Exercise 7, § 5

Algebra, III, 630

Bourbaki marks this one as harder than the rest.

Let E be a vector space of finite dimension n over a field K. For every integer r $\geqslant 1$ , the symmetric group $S_r$ operates linearly on the tensor product $E^{\otimes r}$ by the action $(\sigma, z) \mapsto \sigma . z$ such that

\sigma . (x_1 \otimes x_2 \otimes \cdots \otimes x_r) = x_{\sigma^{-1}(1)} \otimes x_{\sigma^{-1}(2)} \otimes \cdots \otimes x_{\sigma^{-1}(r)}

for every sequence of r vectors $(x_i)_{1 \leq i \leq r}$ of E. Let $u_\sigma(z) = \sigma . z$ . Show that for every sequence $(v_i)_{1 \leq i \leq r}$ of r endomorphisms of E,

(*)

\operatorname{Tr}(u_\sigma \circ (v_1 \otimes v_2 \otimes \cdots \otimes v_r)) = \operatorname{Tr}(w_1) \operatorname{Tr}(w_2) \ldots \operatorname{Tr}(w_k)

where the endomorphisms $w_j$ of E are defined as follows: $\sigma^{-1}$ is decomposed into cycles, $\sigma^{-1} = \lambda_1 \lambda_2 \ldots \lambda_k$ and, if $\lambda_j = (i_1 \ i_2 \ldots i_n)$ , we write

w_j = v_{i_1} \circ v_{i_2} \circ \cdots \circ v_{i_n}.

(Reduce it to calculating the left hand side of (*) when each $v_i$ is of the form $x \mapsto \langle x, a_h^* \rangle a_k$ , where $(a_j)_{1 \leq j \leq n}$ is a basis of E and $(a_j^*)_{1 \leq j \leq n}$ the dual basis.)

Solution

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