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727
The CalogeroSutherland Model And Generalized Classical Polynomials
 Comm. Math. Phys
, 1997
"... this paper. The first is the discussion of some mathematical properties relating to the eigenfunctions, while the second is the evaluation of the density in the ground state and the exact solution of (1.6) for certain initial conditions. These problems are in fact interrelated; we find that the den ..."
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Cited by 70 (9 self)
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this paper. The first is the discussion of some mathematical properties relating to the eigenfunctions, while the second is the evaluation of the density in the ground state and the exact solution of (1.6) for certain initial conditions. These problems are in fact interrelated; we find that the density for each system can be written in terms of a certain eigenstate and that a summation theorem for the eigenstates gives an exact solution of (1.6). A feature of the Schrodinger operators (1.2) is that after conjugation with the ground state: \Gamma e i @ \Gamma fi @W (1.7) the resulting differential operator has a complete set of polynomial eigenfunctions. In Section 2 we consider the form of the expansion of these polynomials in terms of some different bases of symmetric functions. We note that in the N = 1 case, after a suitable change of variables, the operator (1.7) with W given by (1.3) is the eigenoperator for the classical Hermite, Laguerre and Jacobi polynomials. Previous studies of the operator for general N in the Jacobi case [1] have established an orthogonality relation. Since the polynomials in the Hermite and Laguerre cases are limiting cases of these generalized Jacobi polynomials, we can obtain the corresponding orthogonality relations via the limiting procedure. The generalized Hermite polynomials, which are the polynomial eigenfunctions of (1.4) with W = W as given by (1.3a), are studied in Section 3. Many higherdimensional analogues of properties of the classical Hermite polynomials are obtained, including a generating function formula, differentiation and integration formulas, a summation theorem and recurrence relations. An analogous study of the generalized Laguerre polynomials is performed in Section 4. In Section 5 we relate the...
Modular Operads
 COMPOSITIO MATH
, 1994
"... We develop a "higher genus" analogue of operads, which we call modular operads, in which graphs replace trees in the definition. We study a functor F on the category of modular operads, the Feynman transform, which generalizes Kontsevich's graph complexes and also the bar construction for operads. ..."
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Cited by 68 (5 self)
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We develop a "higher genus" analogue of operads, which we call modular operads, in which graphs replace trees in the definition. We study a functor F on the category of modular operads, the Feynman transform, which generalizes Kontsevich's graph complexes and also the bar construction for operads. We calculate the Euler characteristic of the Feynman transform, using the theory of symmetric functions: our formula is modelled on Wick's theorem. We give applications to the theory of moduli spaces of pointed algebraic curves.
Brownian Motion in a Weyl Chamber, NonColliding Particles, and Random Matrices
, 1997
"... . Let n particles move in standard Brownian motion in one dimension, with the process terminating if two particles collide. This is a specific case of Brownian motion constrained to stay inside a Weyl chamber; the Weyl group for this chamber is An\Gamma1 , the symmetric group. For any starting posit ..."
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Cited by 66 (2 self)
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. Let n particles move in standard Brownian motion in one dimension, with the process terminating if two particles collide. This is a specific case of Brownian motion constrained to stay inside a Weyl chamber; the Weyl group for this chamber is An\Gamma1 , the symmetric group. For any starting positions, we compute a determinant formula for the density function for the particles to be at specified positions at time t without having collided by time t. We show that the probability that there will be no collision up to time t is asymptotic to a constant multiple of t \Gamman(n\Gamma1)=4 as t goes to infinity, and compute the constant as a polynomial of the starting positions. We have analogous results for the other classical Weyl groups; for example, the hyperoctahedral group Bn gives a model of n independent particles with a wall at x = 0. We can define Brownian motion on a Lie algebra, viewing it as a vector space; the eigenvalues of a point in the Lie algebra correspond to a point ...
Combinatorial Hopf algebras and generalized DehnSommerville relations
, 2003
"... A combinatorial Hopf algebra is a graded connected Hopf algebra over a field k equipped with a character (multiplicative linear functional) ζ: H → k. We show that the terminal object in the category of combinatorial Hopf algebras is the algebra QSym of quasisymmetric functions; this explains the u ..."
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Cited by 65 (16 self)
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A combinatorial Hopf algebra is a graded connected Hopf algebra over a field k equipped with a character (multiplicative linear functional) ζ: H → k. We show that the terminal object in the category of combinatorial Hopf algebras is the algebra QSym of quasisymmetric functions; this explains the ubiquity of quasisymmetric functions as generating functions in combinatorics. We illustrate this with several examples. We prove that every character decomposes uniquely as a product of an even character and an odd character. Correspondingly, every combinatorial Hopf algebra (H, ζ) possesses two canonical Hopf subalgebras on which the character ζ is even (respectively, odd). The odd subalgebra is defined by certain canonical relations which we call the generalized DehnSommerville relations. We show that, for H = QSym, the generalized DehnSommerville relations are the BayerBillera relations and the odd subalgebra is the peak Hopf algebra of Stembridge. We prove that QSym is the product (in the categorical sense) of its even and odd Hopf subalgebras. We also calculate the odd subalgebras of various related combinatorial Hopf algebras: the MalvenutoReutenauer Hopf algebra of permutations, the
Inhomogeneous lattice paths, generalized Kostka polynomials and A_n1 supernomials
"... Inhomogeneous lattice paths are introduced as ordered sequences of rectangular Young tableaux thereby generalizing recent work on the Kostka polynomials by Nakayashiki and Yamada and by Lascoux, Leclerc and Thibon. Motivated by these works and by Kashiwara's theory of crystal bases we define a st ..."
