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Erschienen in:
2008 | OriginalPaper | Buchkapitel
Orbit Sums and Modular Vector Invariants
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Let
m, n
positive integers,
R
a commutative ring with the unit element 1, and
$$ A_{mn} = R\left[ {x_{11} , \ldots ,x_{m1} ; \ldots ;x_{1n} , \ldots ,x_{mn} } \right] $$
the algebra of polynomials in
mn
variables x
ij
over
R.
The symmetric group
S
n
operates on the algebra
A
mn
as a group of
R
-automorphisms by the rule: σ(x
ij
)
=
x
i
,σ(
j
),
σ
∈
G.
Denote by
$$ A_{mn}^{S_n } $$
the subalgebra of invariants of the algebra
A
mn
with respect to
S
n
and define polarized elementary symmetric polynomials
$$ u_{r_1 , \ldots ,r_m } \in A_{mn}^{S_n } $$
in
n
vector variables (x
11
,..., x
m1
),..., (x
1n
,..., x
mn
) by means of the following formal identity
$$ \prod\limits_{j = 1}^n {\left( {1 + x_1 jz_1 + \cdots + x_{mj} z_m } \right) = 1 + \sum\limits_{1 \leqslant r_1 + \cdots + r_m \leqslant n} {u_{r1, \ldots rm} z_1^{r_1 } \cdots z_m^{r_m } .} } $$
