Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 1076.16019
Aslaksen, H.; Drensky, V.; Sadikova, L.
Defining relations of invariants of two $3\times 3$ matrices. (English)
[J] C. R. Acad. Bulg. Sci. 58, No. 6, 617-622 (2005). ISSN 0861-1459

Let $K$ be a field of characteristic 0 and let $C_{nd}$ be the algebra of invariants of the general linear group $GL_n(K)$ acting by simultaneous conjugation on $d$ matrices of size $n\times n$. The algebra $C_{nd}$ is generated by the traces of a finite number of products $\text{tr}(X_{i_1}\cdots X_{i_k})$, where $X_i$ are $d$ generic $n\times n$ matrices. By a result of {\it Y. Teranishi} [Nagoya Math. J. 104, 149-161 (1986; Zbl 0615.16013)], $C_{32}$ is generated by the traces of all nine products of degree $\leq 3$ of the generic $3\times 3$ matrices $X,Y$ and by $\text{tr}(X^2Y^2),\text{tr}(X^2Y^2XY)$. The only defining relation of $C_{32}$ is a quadratic equation of $\text{tr}(X^2Y^2XY)$ with coefficients depending on the other ten generators. The explicit (but very complicated) form of the equation was found by {\it K. Nakamoto} [J. Pure Appl. Algebra 166, No. 1-2, 125-148 (2002; Zbl 1001.15022)] with respect to a slightly different system of generators.\par In the paper under review the authors give another system of generators of $C_{32}$ and a defining relation which is much simpler than that of Nakamoto. The proofs are sketched only and will be published elsewhere. (The complete version is posted at the preprint server of Cornell University as the preprint {\tt http://xxx.lanl.gov/abs/math.RA/0405389}.) They use representation theory of $GL_2(K)$ to determine the hypothetic candidate for the relation and standard procedures of Maple to verify that it is really the relation.
[Vesselin Drensky (Sofia)]

MSC 2000:: *16R30 Trace rings and invariant theory (assoc. rings and algebras)
13A50 Invariant theory
15A72 Vector and tensor algebra
13P10 Polynomial ideals, Groebner bases

Keywords: matrix invariants; defining relations; Hilbert series; algebras of invariants

Citations: Zbl 0615.16013; Zbl 1001.15022

Cited in: Zbl 1142.16009

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]