arXiv subjects

Teije Kuijper

Publications and source records attributed to Teije Kuijper.

5 recordsLinked to original sources

arXiv2026

Jacobi polynomials on the simplex of Hermitian matrices and zonal spherical functions on partial flag manifolds

We introduce the radial part of the Laplace--Beltrami operator on partial flag manifolds and study its eigenfunctions, the elementary zonal spherical functions. These eigenfunctions can be related to orthogonal polynomials on the simplex of Hermitian matrix invariant under simultaneous conjugation by unitary matrices with respect to the complex matrix variate Dirichlet distribution. Using a matrix variate version of Koornwinders method for constructing orthogonal polynomials in multiple variables from orthogonal polynomials in one variable, we construct a family of pairwise orthogonal polynomials in terms of Hermitian Jacobi polynomials and conjecture that these are elementary zonal spherical functions. Furthermore, we recall the definition of multivariate Schur polynomials and show how they can be used to obtain information about elementary zonal spherical functions. In particular, we give some partial results in the direction that the radial part of the Laplace--Beltrami operator on partial flag manifolds is strongly triangular with respect to the multivariate Schur polynomials. The elementary zonal spherical functions can then be constructed by applying the Gram--Schmidt orthogonalisation process to the multivariate Schur functions.

math.DG↗

arXiv2025

Quaternionic stochastic areas on quaternionic full flag manifolds and applications

We show that a Brownian motion on the quaternionic full flag manifold can be represented as a matrix-valued diffusion obtained in a simple way from a symplectic Brownian motion. By relating its radial dynamics to the Brownian motion on the quaternionic sphere, an explicit formula for the characteristic function of the joint distribution of the quaternionic stochastic areas is obtained. The limit law of these quaternionic stochastic areas is shown to be a multivariate normal distribution with zero mean and non-diagonal covariance matrix. These results are subsequently applied to establish new results about simultaneous quaternionic windings on the quaternionic spheres.

math.PR↗

arXiv2025

Brownian motion and stochastic areas on complex partial flag manifolds with blocks of equal size

We construct a Brownian motion on complex partial flag manifolds with blocks of equal size as a matrix-valued diffusion from a Brownian motion on the unitary group. This construction leads to an explicit expression for the characteristic function of the joint distribution of the stochastic areas on these manifolds. The limit law of these stochastic areas is shown to be a multivariate Cauchy distribution with independent and identically distributed entries. By relating the area functionals on flag manifolds to the winding functional on the complex Stiefel manifold, we establish new results about simultaneous Brownian windings on the complex Stiefel manifold. To establish these results, this work introduces a new family of diffusions, which generalise both the Jacobi processes on the simplex and the Hermitian Jacobi processes.

math.PR↗

arXiv2025

Brownian motion and stochastic areas on complex full flag manifolds

We show that the Brownian motion on the complex full flag manifold can be represented by a matrix-valued diffusion obtained from the unitary Brownian motion. This representation actually leads to an explicit formula for the characteristic function of the joint distribution of the stochastic areas on the full flag manifold. The limit law for those stochastic areas is shown to be a multivariate Cauchy distribution with independent and identically distributed entries. Using a deep connection between area functionals on the flag manifold and winding functionals on complex spheres, we establish new results about simultaneous Brownian windings on the complex sphere and their asymptotics. As a byproduct, our work also unveils a new probabilistic interpretation of the Jacobi operators and polynomials on simplices.

math.PR↗

arXiv2024

On the law of the index of Brownian loops related to the Hopf and anti-de Sitter fibrations

We give explicit formulas and asymptotics for the distribution of the index of the Brownian loop in the following geometrical settings: the complex projective line from which two points have been removed; the complex hyperbolic line from which one point has been removed; the odd dimensional spheres from which a great hypersphere has been removed; and the complex anti-de Sitter spaces. Our analysis is based on the geometry of the Hopf and anti-de Sitter fibrations, and on the relationship between winding and area forms.

math.PR↗