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3. For $G_1$ and $G_2$ finite groups and $V_1$, $V_2$ finite dimensional complex representations of $G_1$, $G_2$ respectively, the tensor product $V_1 \otimes V_2$ is given an action of $G_1 \times G_2$ by $(g_1, g_2). (v_1 \otimes v_2) = (g_1.v_1) \otimes (g_2.v_2).$ (a) Show that this is a representation. (b) Calculate its character in terms of $\chi_{V_1}$ and $\chi_{V_2}$. (c) If $V_1$, $V_2$ are irreducible, show that $V_1 \otimes V_2$ is irreducible. (d) Show that all irreducible representations of $G_1 \times G_2$ arise this way.

          3. For $G_1$ and $G_2$ finite groups and $V_1$, $V_2$ finite dimensional complex representations of $G_1$, $G_2$ respectively, the tensor product $V_1 \otimes V_2$ is given an action of $G_1 \times G_2$ by
$(g_1, g_2). (v_1 \otimes v_2) = (g_1.v_1) \otimes (g_2.v_2).$
(a) Show that this is a representation.
(b) Calculate its character in terms of $\chi_{V_1}$ and $\chi_{V_2}$.
(c) If $V_1$, $V_2$ are irreducible, show that $V_1 \otimes V_2$ is irreducible.
(d) Show that all irreducible representations of $G_1 \times G_2$ arise this way.

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3. For G1 and G2 finite groups and V1, V2 finite dimensional complex representations of G1, G2 respectively, the tensor product V1 ⊗ V2 is given an action of G1 × G2 by
(g1, g2). (v1 ⊗ v2) = (g1.v1) ⊗ (g2.v2).
(a) Show that this is a representation.
(b) Calculate its character in terms of χV1 and χV2.
(c) If V1, V2 are irreducible, show that V1 ⊗ V2 is irreducible.
(d) Show that all irreducible representations of G1 × G2 arise this way.

Added by Kayla O.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

03/16/2023

Step 1

(v₁ ⊗ v₂) = (g₁.v₁) ⊗ (g₂.v₂) satisfies the following properties: 1. **Identity:** (e₁, e₂).(v₁ ⊗ v₂) = v₁ ⊗ v₂ for all v₁ ∈ V₁ and v₂ ∈ V₂. 2. **Compatibility:** (g₁, g₂).((h₁, h₂).(v₁ ⊗ v₂)) = ((g₁, g₂)(h₁, h₂)).(v₁ ⊗ v₂) for all g₁, h₁ ∈ G₁, g₂, h₂ ∈ G₂, and v₁ Show more…

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3. For G₁ and G₂ finite groups and V₁, V₂ finite dimensional complex representations of G₁, G₂ respectively, the tensor product V₁ ⊗ V₂ is given an action of G₁ × G₂ by $$(g_1, g_2).(v_1 \otimes v_2) = (g_1.v_1) \otimes (g_2.v_2).$$ (a) Show that this is a representation. (b) Calculate its character in terms of χv₁ and χv₂. (c) If V₁, V₂ are irreducible, show that V₁ ⊗ V₂ is irreducible. (d) Show that all irreducible representations of G₁ × G₂ arise this way.

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