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The Cartesian product in the category of groups is often called the direct product of groups. For abelian groups and a finite number of factors, this is also the direct sum of groups.
(Compare the dual notion of “free products” of groups, which are really their category theoretic coproducts.)
(irreps of direct product groups are external tensor products of irreps)
Let $G_1, G_2$ be two groups. Then, over an algebraically closed ground field, every irreducible representation $\rho \in (G_1 \times G_2) Rep_{irr}$ of their direct product group $G_1 \times G_2$ is the external tensor product of irreducible representations $\rho_i \in G_i Rep_{irr}$ of the two groups separately:
Here the external tensor product has as underlying vector space the corresponding tensor product of vector spaces, equipped with the evident action
By Schur's lemma see e.g. here.
The statement of Prop. is in general false if the ground field is not algebraically closed. A counterexample is given im Kowalski 13, Example 2.7.31.
Also the converse to Prop. is false in general. The external tensor product of irreducible representations need not be irreducible itself. For more see Fein 67.
E. Kowalski, around p. 41 of Representation theory, 2013 (KowalskiReptheory2013.pdf)
Burton Fein, Representations of direct products of finite groups, Pacific J. Math. Volume 20, Number 1 (1967), 45-58 (Euclid:1102992967) See also
Wikipedia, Direct product of groups
Last revised on May 16, 2023 at 14:06:11. See the history of this page for a list of all contributions to it.