arXiv subjects

Charles F. Dunkl

Publications and source records attributed to Charles F. Dunkl.

50 records · Page 3Linked to original sources

arXiv2010

Numerical shadows: measures and densities on the numerical range

For any operator $M$ acting on an $N$-dimensional Hilbert space $H_N$ we introduce its numerical shadow, which is a probability measure on the complex plane supported by the numerical range of $M$. The shadow of $M$ at point $z$ is defined as the probability that the inner product $(Mu,u)$ is equal to $z$, where $u$ stands for a random complex vector from $H_N$, satisfying $||u||=1$. In the case of N=2 the numerical shadow of a non-normal operator can be interpreted as a shadow of a hollow sphere projected on a plane. A similar interpretation is provided also for higher dimensions. For a hermitian $M$ its numerical shadow forms a probability distribution on the real axis which is shown to be a one dimensional $B$-spline. In the case of a normal $M$ the numerical shadow corresponds to a shadow of a transparent solid simplex in $R^{N-1}$ onto the complex plane. Numerical shadow is found explicitly for Jordan matrices $J_N$, direct sums of matrices and in all cases where the shadow is rotation invariant. Results concerning the moments of shadow measures play an important role. A general technique to study numerical shadow via the Cartesian decomposition is described, and a link of the numerical shadow of an operator to its higher-rank numerical range is emphasized.

math.FA↗

arXiv2010

Symmetric and Antisymmetric Vector-valued Jack Polynomials

Polynomials with values in an irreducible module of the symmetric group can be given the structure of a module for the rational Cherednik algebra, called a standard module. This algebra has one free parameter and is generated by differential-difference ("Dunkl") operators, multiplication by coordinate functions and the group algebra. By specializing Griffeth's (arXiv:0707.0251) results for the G(r,p,n) setting, one obtains norm formulae for symmetric and antisymmetric polynomials in the standard module. Such polynomials of minimum degree have norms which involve hook-lengths and generalize the norm of the alternating polynomial.

math.CO↗

arXiv2008

Some Orthogonal Polynomials in Four Variables

The symmetric group on 4 letters has the reflection group $D_{3}$ as an isomorphic image. This fact follows from the coincidence of the root systems $A_{3}$ and $D_{3}$. The isomorphism is used to construct an orthogonal basis of polynomials of 4 variables with 2 parameters. There is an associated quantum Calogero-Sutherland model of 4 identical particles on the line.

math.CA↗

arXiv2007

Polynomials Associated with Dihedral Groups

There is a commutative algebra of differential-difference operators, with two parameters, associated to any dihedral group with an even number of reflections. The intertwining operator relates this algebra to the algebra of partial derivatives. This paper presents an explicit form of the action of the intertwining operator on polynomials by use of harmonic and Jacobi polynomials. The last section of the paper deals with parameter values for which the formulae have singularities.

math.CA↗

arXiv2006

An Intertwining Operator for the Group B2

There is a commutative algebra of differential-difference operators, acting on polynomials on R_2, associated with the reflection group B2. This paper presents an integral transform which intertwines this algebra, allowing one free parameter, with the algebra of partial derivatives. The method of proof depends on properties of a certain class of balanced terminating hypergeometric series of 4F3-type. These properties are in the form of recurrence and contiguity relations and are proved herein.

math.CA↗

arXiv2005

Hook-lengths and Pairs of Compositions

The monomial basis for polynomials in N variables is labeled by compositions. To each composition there is associated a hook-length product, which is a product of linear functions of a parameter. The zeroes of this product are related to "critical pairs" of compositions; a concept defined in this paper. This property can be described in an elementary geometric way; for example: consider the two compositions (2,7,8,2,0,0) and (5,1,2,5,3,3), then the respective ranks, permutations of the index set {1,2,...,6} sorting the compositions, are (3,2,1,4,5,6) and (1,6,5,2,3,4), and the two vectors of differences (between the compositions and the ranks, respectively) are (-3,6,6,-3,-3,-3) and (2,-4,-4,2,2,2), which are parallel, with ratio -3/2. For a given composition and zero of its hook-length product there is an algorithm for constructing another composition with the parallelism property and which is comparable to it in a certain partial order on compositions, derived from the dominance order. This paper presents the motivation from the theory of nonsymmetric Jack polynomials and the description of the algorithm, as well as the proof of its validity.

math.CO↗

arXiv2005

Singular Polynomials and Modules for the Symmetric Groups

For certain negative rational numbers k0, called singular values, and associated with the symmetric group S_N on N objects, there exist homogeneous polynomials annihilated by each Dunkl operator when the parameter k = k0. It was shown by de Jeu, Opdam and the author (TAMS 346(1994),237-256) that the singular values are exactly the values m/n with 2<=n<=N, m = 1,2... and m/n is not an integer. For each pair (m,n) satisfying these conditions there is a unique irreducible S_N-module of singular polynomials for the singular value -m/n. The existence of these polynomials was established by the author (IMRN 2004,#67,3607-3635). The uniqueness is proven in the present paper. By using Murphy's (J. Alg. 69(1981), 287-297) results on the eigenvalues of the Murphy elements, the problem of existence of singular polynomials is first restricted to the isotype of a partition of N (corresponding to an irreducible representation of S_N) such that (n/gcd(m,n)) divides t+1 for each part t of the partition except the last one. Then by arguments involving nonsymmetric Jack polynomials it is shown that the assumption that the second part of the partition is greater than or equal to n/gcd(m,n) leads to a contradiction.This shows that the singular polynomials are exactly those already determined.

