investigation of continuous time quantum walk via modules of bose mesner and terwilliger algebras the continuous time quantum walk on the underlying graphs of association schemes has been studied via the algebraic combinatorics structures of association schemes namely semi simple modules of their bose mesner and terwilliger algebras it is shown that the terwilliger algebra stratifies the graph into a d 1 disjoint union of strata which is different from the stratification based on distance except for distance regular graphs in underlying graphs of association schemes the probability amplitudes and average probabilities are given in terms of dual eigenvalues of association schemes such that the amplitudes of observing the continuous time quantum walk on all sites belonging to a given stratum are the same therefore there are at most d 1 different observing probabilities the importance of association scheme in continuous time quantum walk is shown by some worked out examples such as arbitrary finite group association schemes followed by symmetric sn dihedral d2m and cyclic groups at the end it is shown that the highest irreducible representations of terwilliger algebras pave the way to use the spectral distributions method of jafarizadeh and salimi 2005 preprint quant ph 0510174 in studying quantum walk on some rather important graphs called distance regular graphs

equitable switching and spectra of graphs abstract let g be a simple graph and an equitable partition of the vertex set v g it is well known that the characteristic polynomial g of a quotient graph g divides that of g the seidel swiching g of g with respect to a proper subset v of v g is called equitable if v is a union of some cells in we show that if g is equitable g g g g some applications of this theorem to generalized composition graphs isospectral graphs integral graphs and ramanujan graphs are contained

some combinatorial properties of jack symmetric functions if 1 l n then write 2 n or 1 n if p is another partition then write p c j if pi 6 for all i i e if the diagram of 1 contains the diagram of p if ipl ii 1 then write p 2 1 reverse lexicographic order if either j i or the first nonvanishing difference e pi is positive for instance writing 2 1 i for ai i it 5 5 41 32 5 311 221 2111 11111 finally write p i if ipl lal and p1 pl2 pli a1 a2 li for all i macdonald m p 63 call the partial ordering the natural ordering but we will call it the dominance ordering let x x x2 be an infinite set of indeterminates as in mi we

explicit formulae for the generalized hermite polynomials in superspace we provide explicit formulae for the orthogonal eigenfunctions of the supersymmetric extension of the rational calogero moser sutherland model with harmonic confinement i e the generalized hermite or hi jack polynomials in superspace the construction relies on the triangular action of the hamiltonian on the supermonomial basis this translates into determinantal expressions for the hamiltonian s eigenfunctions

extensions and applications of equitable decompositions for graphs with symmetries abstract we extend the theory of equitable decompositions introduced in 2 where it was shown that if a graph has a particular type of symmetry i e a uniform or basic automorphism it is possible to use to decompose a matrix m appropriately associated with the graph the result is a number of strictly smaller matrices whose collective eigenvalues are the same as the eigenvalues of the original matrix m we show here that a large class of automorphisms which we refer to as separable can be realized as a sequence of basic automorphisms allowing us to equitably decompose m over any such automorphism we also show that not only can a matrix m be decomposed but that the eigenvectors of m can also be equitably decomposed additionally we prove under mild conditions that if a matrix m is equitably decomposed the resulting divisor matrix which is the divisor matrix of the associated equitable partition will have the same spectral radius as the original matrix m last we describe how an equitable decomposition effects the gershgorin region m of a matrix m which can be used to localize the eigenvalues of m we show that the gershgorin region of an equitable decomposition of m is contained in the gershgorin region m of the original matrix we demonstrate on a real world network that by a sequence of equitable decompositions it is possible to significantly reduce the size of a matrix gershgorin region

distance regularity and the spectrum of graphs abstract we deal with the question can one see from the spectrum of a graph whether it is distance regular or not up till now the answer has not been known when has precisely four distinct eigenvalues the diameter 3 case we show that in this case the answer is negative we also give positive answers in some special situations for instance if has the spectrum of a distance regular graph with diameter 3 and 1 then is distance regular our main tools are eigenvalue techniques for partitioned matrices

