Induced Representations, Frobenius Reciprocity and Applications

Because $T$ meets each left coset $gH$ in exactly one point, every $g\in G$ can be written uniquely as $g=t_{i}h$ with $t_i\in T$ and $h\in H$.

The map ${\mathrm{ev}}_T$ is $R$-linear because [F1] and [F3] define both module structures coordinatewise.

Define $\mathrm{\Phi }:{⨁}_{i=1}^{n}W⟶{\mathrm{Ind}}_H^{G}W$ by $\mathrm{\Phi }(w_1,\dots ,w_n)(t_{i}h):=h^{-1}\cdot w_i$. Step 1.1 makes this well defined, and the displayed formula satisfies the covariance condition of [F1], so $\mathrm{\Phi }(w_1,\dots ,w_n)\in {\mathrm{Ind}}_H^{G}W$.

The map $\mathrm{\Phi }$ is $R$-linear because the $H$-action on $W$ is $R$-linear and the formula of step 2.1 is coordinatewise in the tuple entries.

For $(w_1,\dots ,w_n)\in {⨁}_{i}W$, ${\mathrm{ev}}_T(\mathrm{\Phi }(w_1,\dots ,w_n))=(w_1,\dots ,w_n)$ because $\mathrm{\Phi }(w_1,\dots ,w_n)(t_i)=w_i$.

For $f\in {\mathrm{Ind}}_H^{G}W$ and $g=t_{i}h$ as in step 1.1, one has $\mathrm{\Phi }({\mathrm{ev}}_T(f))(g)=\mathrm{\Phi }(f(t_1),\dots ,f(t_n))(t_{i}h)=h^{-1}\cdot f(t_i)=f(t_{i}h)=f(g)$, where the third equality is the covariance condition from [F1]. Hence $\mathrm{\Phi }({\mathrm{ev}}_T(f))=f$.

Steps 3.2 and 3.3 show that $\mathrm{\Phi }$ and ${\mathrm{ev}}_T$ are inverse $R$-module isomorphisms. Since $T$ has one element on each left coset, its cardinality is $[G:H]$.

Choose a left transversal $T=\{t_1,\dots ,t_n\}$ for $G\mathrm{/}H$; since $G$ is finite, $n=[G:H]$. By [F1], ${\mathrm{Ind}}_H^{G}W\cong {⨁}_{i=1}^{n}W$ as $k$-vector spaces.

Let $B$ be a basis of $W$ with $\mathrm{\mid }B\mathrm{\mid }={\dim }_{k}W$ by [F2]. The vectors supported in one summand and equal there to a basis element of $B$ form a basis of ${⨁}_{i=1}^{n}W$, so that direct sum has $n{\dim }_{k}W$ basis vectors. Hence ${\dim }_k{\mathrm{Ind}}_H^{G}W=n{\dim }_{k}W=[G:H]{\dim }_{k}W$.

The induced module already carries a $k$-linear $G$-action by its definition, and step 2.1 shows that its underlying vector space is finite-dimensional. Therefore it is a finite-dimensional representation of $G$ over $k$ in the sense of [F3].

Steps 2.1 and 3.1 prove the stated dimension formula and finite-dimensionality claim.

Choose a left transversal $T$ for $G\mathrm{/}H$. By [F2], ${\mathrm{Ind}}_H^{G}W\cong {⨁}_{t\in T}{W}_t$, where each $W_t$ is one copy of $W$ indexed by the coset representative $t$.

For $t\in T$, write $gt=t^{\mathrm{\prime }}h$ with $t^{\mathrm{\prime }}\in T$ and $h\in H$. Under the identification of step 1.1, the action of $g$ sends the $t$-summand to the $t^{\mathrm{\prime }}$-summand; if $t^{\mathrm{\prime }}=t$, so $t^{-1}gt=h\in H$, then this action on $W_t$ is exactly the action of $h=t^{-1}gt$ on $W$. Therefore the contribution of the $t$-summand to the trace is $\chi (t^{-1}gt)$ when $t^{-1}gt\in H$, and $0$ otherwise.

The trace of $g$ on the direct sum of step 1.1 is the sum of the traces on the summands fixed by the permutation it induces on $T$. Hence ${\mathrm{Ind}}_H^G\chi (g)={\sum }_{t\in T,t^{-1}gt\in H}\chi (t^{-1}gt)$.

