Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0708.3583

Defining Relations of Low Degree of Invariants of Two $4 \times 4$ Matrices

Vesselin Drensky·Roberto La Scala
View original source ↗Read source document ↗Export metadata ↓

Abstract

Over a field K of characteristic 0, we study the algebra of invariants of the general linear group GL(4,K) acting by simultaneous conjugation on two matrices of order 4. It coincides with the trace algebra generated by all traces of products of two generic matrices of order 4. It is known that the minimal degree of the defining relations of any homogeneous minimal generating set of this algebra is equal to 12. Starting with the generating set given recently by Drensky and Sadikova, we have determined all relations of degree < 15. For this purpose we have developed further algorithms based on representation theory of the general linear group and easy computer calculations with standard functions of Maple.

Explore related subjects

math.RAmath.AC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Vesselin Drensky, Roberto La Scala. 2007-08-27. Defining Relations of Low Degree of Invariants of Two $4 \times 4$ Matrices. https://arxiv.org/abs/0708.3583

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Normal Quaternionic Matrices and Finitely Generated Witt Rings

We present a new approach to verify the Elementary Type Conjecture for abstract Witt rings with small number of square classes. To do so, we make use of an abstract analogue of the 2-torsion part of the Brauer group. We develop a description of the entire structure of an abstract Witt ring with $2^n$ square classes in terms of a unique $n\times n$ matrix satisfying a small additional condition that particularly holds for Witt rings of fields. Via computational search, we find all these matrices for $n$ up to $7$. This verifies that all Witt rings of fields with up to $128$ square classes are of elementary type.

math.RA↗
arXiv2026

Graded differential polynomial rings

We study differential polynomial rings $R[t;δ]$ over $Γ$-graded rings, where $Γ$ is an arbitrary group. We show that $R[t;δ]$ admits a $Γ$-grading compatible with that of $R$ if and only if $δ$ is a $γ$-derivation for some $γ\in C_Γ(Γ_R)$, and that this grading is unique once $°(t)=γ$ is fixed; if $δ\neq0$, then $γ$ is itself uniquely determined by $δ$. We characterize the resulting graded ring by a universal property. We prove a characteristic-free center criterion for gr-simplicity whenever $Z(R[t;δ])$ is a graded subring; in characteristic zero, gr-simplicity is equivalent to $δ$-gr-simplicity of $R$ and $γ$-outerness of $δ$, extending Jordan's simplicity criterion to the graded setting. We further show that $R[t;δ]$ is gr-prime if and only if $R$ is $δ$-gr-prime, and that gr-Noetherianity of $R$ passes to $R[t;δ]$, recovering a graded Hilbert basis theorem as a special case. When $Γ$ is abelian, gr-simplicity and gr-primality are shown to be invariants of homogeneous graded Morita equivalence, and every ring homogeneously graded equivalent to $R[t;δ]$ via a compatible idempotent is again a graded differential polynomial ring.

math.RA↗
arXiv2026

Affinization of algebraic structures: Poisson algebras

An affinization of the notion of a Poisson algebra is presented. This is termed a Poisson affgebra and consists of an affine space together with an associative bi-affine multiplication and a bi-affine Lie bracket that acts as an affine derivation for the associative product. The constructive relation between Poisson affgebras and Poisson algebras is described and several low-dimensional examples are studied in detail.

math.RA↗