Algebraic and combinatorial aspects of the general theory of hypergeometric functions
type: Item
instance of: scholarly article
title: Algebraic and combinatorial aspects of the general theory of hypergeometric functions
zbMATH Open document ID: 0619.33004
DOI: 10.1007/BF01078470
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 33C60
Mathematics Subject Classification ID: 35N10
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 32S60
zbMATH DE Number: 4004621
zbMATH Keywords: Grassmann manifold
zbMATH Keywords: homology of complement of hyperplanes
zbMATH Keywords: Kostant partition function
zbMATH Keywords: homology group of the complement of hyperplanes
author: I. M. Gel'fand
author: Andrei Zelevinsky
reviewed by: Akira Kaneko
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01078470
OpenAlex ID: W1989488790
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 4004621