Dualizing, projecting, and restricting GKZ systems

TL;DR

This paper explores how hypergeometric systems associated with an integer matrix behave under projection, restriction, and duality, especially when the matrix is normal and homogeneous, extending previous results and confirming conjectures.

Contribution

It demonstrates that projections and restrictions of $A$-hypergeometric systems are essentially $F$-hypergeometric and proves the holonomic dual of such systems is also hypergeometric under certain conditions.

Findings

Projection and restriction preserve hypergeometric structure.

At most one of the projected or restricted systems is nonzero.

Holonomic duality preserves hypergeometric systems in the homogeneous case.

Abstract

Let $A$ be an integer matrix, and assume that its semigroup ring $\mathbb{C}[\mathbb{N}A]$ is normal. Fix a face $F$ of the cone of $A$. We show that the projection and restriction of an $A$-hypergeometric system to the coordinate subspace corresponding to $F$ are essentially $F$-hypergeometric; moreover, at most one of them is nonzero. We also show that, if $A$ is in addition homogeneous, the holonomic dual of an $A$-hypergeometric system is itself $A$-hypergeometric. This extends a result of Uli Walther, proving a conjecture of Nobuki Takayama in the normal homogeneous case.