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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
Content evaluation and class symmetric functions
AbstractIn this article we study the evaluation of symmetric functions on the alphabet of contents of a partition. Applying this notion of content evaluation to the computation of central characters of the symmetric group, we are led to the definition of a new basis of the algebra Λ of symmetric functions over Q(n) that we call the basis of class symmetric functions.By definition this basis provides an algebra isomorphism between Λ and the Farahat–Higman algebra FH governing for all n the products of conjugacy classes in the center Zn of the group algebra of the symmetric group Sn. We thus obtain a calculus of all connexion coefficients of Zn inside Λ. As expected, taking the homogeneous components of maximal degree in class symmetric functions, we recover the symmetric functions introduced by Macdonald to describe top connexion coefficients.We also discuss the relation of class symmetric functions to the asymptotic of central characters and of the enumeration of standard skew young tableaux. Finally we sketch the extension of these results to Hecke algebras
Endomorphisms of spaces of virtual vectors fixed by a discrete group
Consider a unitary representation of a discrete group , which, when
restricted to an almost normal subgroup , is of type II. We
analyze the associated unitary representation of
on the Hilbert space of "virtual" -invariant vectors, where
runs over a suitable class of finite index subgroups of .
The unitary representation of is uniquely
determined by the requirement that the Hecke operators, for all , are
the "block matrix coefficients" of .
If is an integer multiple of the regular representation, there
exists a subspace of the Hilbert space of the representation , acting
as a fundamental domain for . In this case, the space of
-invariant vectors is identified with .
When is not an integer multiple of the regular representation,
(e.g. if , is the modular group,
belongs to the discrete series of representations of ,
and the -invariant vectors are the cusp forms) we assume that is
the restriction to a subspace of a larger unitary representation having a
subspace as above.
The operator angle between the projection onto (typically the
characteristic function of the fundamental domain) and the projection
onto the subspace (typically a Bergman projection onto a space of
analytic functions), is the analogue of the space of - invariant
vectors.
We prove that the character of the unitary representation
is uniquely determined by the character of the
representation .Comment: The exposition has been improved and a normalization constant has
been addressed. The result allows a direct computation for the characters of
the unitary representation on spaces of invariant vectors (for example
automorphic forms) in terms of the characters of the representation to which
the fixed vectors are associated (e.g discrete series of PSL(2, R) for
automorphic forms
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