## Constructing new representations

Posted: February 10, 2011 in Representations of Finite Groups
Tags: , ,

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.