A Basis Construction for the Shi Arrangement of the TypeBℓorCℓ

Suyama, Daisuke

doi:10.1080/00927872.2013.865051

Cited by 5 publications

(4 citation statements)

References 10 publications

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“…First, in [19], a basis for D(c Shi (1)) was constructed using the Bernoulli polynomial. Subsequently, in [10] and [18], similar bases were constructed for root systems of type B, C, and D. Note that these works are for Shi arrangements with m = 1. Catalan arrangements and Shi arrangements with m > 1 have not been covered.…”

Section: Constructions Of Explicit Basesmentioning

confidence: 99%

The Primitive Derivation and Discrete Integrals

Suyama¹,

Yoshinaga

2021

SIGMA

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he 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.

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Section: Constructions Of Explicit Basesmentioning

confidence: 99%

The Primitive Derivation and Discrete Integrals

Suyama¹,

Yoshinaga

2021

SIGMA

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“…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. Catalan arrangements and Shi arrangements with m > 1 have not been covered.…”

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confidence: 99%

The Primitive Derivation and Discrete Integrals

Suyama,

Yoshinaga

2020

Preprint

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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.

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“…However, there have been few researches how to construct their explicit bases. Recently, in the case of types A ℓ , B ℓ , C ℓ , D ℓ , explicit bases for the cone cShi 1 over the Shi arrangements were constructed ( [7], [8], [3]). Also, a nice basis for the extended Shi arrangements was determined in [2].…”

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A basis construction of the extended Catalan and Shi arrangements of the type A2

Abe

Suyama

2018

Journal of Algebra

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In [9], Terao proved the freeness of multi-Coxeter arrangements with constant multiplicities by giving an explicit construction of bases.Combining it with algebro-geometric method, Yoshinaga proved the freeness of the extended Catalan and Shi arrangements in [11]. However, there have been no explicit constructions of the bases for the logarithmic derivation modules of the extended Catalan and Shi arrangements. In this paper, we give the first explicit construction of them when the root system is of the type A 2 .

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A Basis Construction for the Shi Arrangement of the TypeB_ℓorC_ℓ

Cited by 5 publications

References 10 publications

The Primitive Derivation and Discrete Integrals

The Primitive Derivation and Discrete Integrals

The Primitive Derivation and Discrete Integrals

A basis construction of the extended Catalan and Shi arrangements of the type A2

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