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Skew-symmetric tensor decomposition. (English) Zbl 1458.15044

Let \(V\) be a vector space of dimension \(n+1\) defined over \(K\), an algebraically closed field of characteristic zero. Denote by \(\wedge^d V\) the \(d\)-th exterior power of \(V\), where \(d=1, \dots, n+1\), i.e., generated by the elements of the form \(e^\wedge= e_{i_1}\wedge \cdots \wedge e_{i_d}\) which are known as decomposable skew-symmetric tensors (see [M. Marcus, Finite dimensional multilinear algebra. Part I. New York, NY: Marcel Dekker, Inc. (1973; Zbl 0284.15024); Part II. New York, NY: Marcel Dekker, Inc. (1975; Zbl 0339.15003); R. Merris, Multilinear algebra. Langhorne, PA: Gordon & Breach (1997; Zbl 0892.15020)]). The vectors \(v\in\wedge^d V\) are called skew-symmetric tensors. The exterior powers are venue for Élie Cartan’s powerful exterior derivative of a differential form in differential geometry. A basic question is: given \(v\in\wedge^d V\), how can we find a decomposition \[ v =\sum_{i=1}^r\lambda_i v_i^\wedge, \] into a summand of decomposable skew-symmetric tensors with \(r\), the number of summands, minimum among all such decompositions? Such \(r\) clearly exists and is called the rank of \(v\). From the algebraic geometry point of view, it corresponds to find the minimum number \(r\) of distinct points on a Grassmannian \({\mathbb G}(d, V)\) whose span contains the given tensor \(v\).
The authors introduce the “skew apolarity lemma” and use it to give algorithms for the skew-symmetric rank and the decompositions of tensors in \(\wedge^d V\). Some examples of ideals of points in the skew-symmetric algebra are presented and analyzed. Some nice diagrams are given for illustration. A complete analysis for tensors in \(\wedge^3 V\) with \(\dim v\le 8\) and algorithms are given.
Note: The authors gives another proof of a trivector result of R. Westwick [Pac. J. Math. 80, 575–579 (1979; Zbl 0365.15012)]. Readers may be interested in two other papers of R. Westwick [Linear Multilinear Algebra 10, 183–204 (1981; Zbl 0464.15001); Linear Multilinear Algebra 9, 1–4 (1980; Zbl 0439.15014)].
Reviewer: Tin Yau Tam (Reno)

MSC:

15A69 Multilinear algebra, tensor calculus
15A75 Exterior algebra, Grassmann algebras
15A23 Factorization of matrices
14M15 Grassmannians, Schubert varieties, flag manifolds
16W22 Actions of groups and semigroups; invariant theory (associative rings and algebras)
16Z05 Computational aspects of associative rings (general theory)
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