weighted zeta functions of digraphs abstract we define a weighted zeta function of a digraph and a weighted l function of a symmetric digraph and give determinant expressions of them furthermore we give a decomposition formula for the weighted zeta function of a g cyclic cover of a symmetric digraph for any finite group and g a decomposition formula for the weighted zeta function of an oriented line graph l g of a regular covering g of a graph g is given furthermore we define a weighted l function of an oriented line graph l g of g and present a factorization formula for the weighted zeta function of l g by weighted l functions of l g as a corollay we obtain a factorization formula for the multiedge zeta function of g given by stark and terras

spectra of digraph transformations let d be a directed graph without loops or multiple arcs with vertex set v d v and arc set e d e let d0 be the digraph with vertex set v and with no arcs d1 the complete digraph with vertex set v d d and d the complement dc of d for e x y e let x t e and y h e let t d tcb d be the digraph with vertex set v e and such that v e is an arc in t d resp in tcb d if and only if v v e e and vertex v t e resp v t e in d similarly let h d hcb d be the digraph with vertex set v e and such that e v is an arc in h d resp in hcb d if and only if v v e e and vertex v h e resp v h e in d given a digraph d and three variables x y z 0 1 the xyz transformation of d is the digraph dxyz such that dxy0 dx dl y and dxyz dxy0 w where w t d h d if z w tcb d hcb d if z and w is the complete bipartite digraph with parts v and e if z 1 in this paper we obtain the adjacency characteristic polynomials of some xyz transformations of an r regular digraph d in terms of r the number of vertices and the adjacency spectrum of d similar results are obtained for some non regular digraphs named digraph functions we also give various constructions of non isomorphic cospectral digraphs using xyz transformations our notion of xyz transformation is also valid for digraphs with multiple arcs and loops if x y z 0 and z 0 1 we also extend the notion of xyz transformation to digraphs v e with loops and no multiple arcs

supersymmetric ruijsenaars schneider model an integrable supersymmetric generalization of the trigonometric ruijsenaars schneider model is presented whose symmetry algebra includes the super poincar e algebra moreover its hamiltonian is shown to be diagonalized by the recently introduced macdonald superpolynomials somewhat surprisingly the consistency of the scalar product forces the discreteness of the hilbert space

characteristic polynomials of graphs having a semifree action j h kwak and j lee linear and multilinear algebra 32 1992 61 73 computed the characteristic polynomial of a finite graph g having an abelian automorphism group which acts freely on g for a finite weighted symmetric pseudograph g having an abelian automorphism group which acts semifreely on g k wang linear algebra appl 51 1983 121 125 showed that the characteristic polynomial of g is factorized into a product of a polynomial associated to the orbit graph and a polynomial associated to the free part of the action but he did not explicitly compute the characteristic polynomial of such a graph g in this paper we introduce a new method to construct a finite pseudograph g having an automorphism group which acts semifreely on g and obtain an explicit formula to compute the characteristic polynomial of such a graph by using the construction method 2000 elsevier science inc all rights reserved

interlacing eigenvalues and graphs abstract we give several old and some new applications of eigenvalue interlacing to matrices associated to graphs bounds are obtained for characteristic numbers of graphs such as the size of a maximal co clique the chromatic number the diameter and the bandwidth in terms of the eigenvalues of the standard adjacency matrix or the laplacian matrix we also deal with inequalities and regularity results concerning the structure of graphs and block designs

extended trigonometric cherednik algebras and nonstationary schrodinger equations with delta potentials we realize an extended version of the trigonometric cherednik algebra as affine dunkl operators involving heaviside functions we use the quadratic casimir element of the extended trigonometric cherednik algebra to define an explicit nonstationary schrodinger equation with delta potential we use coordinate bethe ansatz methods to construct solutions of the nonstationary schrodinger equation in terms of generalized bethe wave functions it is shown that the generalized bethe wave functions satisfy affine difference knizhnik zamolodchikov equations as functions of the momenta the relation to the vector valued root system analogs of the quantum bose gas on the circle with delta function interactions is indicated

