1. Th e Superalgebra Supe r [A]

0. Introduction . W e define in this chapter a generalization of the ordinary

algebra o f polynomials i n a set of A variable s wit h intege r coefficients . Ou r

variables shall be of three kinds: positively signed, neutral, and negatively signed:

A = A+UA

0\JA~.

Positively signed variables are the least familiär: the y are the divided powers.

To every positively signe d variable a we assign a sequence a^\a^

2\a^3\...

o f

divided powers, which behave algebraically "a s if' a^ wer e to equal

a%

ji\ The n

for example we have the "rules"

o«aü = (*' +

J'W'-

and

(a + 6)W= £ flÜ)6(*).

j+k=i

Positively signe d variables and their divide d power s commute. Thi s seemingly

artificial devic e is essential in making invariant theory characteristic-free .

Neutral variables behave like ordinary polynomial variables; in particular they

also commute.

Negatively signe d variable s a, 6 anticommute: ab = —ba an d

a2

= b

2

= 0.

However, we have ac = ca when a is negatively signed and c is a variable (o r a

divided power) of either of the two other kinds.

The superalgebra Super[A] is the algebra spanned by monomials obtained by

multiplying variables of each of the three kinds—for positivel y signed variables,

one multiplies divide d powers . Th e variables appearin g i n a monomial can be

permuted a t will , subjec t onl y t o the condition tha t a minus sig n be prefixed

every time two negatively signed variables are permuted.

i

http://dx.doi.org/10.1090/cbms/069/01