The Shi arrangements and the Bernoulli polynomials

Abstract

The braid arrangement is the Coxeter arrangement of the type Aℓ. The Shi arrangement is an affine arrangement of hyperplanes consisting of the hyperplanes of the braid arrangement and their parallel translations. In this paper, we give an explicit basis construction for the derivation module of the cone over the Shi arrangement. The essential ingredient of our recipe is the Bernoulli polynomials.

Simple-root bases for Shi arrangements

“…This problem has been solved in [15,14,8] for the SRB− when k = 1 and the root system is either of the type A ℓ , B ℓ , C ℓ , D ℓ or G 2 . The SRB+ can be obtained from the SRB− by Definition 1.2 (2).…”

Supersolvable orders and inductively free arrangements

“…Let S D S.V / be the symmetric algebra of V and Der K .S/ be the module of derivations A is called free if D.A/ is free. There are a lot of works on the freeness of central arrangements, especially on Coxeter arrangements and the cones over Catalan and Shi arrangements [1][2][3][4][5][6]. For proving the freeness of arrangements, Terao's Addition Theorem [7] provides a standard tool and this theorem leads to the notion of inductively freeness.…”

The Primitive Derivation and Discrete Integrals

“…Since then, a number of efforts have been made to construct explicit bases for D(c Cat ℓ (m)) and D(c Shi ℓ (m)). First, in [18], a basis for D(c Shi ℓ (1)) was constructed using the Bernoulli polynomial. Subsequently, in [10] and [19], similar bases were constructed for root systems of type B, C, and D. Note that these works are for Shi arrangements with m = 1.…”