Tensor product of linear representations

Definition

Suppose $G$ is a group and $\alpha :G\to GL(V)$ and $\beta :G\to GL(W)$ are linear representations of $G$ over a field $K$. The tensor product of the representations, denoted $\alpha \otimes \beta$ is a linear representation $\alpha \otimes \beta :G\to GL(V\otimes W)$ on the tensor product of the vector spaces $V\otimes _{K}W$ defined as follows: for $g\in G$, $(\alpha \otimes \beta )(g)=\alpha (g)\otimes \beta (g)$. Here, $\alpha (g)\otimes \beta (g)$ is the image of the pair $(\alpha (g),\beta (g))$ in the natural homomorphism $GL(V)\otimes GL(W)\to GL(V\otimes W)$

Conceptually, the mapping:

$GL(V)\otimes GL(W)\to GL(V\otimes W)$

is described as follows: we know that $V,W$ up to isomorphism determine $V\otimes W$ up to isomorphism. This means that any choice of automorphism of $V$ along with automorphism of $W$ induces an automorphism of $V\otimes W$. The mapping describes how.

Explicit definition in terms of block matrices

We use the same notation as in the previous definition, but assume further that $V=K^{m}$ and $W=K^{n}$. Then $V\otimes W$ can be identified with $K^{mn}$ where the first $m$ coordinates represent one copy of $K^{m}$, the next $m$ copies represent the next copy of $K^{m}$, and so on. The explicit definition is now given as follows: for $g\in G$, first write the $n\times n$ matrix for $\beta (g)$. Then, replace each cell of the matrix by a $n\times n$ matrix that equals the cell value times $\alpha (g)$. Overall, we get a $mn\times mn$ matrix.

Facts

• Degree of tensor product of representations is product of degrees
• Character of tensor product of representations is product of characters
• Tensor product of irreducible representation and one-dimensional representation is irreducible
• Tensor product of irreducible representations need not be irreducible