Triality for Homogeneous Polynomials

Symmetry Integrability and Geometry Methods and Applications ◽

10.3842/sigma.2021.079 ◽

2021 ◽

Author(s):

Laura P. Schaposnik ◽

◽

Sebastian Schulz ◽

Keyword(s):

Lie Algebras ◽

Homogeneous Polynomials ◽

Invariant Polynomials ◽

Homogeneous Basis

Through the triality of SO(8,C), we study three interrelated homogeneous basis of the ring of invariant polynomials of Lie algebras, which give the basis of three Hitchin fibrations, and identify the explicit automorphisms that relate them.

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Rings of invariant polynomials for a class of Lie algebras

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-1971-0281839-9 ◽

1971 ◽

Vol 160 ◽

pp. 249-249 ◽

Cited By ~ 17

Author(s):

S. J. Takiff

Keyword(s):

Lie Algebras ◽

Invariant Polynomials

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Solvability of Lie Algebras Arising from Isolated Singularities and Nonisolatedness of Singularities Defined by sℓ(2, C) Invariant Polynomials

American Journal of Mathematics ◽

10.2307/2374785 ◽

1991 ◽

Vol 113 (5) ◽

pp. 773 ◽

Cited By ~ 14

Author(s):

Stephen S.-T. Yau

Keyword(s):

Lie Algebras ◽

Invariant Polynomials ◽

Isolated Singularities

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On Modular Vector Invariant Fields

Algebra Colloquium ◽

10.1142/s1005386720000620 ◽

2020 ◽

Vol 27 (04) ◽

pp. 749-752

Author(s):

Ying Han ◽

Runxuan Zhang

Keyword(s):

Finite Field ◽

Linear Group ◽

Dual Space ◽

General Linear Group ◽

Homogeneous Polynomials ◽

Invariant Polynomials ◽

Standard Representation ◽

Direct Sum ◽

Invariant Ring ◽

Vector Invariant

Let [Formula: see text] be a finite field of any characteristic and [Formula: see text] be the general linear group over [Formula: see text]. Suppose W denotes the standard representation of [Formula: see text], and [Formula: see text] acts diagonally on the direct sum of W and its dual space W∗. Let G be any subgroup of [Formula: see text]. Suppose the invariant field [Formula: see text], where [Formula: see text] in [Formula: see text] are homogeneous invariant polynomials. We prove that there exist homogeneous polynomials [Formula: see text] in the invariant ring [Formula: see text] such that the invariant field [Formula: see text] is generated by [Formula: see text] over [Formula: see text].

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Discriminants in the invariant theory of reflection groups

Nagoya Mathematical Journal ◽

10.1017/s0027763000002749 ◽

1988 ◽

Vol 109 ◽

pp. 23-45 ◽

Cited By ~ 22

Author(s):

Peter Orlik ◽

Louis Solomon

Keyword(s):

Invariant Theory ◽

Reflection Group ◽

Module Structure ◽

Polynomial Functions ◽

Reflection Groups ◽

Homogeneous Polynomials ◽

Complex Vector ◽

Invariant Polynomials ◽

Basic Set ◽

Finite Reflection Group

Let V be a complex vector space of dimension l and let G ⊂ GL(V) be a finite reflection group. Let S be the C-algebra of polynomial functions on V with its usual G-module structure (gf)(v) = f{g-1v). Let R be the subalgebra of G-invariant polynomials. By Chevalley’s theorem there exists a set ℬ = {f1, …, fl} of homogeneous polynomials such that R = C[f1, …, fl]. We call ℬ a set of basic invariants or a basic set for G. The degrees di = deg fi are uniquely determined by G. We agree to number them so that d1 ≤ … ≤ di. The map τ: V/G → C1 defined byis a bijection. Each reflection in G fixes some hyperplane in V.

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Singularities Defined by sℓ(2, C) Invariant Polynomials and Solvability of Lie Algebras Arising From Isolated Singularities

American Journal of Mathematics ◽

10.2307/2374603 ◽

1986 ◽

Vol 108 (5) ◽

pp. 1215 ◽

Cited By ~ 6

Author(s):

Stephen S.-T. Yau

Keyword(s):

Lie Algebras ◽

Invariant Polynomials ◽

Isolated Singularities

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Lie algebras of infinitesimal CR automorphisms of weighted homogeneous and homogeneous CR-generic submanifolds of CN

Filomat ◽

10.2298/fil1606387s ◽

2016 ◽

Vol 30 (6) ◽

pp. 1387-1411 ◽

Cited By ~ 5

Author(s):

Masoud Sabzevari ◽

Amir Hashemi ◽

Benyamin Alizadeh ◽

Joël Merker

Keyword(s):

Lie Algebras ◽

Divide And Conquer ◽

Homogeneous Polynomials ◽

Generic Submanifolds ◽

Pde Systems ◽

Set Up ◽

Infinitesimal Cr Automorphisms ◽

Significant Class ◽

Homogeneous Cr Manifolds ◽

Natural Gradation

We consider the significant class of homogeneous CR manifolds represented by some weighted homogeneous polynomials and we derive some plain and useful features which enable us to set up a fast effective algorithm to compute homogeneous components of their Lie algebras of infinitesimal CR automorphisms. This algorithm mainly relies upon a natural gradation of the sought Lie algebras, and it also consists in treating separately the related graded components. While classical methods are based on constructing and solving some associated PDE systems which become time consuming as soon as the number of variables increases, the new method presented here is based on plain techniques of linear algebra. Furthermore, it benefits from a divide-and-conquer strategy to break down the computations into some simpler subcomputations. Also, we consider the new and effective concept of comprehensive Gr?bner systems which provides us some powerful tools to treat the computations in the much complicated parametric case. The designed algorithm is also implemented in the Maple software, what required also implementing a recently designed algorithm of Kapur et al.

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