nLab direct product group (Rev #11, changes)

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Context

Group Theory

Idea

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.)

Properties

Representations

Proposition

(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:

\rho \;=\; \rho_1 \boxtimes \rho_2 \,.

Here the external tensor product has as underlying vector space the corresponding tensor product of vector spaces, equipped with the evident action

(g_1, g_2)( (v_1 \otimes v_2) ) \;=\; \big( g_1(v_1) \otimes g_2(v_2) \big) \,.

(cf. Fulton & Harris 1991 Ex. 2.36)

Proof

By Schur's lemma see e.g. here.

Remark

The statement of Prop. 1 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. 1 is false in general. The external tensor product of irreducible representations need not be irreducible itself. For more see Fein 67.

Related concepts

References

The universal property of direct product groups understood as Cartesian products of groups (and its formal duality to the coproduct for the case of free product groups) was first identified in:

Saunders MacLane: Groups, Categories and Duality, Proceedings of the National Academy of Sciences 34 6 (1948) [doi:10.1073/pnas.34.6.263]
Saunders MacLane; (3.1) in: Duality for Groups, Bulletin of the American Mathematical Society 56 6 (1950) 485–516 [[doi:10.1090/S0002-9904-1950-09427-0](https://doi.org/10.1090/S0002-9904-1950-09427-0)]