Algebraic and combinatorial aspects of the general theory of hypergeometric functions
This is the first detailed publication of a series of studies by I. M. Gel'fand and his school on generalized hypergeometric functions. These are defined on the Grassmannian \(G_ k({\mathbb{R}}^ n)\) by \[ \Phi (\alpha;\zeta,w)=\int_{S(\zeta)}\prod^{n}_{i=1}(x_ i)_+^{\alpha_ i-1}d{\tilde \omega}, \] where \(\zeta \cong R^ k_ t\in G_ k({\mathbb{R}}^ n)\), S(\(\zeta)\) the unit sphere of \(\zeta\) and \({\tilde \omega}=\sum t_ i\partial /\partial t_ i\rfloor \omega\). This article studies combinatorial and geometric aspects, mainly when \(\alpha\) has components in nonnegative integers, hence \(\Phi\) becomes piecewise rational. The main result of this article is the construction of a canonical decomposition \(\Phi (\alpha;\zeta,\omega)=\sum_ J\Phi_ J(\alpha;\zeta,\omega)\) along the strata J of \(\zeta\). This is equivalent with the calculus of a canonical basis in the homology group of the complement of hyperplanes. The foundation of the combinatorial argument needed here is found to be essentially the matroid theory developed in the thirties.
homology group of the complement of hyperplanes
scientific article; zbMATH DE number 4004621