Tensor product of linear representations

This article gives a basic definition in the following area: linear representation theory
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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 $R$.

Related notions

• Generalization: tensor product of representations over a symmetric bimonoidal category
• Tensor product of permutation representations

Facts

• Degree of tensor product of linear representations is product of degrees
• Character of tensor product of linear 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