Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 038, 9 pages      arXiv:2009.13710      https://doi.org/10.3842/SIGMA.2021.038
Contribution to the Special Issue on Primitive Forms and Related Topics in honor of Kyoji Saito for his 77th birthday

The Primitive Derivation and Discrete Integrals

Daisuke Suyama a and Masahiko Yoshinaga b
a) Faculty of Integrated Media, Wakkanai Hokusei Gakuen University, 1-2290-28 Wakabadai, Wakkanai, Hokkaido 097-0013, Japan
b) Department of Mathematics, Faculty of Science, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo 060-0810, Japan

Received September 30, 2020, in final form April 09, 2021; Published online April 13, 2021

Abstract
The modules of logarithmic derivations for the (extended) Catalan and Shi arrangements associated with root systems are known to be free. However, except for a few cases, explicit bases for such modules are not known. In this paper, we construct explicit bases for type $A$ root systems. Our construction is based on Bandlow-Musiker's integral formula for a basis of the space of quasiinvariants. The integral formula can be considered as an expression for the inverse of the primitive derivation introduced by K. Saito. We prove that the discrete analogues of the integral formulas provide bases for Catalan and Shi arrangements.

Key words: hyperplane arrangements; freeness; Catalan arrangements; Shi arrangements.

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