# Tensor product of linear representations

## Definition

Suppose $G$ is a group and $\rho_1:G \to GL(V_1)$ and $\rho_2:G \to GL(V_2)$ are linear representations of $G$ over a field $K$. The tensor product of the representations, denoted $\rho_1 \otimes_K \rho_2$ is a linear representation $\rho_1 \otimes \rho_2:G \to GL(V_1 \otimes V_2)$ on the tensor product of the vector spaces $V_1 \otimes_K V_2$ defined in the following equivalent ways.

### Definition using tensor product of linear maps

as follows: for $g \in G$, $(\rho_1 \otimes \rho_2)(g) = \rho_1(g) \otimes \rho_2(g)$. Here, $\rho_1(g) \otimes \rho_2(g)$ is the image of the pair $(\rho_1(g),\rho_2(g))$ in the natural homomorphism $GL(V_1) \otimes GL(V_2) \to GL(V_1 \otimes V_2)$.

Conceptually, the mapping:

$GL(V_1) \otimes GL(V_2) \to GL(V_1\otimes V_2)$

is described as follows: we know that $V_1,V_2$ up to isomorphism determine $V_1 \otimes V_2$ up to isomorphism. This means that any choice of automorphism of $V_1$ along with automorphism of $V_2$ induces an automorphism of $V_1 \otimes V_2$. The mapping describes how.

### Definition using outer tensor product

The tensor product $\rho_1 \otimes \rho_2$ can be defined as the composite of the outer tensor product of linear representations of $\rho_1$ and $\rho_2$ (which gives a linear representation of $G \times G$) with the diagonal inclusion map $g \mapsto (g,g)$, a homomorphism from $G$ to $G \times G$.

### Definition in terms of symmetric bimonoidal category

The tensor product of linear representations over a field $K$ can be defined as the tensor product of representations over a symmetric bimonoidal category where the category is the category of $K$-vector spaces, the additive operation is direct sum of vector spaces, and the multiplicative operation is tensor product of vector spaces.

### Explicit definition in terms of block matrices

This definition works for finite-dimensional linear representations, though it also has infinite-dimensional analogues if we use infinitary matrices.

We use the same notation as in the previous definition, but assume further that $V_1 = K^m$ and $V_2 = K^n$. Then $V_1 \otimes V_2$ 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 $\rho_2(g)$. Then, replace each cell of the matrix by a $n \times n$ matrix that equals the cell value times $\rho_1(g)$. Overall, we get a $mn \times mn$ matrix.

## Definition over a commutative unital ring

Suppose $G$ is a group and $\rho_1:G \to GL(V_1)$ and $\rho_2:G \to GL(V_2)$ are linear representations of $G$ over a commutative unital ring $R$. The tensor product of the representations, denoted $\rho_1 \otimes_R \rho_2$ is a linear representation $\rho_1 \otimes \rho_2:G \to GL(V_1 \otimes V_2)$ on the tensor product of the modules $V_1 \otimes_R V_2$ defined as follows: for $g \in G$, $(\rho_1 \otimes \rho_2)(g) = \rho_1(g) \otimes \rho_2(g)$. Here, $\rho_1(g) \otimes \rho_2(g)$ is the image of the pair $(\rho_1(g),\rho_2(g))$ in the natural homomorphism $GL(V_1) \otimes GL(V_2) \to GL(V_1 \otimes V_2)$.

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.

Note that the explicit matrix description of tensor product is available only if the modules $V$ and $W$ are free modules of finite rank over $R$.