jack superpolynomials superpartition ordering and determinantal formulas we call superpartitions the indices of the eigenfunctions of the supersymmetric extension of the trigonometric calogero moser sutherland model we obtain an ordering on superpartitions from the explicit action of the model s hamiltonian on monomial superfunctions this allows to define jack superpolynomials as the unique eigenfunctions of the model that decompose triangularly with respect to this ordering on the basis of monomial superfunctions this further leads to a simple and explicit determinantal expression for the jack superpolynomials

zeta functions of digraphs abstract we define a zeta function of a digraph and an l function of a symmetric digraph and give determinant expressions of them furthermore we give a decomposition formula for the zeta function of a g cyclic cover of a symmetric digraph for any finite group and g

characteristic polynomials of graph bundles having voltages in a dihedral group abstract in this paper we compute the characteristic polynomial of a graph bundle when its voltages lie in a dihedral group as the first attempt to compute the characteristic polynomial of a graph bundle also of a graph covering having voltages in a nonabelian group as a result we compute the characteristic polynomial of a graph bundle having a circulant graph as a fibre it is applied for the characteristic polynomials of a discrete torus and a discreteklein bottle

weighted zeta functions for quotients of regular coverings of graphs abstract let g be a connected graph we reformulate stark and terras galois theory for a quotient h of a regular covering k of a graph g by using voltage assignments as applications we show that the weighted bartholdi l function of h associated to the representation of the covering transformation group of h is equal to that of g associated to its induced representation in the covering transformation group of k furthermore we express the weighted bartholdi zeta function of h as a product of weighted bartholdi l functions of g associated to irreducible representations of the covering transformation group of k we generalize stark and terras galois theory to digraphs and apply to weighted bartholdi l functions of digraphs

characteristic polynomials and zeta functions of equitably partitioned graphs abstract let v 1 v r be an equitable partition of the vertex set of a directed graph digraph x it is well known that the characteristic polynomial x x of a quotient graph x divides that of x but the remainder part is not well investigated in this paper we define a deletion graph x over an equitable partition which is a signed directed graph defined for a fixed set of deleting vertices v i v i i 1 r and give a similarity transformation exchanging the adjacency matrix a x which is compatible with the equitable partition for a block triangular matrix whose diagonal blocks are the adjacency matrix of the quotient graph and the deletion graph in fact we show the result for more general matrices including adjacency matrix of graphs and as corollaries we show the followings i a decomposition formula of the reciprocal of the ihara bartholdi zeta function over an equitably partitioned undirected graph into the quotient graph part and the deletion graph part and ii chen and chen s result 4 theorem 3 1 on the ihara bartholdi zeta functions on generalized join graphs and iii teranishi s result 32 theorem 3 3

exchange operator formalism for n body spin models with near neighbours interactions we present a detailed analysis of the spin models with near neighbours interactions constructed in our previous paper enciso et al 2005 phys lett b 605 214 by a suitable generalization of the exchange operator formalism we provide a complete description of a certain flag of finite dimensional spaces of spin functions preserved by the hamiltonian of each model by explicitly diagonalizing the hamiltonian in the latter spaces we compute several infinite families of eigenfunctions of the above models in closed form in terms of generalized laguerre and jacobi polynomials

characteristic polynomials of symmetric graphs abstract in this paper we study the characteristic polynomials of graphs which admit semifree actions of an abelian group using the method of group matrices we are able to show that the characteristic polynomial of a such a graph is factorized into a product of a polynomial associated to the orbit of the action and a polynomial associated to the free part of the action