math.RT↗

arXiv2004

Singular polynomials for the symmetric groups

For certain negative rational numbers k, called singular values, and associated with the symmetric group S_N on N objects, there exist homogeneous polynomials annihilated by each Dunkl operator when the parameter equals k. It was shown by the author, de Jeu and Opdam (Trans. Amer. Math. Soc. 346 (1994), 237-256) that the singular values are exactly the values -m/n with 2 <= n <= N, m = 1,2,... and m/n is not an integer. This paper constructs for each pair (m,n) satisfying these conditions an irreducible S_N-module of singular polynomials for the singular value -m/n. The module is of isotype (n-1,(n1-1)^p,r) where n1 = n/gcd(m,n), r = N-n+1-p(n1-1) and 1 <= r <= n1-1 (thus defining p). The singular polynomials are special cases of nonsymmetric Jack polynomials. The paper presents some formulae for the action of Dunkl operators on these polynomials valid in general, and a method for showing the dependence of poles (in the parameter) on the number of variables (N). Murphy elements are used to analyze the representation of S_N on irreducible spaces of singular polynomials.

math.RT↗

arXiv2003

Singular Polynomials for the Symmetric Group and Krawtchouk Polynomials

A singular polynomial is one which is annihilated by all Dunkl operators for a certain parameter value. These polynomials were first studied by Dunkl, de Jeu and Opdam, (Trans. Amer. Math. Soc. 346 (1994), 237-256). This paper constructs a family of such polynomials associated to the irreducible representation (N-2,1,1) of the symmetric group S_N for odd N and parameter values -1/2, -3/2, -5/2,... . The method depends on the use of Krawtchouk polynomials to carry out a change of variables in a generating function involved in the construction of nonsymmetric Jack polynomials labeled by (m,n,0,....), m>=n.

math.QA↗

arXiv2002

Symmetric Functions and B_N-invariant Spherical Harmonics

The wave functions of a quantum isotropic harmonic oscillator in N-space modified by barriers at the coordinate hyperplanes can be expressed in terms of certain generalized spherical harmonics. These are associated with a product-type weight function on the sphere. Their analysis is carried out by means of differential-difference operators. The symmetries of this system involve the Weyl group of type B, generated by permutations and changes of sign of the coordinates. A new basis for symmetric functions as well as an explicit transition matrix to the monomial basis is constructed. This basis leads to a basis for invariant spherical harmonics. The determinant of the Gram matrix for the basis in the natural inner product over the sphere is evaluated, and there is a formula for the evaluation of the basis elements at (1,1,...,1). When the underlying parameter is specialized to zero, the basis consists of ordinary spherical harmonics with cube group symmetry, as used for wave functions of electrons in crystals. The harmonic oscillator can also be considered as a degenerate interaction-free spin Calogero model.

math.CA↗

arXiv2002

A Laguerre Polynomial Orthogonality and the Hydrogen Atom

The radial part of the wave function of an electron in a Coulomb potential is the product of a Laguerre polynomial and an exponential with the variable scaled by a factor depending on the degree. This note presents an elementary proof of the orthogonality of wave functions with differing energy levels. It is also shown that this is the only other natural orthogonality for Laguerre polynomials. By expanding in terms of the usual Laguerre polynomial basis an analogous strange orthogonality is obtained for Meixner polynomials.

math-ph↗

arXiv1999

Computation of the Generalized F Distribution

Exact expressions are given for the distribution function of the ratio of a weighted sum of independent chi-squared variables to a single chi-square variable, scaled appropriately. This distribution is the generalization of the classical F distribution to mixtures of chi-squared variables. The distribution is given in terms of the Lauricella functions. The truncation error bounds are given in terms of hypergeometric functions. Applications to detecting joint outliers and Hotelling's misspecified T^2 distribution are given.

math.CA↗

arXiv1999

Planar Harmonic Polynomials of Type B

The hyperoctahedral group is the Weyl group of type B and is associated with a two-parameter family of differential-difference operators T_i, i=1,..,N (the dimension of the underlying Euclidean space). These operators are analogous to partial derivative operators. This paper finds all the polynomials in N variables which are annihilated by the sum of the squares (T_1)^2+(T_2)^2 and by all T_i for i>2 (harmonic). They are given explicitly in terms of a novel basis of polynomials, defined by generating functions. The harmonic polynomials can be used to find wave functions for the quantum many-body spin Calogero model.

math.CA↗

arXiv1998

Orthogonal Polynomials of Types A and B and Related Calogero Models

Calogero-Sutherland models associated to the Weyl groups of type A and B with exchange terms included in the Hamiltonians systems have non-symmetric eigenfunctions, which are products of the ground state with members of a family of orthogonal polynomials. These polynomials can be defined and studied by using the differential-difference operators introduced by the author in TAMS 1989 (311), 167-183. There is a study of polynomials which are invariant or alternating for parabolic subgroups of the symmetric group. The detailed analysis depends on using two bases of polynomials, one of which transforms monomially under group actions and the other one is orthogonal. There are formulas for norms and point-evaluations which are simplifications of those of Sahi. For any parabolic subgroup of the symmetric group there is a skew operator on polynomials which leads to evaluation at (1,1,...,1) of the quotient of the unique skew polynomial in a given irreducible subspace by the minimum alternating polynomial, analogously to a Weyl character formula. The last section concerns orthogonal polynomials for the type B Weyl group with an emphasis on the Hermite-type polynomials. A complete basis of eigenfunctions of Yamamoto's B_N spin Calogero model is obtained by multiplying these polynomials by the ground state.

q-alg↗