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
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