Throughout this post G is a finite group.

First Construction. Suppose that \rho_i : G \longrightarrow \text{GL}(V_i), \ i=1,2, are two representations of G. It is straightforward to show that the map \rho_1 \oplus \rho_2 : G \longrightarrow \text{GL}(V_1 \oplus V_2) defined by

(\rho_1 \oplus \rho_2)(g)(v_1,v_2) = (\rho_1(g)(v_1), \rho_2(g)(v_2)),

for all g \in G and v_1 \in V_1, \ v_2 \in V_2 is a representation of G. It is also clear that

\deg (\rho_1 \oplus \rho_2) = \dim_{\mathbb{C}}(V_1 \oplus V_2) = \dim_{\mathbb{C}}V_1 + \dim_{\mathbb{C}} V_2 = \deg \rho_1 + \deg \rho_2.

Second Construction. Suppose that \rho_i : G \longrightarrow \text{GL}(V_i), \ i=1,2, are two representations of G. It is easy to see that the map \rho_1 \otimes \rho_2 : G \longrightarrow \text{GL}(V_1 \otimes_{\mathbb{C}} V_2) defined by

(\rho_1 \otimes \rho_2)(g)(v_1 \otimes_{\mathbb{C}} v_2) = \rho_1(g)(v_1) \otimes_{\mathbb{C}} \rho_2(g)(v_2),

for all g \in G and v_1 \in V_1, \ v_2 \in V_2, and then of course extended linearly to all elements of V_1 \otimes_{\mathbb{C}} V_2, is a representation of G. In this case we have

\deg (\rho_1 \otimes \rho_2) = \dim_{\mathbb{C}} (V_1 \otimes_{\mathbb{C}} V_2) = (\dim_{\mathbb{C}}V_1)(\dim_{\mathbb{C}}V_2) = (\deg \rho_1)(\deg \rho_2).

Third Construction. Suppose that \rho : G \longrightarrow \text{GL}(V) is a representation of G. Let V^* be the dual space of V, i.e. the set of all linear maps f : V \longrightarrow \mathbb{C}. The map \rho^* : G \longrightarrow \text{GL}(V^*) defined by

\rho^*(g)(f)=f \rho(g^{-1}),

for all g \in G and f \in V^*, is a representation of G. We will call \rho^* the dual representation of \rho. Clearly

\deg \rho^* = \dim_{\mathbb{C}} V^* = \dim_{\mathbb{C}}V=\deg \rho.

In the next post I will discuss the dual representation in more details.

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