[[Group representation theory MOC]]
# Tensor product of group representations

Given two representations $\mathfrak{X} : G \to \mathrm{GL}(V)$ and $\mathfrak{Y} : G \to \mathrm{GL}(W)$,
the **tensor product** $\mathfrak{X} \otimes \mathfrak{Y} : G \to \mathrm{GL(V \otimes W)}$ is defined using the [[tensor product of linear maps]] as
$$
\begin{align*}
(\mathfrak{X} \otimes \mathfrak{Y})(G) = \mathfrak{X}(G) \otimes \mathfrak{Y}(G)
\end{align*}
$$
We denote the tensor product of irreps as $\mathfrak{X}^\mu \otimes \mathfrak{X}^\nu = \mathfrak{X}^{\mu \otimes \nu}$

## See also

- [[Tensor product of Lie algebra representations]].

#
---
#state/develop | #lang/en | #SemBr