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Cited by 62 (14 self)
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Inhomogeneous lattice paths are introduced as ordered sequences of rectangular Young tableaux thereby generalizing recent work on the Kostka polynomials by Nakayashiki and Yamada and by Lascoux, Leclerc and Thibon. Motivated by these works and by Kashiwara's theory of crystal bases we define a statistic on paths yielding two novel classes of polynomials.
Functional relations in solvable lattice models. I. Functional relations and representation theory
 Internat. J. Modern Phys. A
, 1994
"... Abstract. Reported are two applications of the functional relations (Tsystem) among a commuting family of rowtorow transfer matrices proposed in the previous paper Part I. For a general simple Lie algebra Xr, we determine the correlation lengths of the associated massive vertex models in the anti ..."
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Cited by 62 (7 self)
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Abstract. Reported are two applications of the functional relations (Tsystem) among a commuting family of rowtorow transfer matrices proposed in the previous paper Part I. For a general simple Lie algebra Xr, we determine the correlation lengths of the associated massive vertex models in the antiferroelectric regime and central charges of the RSOS models in two critical regimes. The results reproduce known values or even generalize them, demonstrating the efficiency of the Tsystem.
Quantum Immanants And Higher Capelli Identities
, 1996
"... We consider remarkable central elements of the universal enveloping algebra U(gl(n)) which we call quantum immanants. We express them in terms of generators E ij of U(gl(n)) and as differential operators on the space of matrices. These expressions are a direct generalization of the classical Capelli ..."
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Cited by 56 (17 self)
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We consider remarkable central elements of the universal enveloping algebra U(gl(n)) which we call quantum immanants. We express them in terms of generators E ij of U(gl(n)) and as differential operators on the space of matrices. These expressions are a direct generalization of the classical Capelli identities. They result in many nontrivial properties of quantum immanants. 1.
Infinite wedge and random partitions
 Selecta Mathematica (new series
"... The aim of this paper is to show that random partitions have a very natural and direct connection to various structures which are well known in integrable systems. This connection is arguably even more natural than, for example, ..."
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Cited by 56 (6 self)
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The aim of this paper is to show that random partitions have a very natural and direct connection to various structures which are well known in integrable systems. This connection is arguably even more natural than, for example,
Schubert Polynomials for the Classical Groups
 J. Amer. Math. Soc
, 1994
"... Introduction The task of a theory of Schubert polynomials is to produce explicit representatives for Schubert classes in the cohomology ring of a flag variety, and to do so in a manner that is as natural as possible from a combinatorial point of view. To explain more fully, let us review a special ..."
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Cited by 55 (4 self)
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Introduction The task of a theory of Schubert polynomials is to produce explicit representatives for Schubert classes in the cohomology ring of a flag variety, and to do so in a manner that is as natural as possible from a combinatorial point of view. To explain more fully, let us review a special case, the Schubert calculus for Grassmannians, where one asks for the number of linear spaces of given dimension satisfying certain geometric conditions. A typical problem is to find the number of lines meeting four given lines in general position in 3space (answer below). For each of the four given lines, the set of lines meeting it is a Schubert variety in the Grassmannian and we want the number of intersection points of these four subvarieties. In the modern solution of this problem, the Schubert varieties induce canonical elements of the cohomology ring of the Grassmannian, called Schubert classes. The product of these Schubert classes is the class of a point times the number of inter
Computing the Singular Value Decomposition with High Relative Accuracy
 Linear Algebra Appl
, 1997
"... We analyze when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy. This means that each computed singular value is guaranteed to have some correct digits, even if the singular values have widely varying magnitudes. This is in contrast to the a ..."
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Cited by 55 (12 self)
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We analyze when it is possible to compute the singular values and singular vectors of a matrix with high relative accuracy. This means that each computed singular value is guaranteed to have some correct digits, even if the singular values have widely varying magnitudes. This is in contrast to the absolute accuracy provided by conventional backward stable algorithms, whichin general only guarantee correct digits in the singular values with large enough magnitudes. It is of interest to compute the tiniest singular values with several correct digits, because in some cases, such as #nite element problems and quantum mechanics, it is the smallest singular values that havephysical meaning, and should be determined accurately by the data. Many recent papers have identi#ed special classes of matrices where high relative accuracy is possible, since it is not possible in general. The perturbation theory and algorithms for these matrix classes have been quite di#erent, motivating us to seek a co...