9 research outputs found
The Efficient Evaluation of the Hypergeometric Function of a Matrix Argument
We present new algorithms that efficiently approximate the hypergeometric
function of a matrix argument through its expansion as a series of Jack
functions. Our algorithms exploit the combinatorial properties of the Jack
function, and have complexity that is only linear in the size of the matrix.Comment: 14 pages, 3 figure
Rank-deficient submatrices of Fourier matrices
AbstractWe consider the maximal rank-deficient submatrices of Fourier matrices with order a power of a prime number. We do this by considering a hierarchical subdivision of these matrices into low rank blocks. We also explore some connections with the fast Fourier transform (FFT), and with an uncertainty principle for Fourier transforms over finite Abelian groups
The Horn Problem for Real Symmetric and Quaternionic Self-Dual Matrices
Horn's problem, i.e., the study of the eigenvalues of the sum of two
matrices, given the spectrum of and of , is re-examined, comparing the
case of real symmetric, complex Hermitian and self-dual quaternionic matrices. In particular, what can be said on the probability distribution
function (PDF) of the eigenvalues of if and are independently and
uniformly distributed on their orbit under the action of, respectively, the
orthogonal, unitary and symplectic group? While the two latter cases (Hermitian
and quaternionic) may be studied by use of explicit formulae for the relevant
orbital integrals, the case of real symmetric matrices is much harder. It is
also quite intriguing, since numerical experiments reveal the occurrence of
singularities where the PDF of the eigenvalues diverges. Here we show that the
computation of the PDF of the symmetric functions of the eigenvalues for
traceless matrices may be carried out in terms of algebraic
functions;- roots of quartic polynomials;- and their integrals. The computation
is carried out in detail in a particular case, and reproduces the expected
singular patterns. The divergences are of logarithmic or inverse power type. We
also relate this PDF to the (rescaled) structure constants of zonal polynomials
and introduce a zonal analogue of the Weyl characters
Polynomial Total Positivity and High Relative Accuracy Through Schur Polynomials
In this paper, Schur polynomials are used to provide a bidiagonal decomposition of polynomial collocation matrices. The symmetry of Schur polynomials is exploited to analyze the total positivity on some unbounded intervals of a relevant class of polynomial bases. The proposed factorization is used to achieve relative errors of the order of the unit round-off when solving algebraic problems involving the collocation matrix of relevant polynomial bases, such as the Hermite basis. The numerical experimentation illustrates the accurate results obtained when using the findings of the paper
Central Charge and Quasihole Scaling Dimensions From Model Wavefunctions: Towards Relating Jack Wavefunctions to W-algebras
We present a general method to obtain the central charge and quasihole
scaling dimension directly from groundstate and quasihole wavefunctions. Our
method applies to wavefunctions satisfying specific clustering properties. We
then use our method to examine the relation between Jack symmetric functions
and certain W-algebras. We add substantially to the evidence that the (k,r)
admissible Jack functions correspond to correlators of the conformal field
theory W_k(k+1,k+r), by calculating the central charge and scaling dimensions
of some of the fields in both cases and showing that they match. For the Jacks
described by unitary W-models, the central charge and quasihole exponents match
the ones previously obtained from analyzing the physics of the edge
excitations. For the Jacks described by non-unitary W-models the central charge
and quasihole scaling dimensions obtained from the wavefunctions differ from
the ones obtained from the edge physics, which instead agree with the
"effective" central charge of the corresponding W-model.Comment: 22 pages, no figure