On deformation and classification of ∨-systems

Abstract

The ∨-systems are special finite sets of covectors which appeared in the theory of the generalized WittenDijkgraaf-Verlinde-Verlinde (WDVV) equations. Several families of ∨-systems are known, but their classification is an open problem. We derive the relations describing the infinitesimal deformations of ∨-systems and use them to study the classification problem for ∨-systems in dimension three. We discuss also possible matroidal structures of ∨-systems in relation with projective geometry and give the catalogue of all known irreducible rank three ∨-systems.

Extended ⋁-systems and trigonometric solutions to the WDVV equations

“…Further examples, too, would be of interest. There has been recent work on the classification of ⋁-systems, 20,21 and it would be interesting to see if the extended version of these systems exists. One could ask, for example, how the matroid for the extended systems can be constructed from the matroid of the original system.…”

Commutativity equations and their trigonometric solutions

“…More generally, solutions of the form (1.4) exist for special configurations of vectors known as ∨-systems introduced by Veselov in [26]. This class of of solutions was studied further in [7,13,14,23]. Thus it was shown that the class is closed under the operations of taking subsystems and projections of A, and such solutions have to do with Dubrovin's almost duality on the discriminant strata.…”

Trigonometric ∨ -systems and solutions of WDVV equations *

“…The class of ∨-systems contains multi-parameter deformations of the root systems A n and B n ( [9], see also [17] for more examples). The underlying matroids were examined in [27]. The problem of classification of ∨-systems remains open.…”