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 $R$.