characteristic polynomials of digraphs having a semi free action abstract let be a semi free action of a group g on a finite digraph the front divisor of has as vertex set the set of the vertex orbits of and there are t arcs going from u to v in if from any vertex u of the orbit u there are t arcs of going towards the vertices in v our main result is that the characteristic polynomial of is a product of the characteristic polynomial of its front divisor and polynomials associated with the free part of under this work extends earlier work of lee and kim characteristic polynomials of graphs having a semifree action linear algebra appl 307 2000 35 46

on the matrix equation al al k jn abstract this paper studies the matrix equation a l a l k j n where l k are nonnegative integers j n is the n n matrix of all l s and a is an unknown 0 1 matrix we shall provide a solution for every odd k and every n which is feasible i e n d l d l k for some nonnegative integer d and show that the equation has no solution in other cases with some trivial cases excluded we also show that for any solution a to this equation there must be a 0 1 matrix c satisfying i c k j d k 1 and a the associated digraph of a is the l th iterated line digraph of c in particular the well known kautz digraph k d l 1 can be characterized as a where a satisfies a l a l 1 j n for n d l d l 1

quantum integrability of the generalized elliptic ruijsenaars models the quantum integrability of the generalized elliptic ruijsenaars models is shown these models are mathematically related to the macdonald operator and the macdonald koornwinder operator which appeared in the q orthogonal polynomial theories we construct these integrable families by using the yang baxter equation and the reflection equation

static modules morita contexts and equivalences abstract the purpose of this paper is to form an interaction between morita contexts and static modules and to establish various equivalences of module categories

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

conserved operators of the generalized elliptic ruijsenaars models we construct the integrable families of the generalized elliptic ruijsenaars models by use of the yang baxter equation and the reflection equation the higher order conserved operators of these models are explicitly presented

calculating two point resistances in distance regular resistor networks an algorithm for the calculation of the resistance between two arbitrary nodes in an arbitrary distance regular resistor network is provided where the calculation is based on stratification introduced in jafarizadeh and salimi 2006 j phys a math gen 39 1 29 and the stieltjes transform of the spectral distribution stieltjes function associated with the network it is shown that the resistances between a node and all nodes belonging to the same stratum with respect to the r i belonging to the ith stratum with respect to the a re the same also the analytical formulae for two point resistances r i i 1 2 3 are given in terms of the size of the network and corresponding intersection numbers in particular the two point resistances in a strongly regular network are given in terms of its parameters v moreover the lower and upper bounds for two point resistances in strongly regular networks are discussed

equitable decompositions of graphs with symmetries abstract we investigate connections between the symmetries automorphisms of a graph and its spectral properties whenever a graph has a symmetry i e a nontrivial automorphism it is possible to use to decompose any matrix m c n n appropriately associated with the graph the result of this decomposition is a number of strictly smaller matrices whose collective eigenvalues are the same as the eigenvalues of the original matrix m some of the matrices that can be decomposed are the graph s adjacency matrix laplacian matrix etc because this decomposition has connections to the theory of equitable partitions it is referred to as an equitable decomposition since the graph structure of many real world networks is quite large and has a high degree of symmetry we discuss how equitable decompositions can be used to effectively bound both the network s spectral radius and spectral gap which are associated with dynamic processes on the network moreover we show that the techniques used to equitably decompose a graph can be used to bound the number of simple eigenvalues of undirected graphs where we obtain sharp results of petersdorf sachs type

zeta functions of finite graphs and coverings iii abstract a graph theoretical analog of brauer siegel theory for zeta functions of number fields is developed using the theory of artin l functions for galois coverings of graphs from parts i and ii in the process we discuss possible versions of the riemann hypothesis for the ihara zeta function of an irregular graph

which graphs are determined by their spectrum for almost all graphs the answer to the question in the title is still unknown here we survey the cases for which the answer is known not only the adjacency matrix but also other types of matrices such as the laplacian matrix are considered

