Synopsis
We prov e tw o distinct bu t closely relate d results . Th e first i s the extensio n
of the Standard basi s theore m t o superalgebras (define d below) . Th e second is
the applieatio n o f the Standard basi s theore m t o the computation o f invariants
(and, mor e generally , o f eovariants) o f Symmetrie an d skew-symmetric tensors .
In thi s Synopsi s w e give a n informal descriptio n o f the main idea s an d result s
which ean be read independentl y o f the body of the work and which can be used
as a guideline to the text.
We begi n b y recalling th e three fundamenta l algebrai c System s o f invarian t
theory: th e Symmetri e algebra , th e divide d power s algebra , an d the exterio r
algebra.
Given a n aiphabe t (tha t is , a se t whos e element s ar e
to be viewed as "variables"), the Symmetrie algebr a Sym.m(A°)
generated by ^4° is the familiär eommutativ e algebr a of polyno-
(1) mial s in the variables A°. Th e coefficients o f these polynomial s
will be integers, althoug h (her e and everywhere below ) a n arbi-
trary eommutativ e rin g with identit y coul d be taken as the ring
of coefficients .
Given a n aiphabet A~, th e exterior algebr a Exi(A~) i s the al-
gebra generate d b y the variables J4~ , subjee t t o the identitie s
ab = —ba and a 2 = 0 for a, & G A~. Thus , Ext(A~ ) i s the
algebra o f "polynomial s i n anticommutative variable s A~. " A
nonzero monomial i n Ext(A~) i s a produet o f a finite sequenc e
of variable s
where no two Ü{ eoineide, and two monomials are related by the
familiär "sig n law"
OIÖ2 '-a
n
= (sgncr)a
CT
iaa2 ' ^ n
for an y permutation o of the set {1,2,..., n}.
vii
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