the non-commutative polynomial ring in the variables $X_1,...,X_n$. Let $R^m$ denote the vector subspace of $R$ consisting of non-commutative monomials in $X_1,...,X_n$ of degree $m$. Let $w_1,...,w_M$ denote an ordered basis of $R^m$ consisting of non-commutative monomials of degree m (words). Note that $M=n^m$. Then $V^{\otimes m}$ can be identified with $R^m$, the identification compatible with the action of $Sl(V)$. For any extension field $[L:K]$ we will think of elements $g \in G(L)$ as $n \times n$ matrices $g_{ij}$ with coefficients in $L$.\
Consider the commutative polynomial ring $B=K[G_{ij}]$. Any $g=g_{ij} \in G(L)$ can thus be thought of as a $L$-valued point of Spec $B$. Let $v=\sum a_iw_i$ be any element in $R^{\otimes m}$ . We define the elementary polynomials associated to $v$ as follows:\
Denote by $K[G_{ij}]
\langle
X_1,...,X_n \rangle$ the noncommutative ring in the variables $X_i$, with coefficients in the commutative polynomial ring $K[G_{ij}]$. Consider the set mapping $\theta: K\langle X_1,...,X_n \rangle
\rightarrow K[G_{ij}] \langle
X_1,...,X_n \rangle$ defined as follows:\
$\theta$ sends a variable $X_i$ in $K\langle X_1,...,X_n \rangle$ to $\sum G_{ij} \cdot X_j$ and extends the action in the obvious way to $K\langle X_1,...,X_n \rangle$. The ordered set of coefficients of the various noncommutative monomials in the 