What does $\mathbb{C}[G ]\otimes_{\mathbb{C}[H]} W$ mean?

Question

I am reading the group representation book by Serre. In chapter 7 (which is about induced representation), he introduces the notation

$$ \mathbb{C}[G] \otimes_{\mathbb{C}[H]} W .$$

What does this notation mean? I understand $W$ as a $H$-module, or a representation of $H$. I also can accept $\mathbb{C}[G] \otimes W$ as a tensor product space of $(\sum_{g\in G} a_g g , w)$, where $w\in W$. But how to understand the $ \mathbb{C}[H] $ in the subscript?

Answer 1

Let $R$ be a ring. Given a right $R$-module $M$ and left $R$-module $N$, we can form the tensor product $M ⊗_R N$. This tensor product can be described as the abelian group with generators $m ⊗ n$, where $m$ and $n$ range through $M$ and $N$ respectively, subject to the the relations \begin{align*} (m_1 + m_2) ⊗ n &= m_1 ⊗ n + m_2 ⊗ n \,, \\ m ⊗ (n_1 + n_2) &= m ⊗ n_1 + m ⊗ n_2 \,, \\ (m r) ⊗ n &= m ⊗ (r n) \end{align*} for all $m, m_1, m_2 ∈ M$, $n, n_1, n_2 ∈ N$, $r ∈ R$.

Let now $S$ be another ring and let $M$ not just be a right $R$-module, but actually an $S$-$R$-bimodule. The tensor product $M ⊗_R N$ can then be made into a left $S$-module such that $$ s ⋅ (m ⊗ n) = (s m) ⊗ n $$ for all $s ∈ S$, $m ∈ M$, $n ∈ N$.

Let us consider the special case that $R$ is a subring of $S$. We can then regard $S$ as an $S$-$R$-bimodule. For every left $R$-module $M$ we can thus form the left $S$-module $S ⊗_R M$. This construction of going from left $R$-modules to $S$-modules is the extension of scalars from $R$ to $S$. It satisfies the adjunction $$ \operatorname{Hom}_S( S ⊗_R M, N ) ≅ \operatorname{Hom}_R( M, N ) $$ for every left $R$-module $M$ and every left $S$-module $N$. (On the right hand side, we regard $N$ as a left $R$-module by restriction of scalars.)

Suppose now that $G$ is a group, $H$ is a subgroup of $G$, and $V$ is a representation of $H$. We would like to extend the representation $V$ into a representation of $G$. In other words, we would like to extend the $ℂ[H]$-module $V$ to a $ℂ[G]$-module. The group algebra $ℂ[H]$ is a subalgebra of the group algebra $ℂ[G]$, so we can use the extension of scalars $$ ℂ[G] ⊗_{ℂ[H]} V $$ to solve this problem. This is then called the induced representation of $V$ from $H$ to $G$, and often denoted by $\operatorname{Ind}^G_H(V)$. This induced representation satisfies the adjunction $$ \operatorname{Hom}_G( \operatorname{Ind}^G_H(V), W ) = \operatorname{Hom}_{ℂ[G]}( ℂ[G] ⊗_{ℂ[H]} V, W ) ≅ \operatorname{Hom}_{ℂ[H]}( V, W ) = \operatorname{Hom}_H( V, W ) $$ for every representation of $V$ of $H$ and every representation $W$ of $G$. In this context of group representations, this adjunction is often referred to as Frobenius reciprocity.

More details about tensor products and extension of scalars can, for example, be found in section 10.4 of Abstract Algebra (3rd edition) by Dummit and Foote.