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2000 Mathematics Subject Classification: 15A69, 15A78.In [3] we present the construction of the semi-symmetric algebra [χ](E) of a module E over a commutative ring K with unit, which generalizes the tensor algebra T(E), the symmetric algebra S(E), and the exterior algebra ∧(E), deduce some of its functorial properties, and prove a classification theorem. In the present paper we continue the study of the semi-symmetric algebra and discuss its graded dual, the corresponding canonical bilinear form,...
The paper studies multilinear algebras, known as comtrans algebras, that are determined by so-called -Hermitian matrices over an arbitrary field. The main result of this paper shows that the comtrans algebra of -dimensional -Hermitian matrices furnishes a simple comtrans algebra.
A trilinear alternating form on dimension can be defined based on a Steiner triple system of order . We prove some basic properties of these forms and using the radical polynomial we show that for dimensions up to nonisomorphic Steiner triple systems provide nonequivalent forms over . Finally, we prove that Steiner triple systems of order with different number of subsystems of order yield nonequivalent forms over .
The multilinear forms, obtained by polarizing the coefficients of the characteristic polynomial of a matrix, are considered. A general relation (formula A) between such forms is proved. It follows in particular a rational expression for the above-mentioned coefficients (formula C), which is in a sense analogous to Newton's formulas, but with the use of the determinant function instead of the trace function.
We introduce the notion of a supersymmetry class of tensors which is the ordinary symmetry class of tensors with a natural ℤ₂-gradation. We give the dimensions of even and odd parts of this gradation as well as their natural bases. Also we give a necessary and sufficient condition for the odd or even part of a supersymmetry class to be zero.
We discuss the existence of an orthogonal basis consisting of decomposable vectors for all symmetry classes of tensors associated with semi-dihedral groups . In particular, a necessary and sufficient condition for the existence of such a basis associated with and degree two characters is given.
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