Abstract
AbstractIt is shown that the description of certain class of representations of the holonomy Lie algebra $\mathfrak g_{\Delta}$ associated with hyperplane arrangement $\Delta$ is essentially equivalent to the classification of $\vee$-systems associated with $\Delta.$ The flat sections of the corresponding $\vee$-connection can be interpreted as vector fields, which are both logarithmic and gradient. We conjecture that the hyperplane arrangement of any $\vee$-system is free in Saito's sense and show this for all known $\vee$-systems and for a special class of $\vee$-systems called harmonic, which includes all Coxeter systems. In the irreducible Coxeter case the potentials of the corresponding gradient vector fields turn out to be Saito flat coordinates, or their one-parameter deformations. We give formulas for these deformations as well as for the potentials of the classical families of harmonic $\vee$-systems.
Highlights
The theory of the hyperplane arrangements has a rich history and is known to be related to many areas of mathematics: combinatorics, topology, singularity theory, and classical analysis, see the Introduction in [37].It was the link with the singularity theory, which was the motivation to study the logarithmic vector fields for Saito, who introduced an important notion of a freeReceived May 20, 2016; Revised November 6, 2016; Accepted November 8, 20162 M
We have shown that the theory of ∨-systems has natural links both with the representation theory of holonomy Lie algebras and with Saito’s theory of logarithmic vector fields, which could be used in both ways
The homological representations of braid groups were first studied by Lawrence [27] in relation with Hecke algebras and later by Bigelow and Krammer, who showed that they provide faithful representations of braid groups [6, 26]
Summary
The theory of the hyperplane arrangements has a rich history and is known to be related to many areas of mathematics: combinatorics, topology, singularity theory, and classical analysis, see the Introduction in [37]. For any finite set of non-collinear covectors A ⊂ V ∗ one can consider the associated arrangement of complex hyperplanes = A := ∪α∈AHα in V given by α(x) = 0, α ∈ A and the corresponding holonomy Lie algebra g with generators {tα}α∈A and the relations [tα , tβ] = 0 , α ∈ A ∩ ,. We have shown that the class of ∨-systems is closed under the restriction [17], so to prove the conjecture it is enough to show that A is free This would imply by Terao’s theorem that the corresponding Poincaré polynomial P A (t) is factorisable in the form (18), which would be already a strong topological restriction of the arrangement. As we will see for the classical series the corresponding potentials turn out to be certain deformations of Saito’s generators of the algebra of invariants
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