Fix $t\in T$ with $t^{-1}gt\in H$. The elements of the left coset $tH$ are $x=th$ with $h\in H$, and then $x^{-1}gx=h^{-1}{t}^{-1}gth$. By [F3], the character value $\chi (x^{-1}gx)$ is therefore the constant $\chi (t^{-1}gt)$ on that whole coset, and every element of $tH$ contributes to the displayed sum exactly when $t^{-1}gt\in H$. So the total contribution of $tH$ to ${\sum }_{x^{-1}gx\in H}\chi (x^{-1}gx)$ is $\mathrm{\mid }H\mathrm{\mid }\chi (t^{-1}gt)$.

Summing the identity of step 4.1 over the distinct cosets indexed by $T$, and reindexing by the finite partition $G={⨆}_{t\in T}tH$, gives ${\sum }_{x\in G,x^{-1}gx\in H}\chi (x^{-1}gx)=\mathrm{\mid }H\mathrm{\mid }{\sum }_{t\in T,t^{-1}gt\in H}\chi (t^{-1}gt)$. By step 3.1 and [F4], dividing by $\mathrm{\mid }H\mathrm{\mid }$ yields the stated Frobenius formula.

In the trivial representation of $H$, every $h\in H$ acts as the identity on $\mathbb{C}$. So the covariance condition of [F1] becomes $f(gh)=f(g)$ for all $g\in G$ and $h\in H$. Therefore $f$ is constant on each left coset $gH$.

Define $\mathrm{\Phi }:{\mathrm{Ind}}_H^G{\mathbf{1}}_H⟶{\mathbb{C}}^{(G\mathrm{/}H)}$ by $\mathrm{\Phi }(f)(gH):=f(g)$. Step 1.1 makes this well defined, and every function on $G\mathrm{/}H$ pulls back uniquely to an $H$-covariant function on $G$, so $\mathrm{\Phi }$ is a linear bijection.

For $x\in G$, one has $\mathrm{\Phi }(x\cdot f)(gH)=(x\cdot f)(g)=f(x^{-1}g)=\mathrm{\Phi }(f)(x^{-1}gH)$, which is exactly the left permutation action of $G$ on the coset set $G\mathrm{/}H$ from [F2]. Hence $\mathrm{\Phi }$ is $G$-equivariant.

The bijection of step 2.1 and the equivariance of step 3.1 identify ${\mathrm{Ind}}_H^G{\mathbf{1}}_H$ with the permutation representation of $G$ on $G\mathrm{/}H$.

For $w\in W$, define ${\eta }_w:G\to W$ by ${\eta }_w(h):=h^{-1}\cdot w$ for $h\in H$ and ${\eta }_w(g):=0$ for $g\mathrm{\notin }H$. If $x\mathrm{\notin }H$, then $xh\mathrm{\notin }H$ for every $h\in H$, so ${\eta }_w(xh)=0=h^{-1}\cdot 0$; if $x\in H$, then ${\eta }_w(xh)=h^{-1}{x}^{-1}\cdot w=h^{-1}\cdot {\eta }_w(x)$. Thus ${\eta }_w\in {\mathrm{Ind}}_H^{G}W$.

Choose a left transversal $T$ for $G\mathrm{/}H$ with $e\in T$. If $\psi :W\to {\mathrm{Res}}_H^{G}V$ is $H$-equivariant, define $\beta (\psi )(f):={\sum }_{t\in T}t\cdot \psi (f(t))$. The sum is finite because $T$ is finite.

If $\mathrm{\Phi }:{\mathrm{Ind}}_H^{G}W\to V$ is $G$-equivariant, define $\alpha (\mathrm{\Phi })(w):=\mathrm{\Phi }({\eta }_w)$. For $h\in H$, one has ${\eta }_{h\cdot w}=h\cdot {\eta }_w$ by the action formula in [F1], so $\alpha (\mathrm{\Phi })(h\cdot w)=\mathrm{\Phi }(h\cdot {\eta }_w)=h\cdot \mathrm{\Phi }({\eta }_w)=h\cdot \alpha (\mathrm{\Phi })(w)$. Hence $\alpha (\mathrm{\Phi })\in {\mathrm{Hom}}_H(W,{\mathrm{Res}}_H^{G}V)$.

