Character of tensor product of linear representations is product of characters

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Statement

Statement for a field

Suppose G is a group (not necessarily finite) and \alpha, \beta are finite-dimensional linear representations of G over a field K. Denote by \chi_\alpha, \chi_\beta respectively the characters of \alpha and \beta. Denote by \alpha \otimes \beta the tensor product of linear representations \alpha and \beta and by \chi_{\alpha \otimes \beta} its character. Then, for any g \in G, we have:

\chi_{\alpha \otimes \beta}(g) = \chi_\alpha(g)\chi_\beta(g)

In point-free notation, this states that \chi_{\alpha \otimes \beta} = \chi_\alpha\chi_\beta.