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Combinatorics of hypergeometric functions associated with positive roots. (English) Zbl 0876.33011

Arnold, V. I. (ed.) et al., The Arnold-Gelfand mathematical seminars: geometry and singularity theory. Boston, MA: Birkhäuser. 205-221 (1997).
Let \(Z_n\) be the group of unipotent matrices of order \(n+1\) and let \(z=(z_{ij})\), \(0\leq i\leq j\leq n\), with \(z_{ii}=1\) be matrices from \(Z_n\). The \(n\)-dimensional torus consisting of diagonal matrices \(t= \text{ diag} (t_0,t_1,\cdots ,t_n)\), \(t_0t_1\cdots t_n=1\), acts on \(Z_n\) by conjugation: \(z\equiv \{z_{ij} \} \to \{z_{ij} t_it_j^{-1} \}\). With this action the following system of differential equations is related \[ -\sum _{i=0}^{j-1} z_{ij}{\partial f\over \partial z_{ij}} + \sum _{k=j+1}^{n} z_{jk}{\partial f\over \partial z_{ij}} =\alpha _j f,\;\;\;j=0,1,2,\cdots ,n, \;\;\;\;{\partial f\over \partial z_{ik}}={\partial ^2f\over \partial z_{ij} \partial z_{jk}},\;\;\;0\leq i<j<k\leq n, \] where \(\alpha =(\alpha _0,\alpha _1,\cdots ,\alpha _n)\) and \(\sum \alpha _i=0\). It is called the hypergeometric system on the group of unipotent matrices. Solutions of this system are called hypergeometric functions on this group. This hypergeometric system gives a holonomic \(D\)-module. The authors find the number of independent solutions of this system at a generic point. This number is equal to the Catalan number. An explicit basis of \(\Gamma\)-series in the solution space of the system is constructed. The restriction of this system to certain strata is considered. Several combinatorial constructions with trees, polyhedra and triangulations, related to this system, are introduced and studied.
For the entire collection see [Zbl 0857.00029].
Reviewer: A.Klimyk (Kiev)

MSC:

33C80 Connections of hypergeometric functions with groups and algebras, and related topics
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