The present work is intended to develop three main results:
(1) An extension of the Standard basis theorem, going back to Doubilet, Rota,
and Stein, and eventually to Capelli and Young, to algebras containing positively
signed and negatively signe d variables, or superalgebras a s we call them. Suc h
an extensio n ha s require d a rethinking o f some of the basi c concept s o f linear
algebra, such as "matrix " an d "coordinat e system," alon g lines that w e believe
to be new, and which we hope will lead to an extension to "signed " modules of
the entir e apparatu s o f linear algebra . Th e Standar d basi s theorem, which we
prove, is characteristic-free an d includes, besides the classical case, straightening
algorithms, which apply to permanents a s well as to determinants, as well as a
mixed generalizatio n o f the notion o f both determinan t an d permanent, calle d
the biproduct, which differs fro m th e Berezin determinant .
(2) A rigorous presentatio n o f th e symboli c metho d o f invarian t theor y fo r
Symmetrie tensors, in characteristic zero. The results here offer no great novelty
over the nineteenth Century, except rigor.
(3) A new symboli c metho d (foreshadowe d b y Weitzenböck ) fo r th e repre -
sentation of invariants of skew-symmetric tensors . Here , the results turn out to
be more satisfactory. Symboli c expressions for the invariants of skew-symmetric
tensors ar e more manageable an d easie r t o comput e tha n thos e fo r Symmetri e
tensors. I n fact, i n contrast t o Symmetrie tensors, the "meaning " o f the vanish-
ing of an invariant can be more easily gleaned from the symbolic representation,
as we show by several examples.
In both instances , th e actua l invarian t i s obtained fro m th e symboli c repre-
sentation by applying an Operator which we call the umbral Operator. Invariant s
of Symmetrie tensor s ar e obtained b y applyin g th e umbral Operato r t o certai n
polynomials i n a commutativ e algebra , wherea s invariant s o f skew-symmetri c
tensors are obtained by applying the umbral Operator to polynomials in an anti-
commutative algebra. Thus , the umbral Operator can be viewed as mapping an
anticommutative algebr a int o a commutative algebra , an d vic e versa. I t i s an
instance of a Schur funetor .