Example 4

Tag 00NH, permanent. Cite it as [Bourbaki, Tag 00NH].

Algebra, VIII, p. 399, § 21. Linear Representations of Finite Groups

Let $(M, \pi )$ and $(M', \pi ')$ be linear representations of G. In Example 1 of VIII, p. 198, we defined a K[G]-module structure on $M\otimes_KM'$ . The corresponding linear representation is called the tensor product of $\pi$ and $\pi '$ and is denoted by $\pi \otimes \pi '$ . For $g\in G,x\in M$ , and $x'\in M'$ , we have $((\pi \otimes \pi ')(g))(x\otimes x') =\pi (g)x\otimes \pi '(g)x'$ .

If M and M' are finite-dimensional free K-modules, then by Proposition 2 of III, §9, No. 2, p. 543, we have

(3)

\chi_{\pi \otimes \pi'} = \chi_{\pi} \chi_{\pi'}.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Proposition 2 of III, §9, No. 2, p. 543 not in this corpus
Example 1 of VIII, p. 198