A Class of Baker–Akhiezer Arrangements

Abstract

Abstract:We study a class of arrangements of lines with multiplicities on the plane which admit the Chalykh-Veselov Baker-Akhiezer function. These arrangements are obtained by adding multiplicity one lines in an invariant way to any dihedral arrangement with invariant multiplicities. We describe all the Baker-Akhiezer arrangements when at most one line has multiplicity higher than 1. We study associated algebras of quasiinvariants which are isomorphic to the commutative algebras of quantum integrals for the generalized Calogero-Moser operators. We compute the Hilbert series of these algebras and we conclude that the algebras are Gorenstein. We also show that there are no other arrangements with Gorenstein algebras of quasi-invariants when at most one line has multiplicity bigger than 1.

Cohen–Macaulay modules over the algebra of planar quasi–invariants and Calogero–Moser systems

“…Remark 3.13. According to a result of Feigin and Johnston [18,Theorem 7.14], the algebra of quasi-invariants from Lemma 3.12 is Gorenstein if and only if k = 0, which matches with our description of a canonical module.…”

Periodic Vortex Streets and Complex Monodromy

“…We will discuss many examples of corresponding relative vortex equilibria in Section 6. Here we will mention only new collinear vortex equilibria (when all the vortices are on a real line), related to Baker-Akhiezer configurations found by M. Feigin and D. Johnston [11].…”

“…. , z l ∈ (0, π) are certain simple real zeros located symmetrically with respect to π/2 (see [11]). This means that the function W = 1 2πi log ψ(z) with…”

Baker-Akhiezer functions and generalised Macdonald-Mehta integrals

“…,m−1, and k m = q(m+m+l). It is known that the corresponding configuration A q (m,m,1 l ) is real and the corresponding lines form a dihedral arrangement of 2q lines with multiplicities m andm together with ql lines of multiplicity 1 that form l/2 dihedral orbits, and that the Baker-Akhiezer function exists, see [11]. In particular, the case l = 0 gives the dihedral configuration with the multiplicities m,m.…”

“…We explicitly compute this value for all known Baker-Akhiezer arrangements. In the twodimensional case we use recent results by Berest et al [2] and by Johnston and one of the authors [11]. For the multidimensional non-Coxeter Baker-Akhiezer arrangements we follow Bombieri's calculation using a version of the Selberg integral found by Dotsenko and Fateev [7].…”