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 A° (tha t is , a se t A° 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}.

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