[Paper Review] A basis construction for the Shi arrangement of the type $B_{\ell}$ or $C_{\ell}$

Daisuke Suyama|arXiv (Cornell University)|May 29, 2012

Advanced Combinatorial Mathematics8 references3 citations

TL;DR

This paper constructs an explicit basis for the derivation module of the cone over the Shi arrangement of type $B_ au$ or $C_ au$ using new Bernoulli-like polynomials $B_{r,s}^B(x)$ and $B_{r,s}^C(x)$. The key result is that the derivations $ heta_E, heta_1^B, heta_2^B, heta_ au^B$ (and similarly for $C_ au$) form a free basis, providing an explicit realization of Yoshinaga’s earlier existence result for free arrangements.

ABSTRACT

The Shi arrangement is an affine arrangement of hyperplanes consisting of the hyperplanes of the Weyl arrangement and their parallel translations. It was introduced by J.-Y. Shi in the study of the Kazhdan-Lusztig representation of the affine Weyl groups. M. Yoshinaga showed that the cone over every Shi arrangement is free. In this paper, we construct an explicit basis for the derivation module of the cone over the Shi arrangements of the type $B_{\ell}$ or $C_{\ell}$.

Motivation & Objective

• To construct an explicit basis for the derivation module of the cone over the Shi arrangement of type $B_ au$ or $C_ au$.
• To provide a constructive realization of Yoshinaga’s result that the cone over the Shi arrangement is free.
• To extend the method of Bernoulli-like polynomials used in type $A_ au$ and $D_ au$ to the classical root systems $B_ au$ and $C_ au$.
• To establish $W$-equivariance and linearity of the basis construction in the context of reflection group actions.
• To prove that the derived derivations form a free basis over the polynomial ring $S$.

Proposed method

• Introduces new Bernoulli-like polynomials $B_{r,s}^B(x)$ and $B_{r,s}^C(x)$ satisfying specific difference equations and normalization conditions.
• Defines homogeneous derivations $ heta_j^B$ and $ heta_j^C$ via homogenized versions of these polynomials and symmetric functions over root subsystems.
• Uses elementary symmetric polynomials $ au_k^{(3,j)}$ and $ au_k^{(3,j)}$ over $x_i^2$ to encode structure of the root system in the derivation formulas.
• Applies the Euler derivation $ heta_E = zrac{ heta}{ heta z} + heta x_i rac{ heta}{ heta x_i}$ as a key component of the basis.
• Verifies that the constructed derivations preserve the defining equations of the cone via divisibility conditions.
• Proves the basis is free by showing linear independence and spanning via $W$-equivariance and dimension counting.

Experimental results

Research questions

• RQ1 Can an explicit basis be constructed for the derivation module of the cone over the Shi arrangement of type $B_ au$?
• RQ2 How can Bernoulli-like polynomials be generalized to construct such bases for non-simply-laced root systems?
• RQ3 Is the derived basis for $D({ m cone}({ m Shi}(B_ au)))$ $W$-equivariant?
• RQ4 Does the construction for $C_ au$ yield a similar free basis as in $B_ au$?
• RQ5 Can the basis be interpreted as a constructive realization of Yoshinaga’s existence theorem for free arrangements?

Key findings

• An explicit basis for $D({ m cone}({ m Shi}(B_ au)))$ is constructed using derivations $ heta_E, heta_1^B, heta_2^B, heta_ au^B$.
• The derivations $ heta_j^B$ are defined via homogenized Bernoulli-like polynomials $B_{r,s}^B(x)$ and symmetric functions over root subsystems.
• The construction for $C_ au$ follows a parallel method using $B_{r,s}^C(x)$, yielding a basis $ heta_E, heta_1^C, heta_2^C, heta_ au^C$.
• The map $ heta_j o heta_j|_{z=0}$ gives a $W$-equivariant isomorphism from $E^*$ to $D({ m Shi}(B_ au), 2)_h$, linking to the Solomon-Terao basis.
• The basis is proven to be free by verifying that the derivations preserve the defining equations of the cone and are linearly independent.
• The result provides an explicit realization of Yoshinaga’s theorem on the freeness of the cone over the Shi arrangement, extending prior constructions from type $A_ au$ and $D_ au$.