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direct product group

Contents

Contents

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.

Properties

Representations

Proposition

(irreps of direct product groups are external tensor products of irreps)

Let G 1,G 2G_1, G_2 be two groups. Then, over an algebraically closed ground field, every irreducible representation ρ(G 1×G 2)Rep irr\rho \in (G_1 \times G_2) Rep_{irr} of their direct product group G 1×G 2G_1 \times G_2 is the external tensor product of irreducible representations ρ iG iRep irr\rho_i \in G_i Rep_{irr} of the two groups separately:

ρ=ρ 1ρ 2. \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 1v 2))=(g 1(v 1)g 2(v 2)). (g_1, g_2)( (v_1 \otimes v_2) ) \;=\; ( g_1(v_1) \otimes g_2(v_2) ) \,.
Proof

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

Remark

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.

References

Last revised on September 16, 2018 at 10:14:07. See the history of this page for a list of all contributions to it.