Orthogonal decompositions and canonical embeddings of multilinear alternating forms. (English) Zbl 1049.15022

Hora, Jan

Linear Multilinear Algebra 52, No. 2, 121-132 (2004).

\(k\)-linear forms \(f\) for \(k \geq 2\) on \(V\) are studied, where \(V\) is a vector space with dim \(V=n < \infty\) over a field \(F\). It is assumed that for every nonzero vector \(u\) there exist vectors \(u_1, \dots,u_{k-1}\) such that \(f(u,u_1, \dots, u_{k-1}) \neq 0\) (i. e., \(f\) is nondegenerate). Two vectors \(u, v\in V\) are orthogonal (\(u \perp v\)) if the value \(f\) is \(0\) whenever \(u\) and \(v\) are two of the arguments. Subspaces \(W_1\), \(W_2\) are called orthogonal if \(w_1 \perp w_2\) for every \(w_1 \in W_1\) and \(w_2 \in W_2\). This paper describes the set of all decompositions \(W_1 \oplus \dots \oplus W_l\) of \(V\) such that \(W_i\) is orthogonal to \(W_j\) for \(i \neq j\). All such decompositions are called orthogonal decompositions of \(f\).
Orthogonal decompositions are the important tool when classifying symmetric and alternating bilinear forms. When investigating \(k\)-linear forms for \(k \geq 3\) there are many forms that are not orthogonally decomposable. It is proved here that for \(k \geq 3\) there always exists the unique minimal (i. e. the finest) orthogonal decomposition of a nondegenerate form \(f\). Let \(h: (F^k)^k \to F\) be a determinant, i. e., a nonzero \(k\)-linear form on a vector space of dimension \(k\). It is shown that there exists \(c \geq 1\) such that \(f\) can be interpreted as a restriction of the form \(h^c\) for a suitable embedding of \(V\) to \((F^k)^c\). Such minimal \(c\) is called the complexity of \(f\). It is proved that the complexity of a form \(f\) is less or equal to its efficiency which notion is also introduced.

Reviewer: Václav Burjan (Praha)

MSC:

15A69

Multilinear algebra, tensor calculus

Keywords:

\(k\)-linear alternating form; orthogonal decomposition; efficiency; complexity; symmetric and alternating bilinear forms

References:

[1]	DOI: 10.1080/00927878808823558 · Zbl 0646.15019 · doi:10.1080/00927878808823558
[2]	Chan W, Birkhauser Boston pp pp63– (1998)
[3]	Rains EM, Handbook of Coding Theory pp pp177– (1998)
[4]	Aschbacher M, Cambridge Tracts in Mathematics 104 (1994)
[5]	DOI: 10.1016/0021-8693(90)90087-5 · Zbl 0695.20040 · doi:10.1016/0021-8693(90)90087-5
[6]	Vojtĕchovský P, Comment. Math. Univ. Carolinae 41 pp 429– (2000)
[7]	DOI: 10.1016/0021-8693(86)90075-X · Zbl 0589.20051 · doi:10.1016/0021-8693(86)90075-X
[8]	DOI: 10.4153/CMB-1978-066-7 · Zbl 0394.15004 · doi:10.4153/CMB-1978-066-7