a recursion and a combinatorial formula for jack polynomials the jack polynomials j x form a remarkable class of symmetric polynomials they are indexed by a partition and depend on a parameter one of their properties is that several classical families of symmetric functions can be obtained by specializing e g the monomial symmetric functions m the elementary functions e 0 the schur functions s 1 and nally the two classes of zonal polynomials 2 1 2 the jack polynomials can be de ned in various ways e g a as an orthogonal family of functions which is compatible with the canonical ltration of the ring of symmetric functions or b as simultaneous eigenfunctions of certain di erential operators the sekiguchi debiard operators recently opdam o constructed a similar family f x of nonsymmetric polynomials the index runs now through all compositions nn they are de ned in a completely similar fashion e g the sekiguchi debiard operators are being replaced by the cherednik di erential re ection operators see sect 3 it is becoming more and more clear that these polynomials are as important as their symmetric counterparts the purpose of this paper is to add to the existing characterizations of jack polynomials two further ones c a recursion formula among the f together with two formulas to obtain j from them d combinatorial formulas of both j and f in terms of certain generalized tableaux

group action graphs and parallel architectures the authors develop an algebraic framework that exposes the structural kinship among the debruijn shujcfle exchange butterjy and cube connected cycles networks and illustrate algorithmic benefits that ensue from the exposed relationships the framework builds on two alge braically specified genres of graphs a group action graph gag for short is given by a set v of vertices and a set h of permutations of v for each v e v and each r eii there is an arc labeled r from vertex v to vertex v a cayley graph is a gag v h where v is the group gr h generated by h and where each r 6 h acts on each g 6 gr h by right multiplication the graphs gr h h and v h are called associated graphs it is shown that every gag is a quotient graph of its associated cayley graph by applying such general results the authors determine the following the butterfly network a cayley graph and the debruijn network a gag are associated graphs the cube connected cycles network a cayley graph and the shuffle exchange network a gag are associated graphs the order n instance of both the butterfly and the cube connected cycles share the same underlying group but have slightly different generator sets ii by analyzing these algebraic results it is delimited for any cayley graph g and associated gag 7 a family of leveled algorithms which run as efficiently on t as they do on the much larger g further analysis of the results yields new surprisingly efficient simulations by the shuffle oriented networks the shuffle exchange and debruijn networks of like sized butterfly oriented networks the butterfly and cube connected cycles networks an n vertex butterfly oriented network can be simulated by the smallest shuffle oriented network that is big enough to hold it with slowdown o log log n this simulation is exponentially faster than the anticipated logarithmic slowdown the mappings that underlie the simulation can be computed in linear time and they afford one an algorithmic tech nique for translating any program developed for a butterfly oriented architecture into an equivalent program for a shuffle oriented architecture the latter program incurring only the indicated slowdown factor

jack polynomials in superspace this work initiates the study of orthogonal symmetric polynomials in superspace here we present two approaches leading to a family of orthogonal polynomials in superspace that generalize the jack polynomials the first approach relies on previous work by the authors in which eigenfunctions of the supersymmetric extension of the trigonometric calogero moser sutherland hamiltonian were constructed orthogonal eigenfunctions are now obtained by diagonalizing the first nontrivial element of a bosonic tower of commuting conserved charges not containing this hamiltonian quite remarkably the expansion coefficients of these orthogonal eigenfunctions in the supermonomial basis are stable with respect to the number of variables the second and more direct approach amounts to symmetrize products of non symmetric jack polynomials with monomials in the fermionic variables this time the orthogonality is inherited from the orthogonality of the non symmetric jack polynomials and the value of the norm is given explicitly

invariant subspace determinant and characteristic polynomials abstract making use of an elementary fact on invariant subspace and determinant of a linear map and the method of algebraic identities we obtain a factorization formula for a general characteristic polynomial of a matrix this answers a question posed in a deng i sato y wu characteristic polynomials of ramified uniform covering digraphs european j combin 28 2007 1099 1114 the approach of this work can be used to supply alternative proofs of several other earlier results including some results of y teranishi equitable switching and spectra of graphs linear algebra appl 359 2003 121 131