zbMATH — the first resource for mathematics

Resonance equals reducibility for \(A\)-hypergeometric systems. (English) Zbl 1251.13023
The paper under review concerns \(A\)-hypergeometric systems \(H_A(\beta)\), a class of holonomic systems of linear partial differential equations introduced by I. M. Gelfand, M. M. Kapranov and A. V. Zelevinskij [Adv. Math. 84, No.2, 255–271 (1990; Zbl 0741.33011)]. \(A\) denotes an integer \(d\times n\)-matrix such that the additive group generated by its columns is equal to \(\mathbb{Z}^d\), and \(\beta\) stands for a parameter in \(\mathbb{C}^d\). One of the fundamental theorems proved by Gel’fand, Graev, Kapranov and Zelevinski is the following, assuming that the toric ring \(\mathbb{C}[\mathbb{N}A]\) is Cohen-Macaulay and standard graded: if \(\beta\) is nonresonant, then the monodromy representation of the solutions of \(H_A(\beta)\) at a generic point is irreducible. Here the authors are able to prove this statement, irrespective on the assumptions made on \(\mathbb{C}[\mathbb{N}A]\). The proof, conceptually simpler than the original one, makes use of the Euler-Koszul functor developed in [L. F. Matusevich, E. Miller and U. Walther, J. Am. Math. Soc. 18, No. 4, 919–941 (2005; Zbl 1095.13033)], and the D-module/representation-theoretic description of hypergeometric systems obtained in [M. Schulze and U. Walther [J. Algebra 322, No. 9, 3392–3409 (2009; Zbl 1181.13023)]. More precisely, the authors give a combinatorial characterization of the irreducibility of the monodromy (also inspired by F. Beukers [Indag. Math., New Ser. 21, No. 1-2, 30–39 (2011; Zbl 1229.33023)]).

13N10 Commutative rings of differential operators and their modules
33C70 Other hypergeometric functions and integrals in several variables
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
Full Text: DOI Link arXiv