The map $\beta (\psi )$ is $G$-equivariant. Indeed, for $x\in G$ and each $t\in T$, write $x^{-1}t=t^{\mathrm{\prime }}h$ with $t^{\mathrm{\prime }}\in T$ and $h\in H$. Then $t\cdot \psi ((x\cdot f)(t))=t\cdot \psi (f(x^{-1}t))=t\cdot \psi (f(t^{\mathrm{\prime }}h))=t\cdot \psi (h^{-1}\cdot f(t^{\mathrm{\prime }}))=t{h}^{-1}\cdot \psi (f(t^{\mathrm{\prime }}))=x{t}^{\mathrm{\prime }}\cdot \psi (f(t^{\mathrm{\prime }}))$, using the covariance from [F1] and the $H$-equivariance of $\psi$. Reindexing the finite sum by $t^{\mathrm{\prime }}$ shows $\beta (\psi )(x\cdot f)=x\cdot \beta (\psi )(f)$.

For $\psi \in {\mathrm{Hom}}_H(W,{\mathrm{Res}}_H^{G}V)$, $\beta (\psi )({\eta }_w)=\psi (w)$ because ${\eta }_w(t)=0$ for $t\mathrm{\ne }e$ and ${\eta }_w(e)=w$. Hence $\alpha (\beta (\psi ))=\psi$.

For $\mathrm{\Phi }\in {\mathrm{Hom}}_G({\mathrm{Ind}}_H^{G}W,V)$ and $f\in {\mathrm{Ind}}_H^{G}W$, one has $\beta (\alpha (\mathrm{\Phi }))(f)={\sum }_{t\in T}t\cdot \alpha (\mathrm{\Phi })(f(t))={\sum }_{t\in T}t\cdot \mathrm{\Phi }({\eta }_{f(t)})=\mathrm{\Phi }({\sum }_{t\in T}t\cdot {\eta }_{f(t)})$. The function inside $\mathrm{\Phi }$ equals $f$, because at a point $th$ it takes the value $h^{-1}\cdot f(t)=f(th)$ by covariance. So $\beta (\alpha (\mathrm{\Phi }))=\mathrm{\Phi }$.

Steps 3.1 and 3.2 show that $\alpha$ and $\beta$ are inverse group isomorphisms, giving the stated adjunction. Via [F3], this is equally the usual $R[G]$-module adjunction.

By [F3] and then [F1], $⟨{\mathrm{Ind}}_H^G\chi ,\psi {⟩}_G=\dim {\mathrm{Hom}}_G({\mathrm{Ind}}_H^{G}W,V)$.

By [F2], this dimension equals $\dim {\mathrm{Hom}}_H(W,{\mathrm{Res}}_H^{G}V)$; applying [F1] again on $H$ gives $\dim {\mathrm{Hom}}_H(W,{\mathrm{Res}}_H^{G}V)=⟨\chi ,{\mathrm{Res}}_H^G\psi {⟩}_H$.

The expressions in steps 1.1 and 1.2 are equal, which is exactly the Frobenius reciprocity identity.

For $F\in {\mathrm{Ind}}_H^G({\mathrm{Ind}}_K^{H}W)$, define $\mathrm{\Psi }(F)(g):=F(g)(e)$. If $k\in K$, then since $k\in H$ as well by [F2], $\mathrm{\Psi }(F)(gk)=F(gk)(e)=(k^{-1}\cdot F(g))(e)=F(g)(k)=k^{-1}\cdot F(g)(e)$, so $\mathrm{\Psi }(F)\in {\mathrm{Ind}}_K^{G}W$.

For $f\in {\mathrm{Ind}}_K^{G}W$, define $\mathrm{\Phi }(f)(g)(h):=f(gh)$ for $g\in G$ and $h\in H$. If $k\in K$, then $\mathrm{\Phi }(f)(g)(hk)=f(ghk)=k^{-1}\cdot f(gh)$, so $\mathrm{\Phi }(f)(g)\in {\mathrm{Ind}}_K^{H}W$; and if $h_0\in H$, then $\mathrm{\Phi }(f)(g{h}_0)(h)=f(g{h}_{0}h)=\mathrm{\Phi }(f)(g)(h_{0}h)$, which is the covariance condition for ${\mathrm{Ind}}_H^G({\mathrm{Ind}}_K^{H}W)$. Thus $\mathrm{\Phi }(f)$ lies in that induced module.

For $f\in {\mathrm{Ind}}_K^{G}W$, $\mathrm{\Psi }(\mathrm{\Phi }(f))(g)=\mathrm{\Phi }(f)(g)(e)=f(g)$, so $\mathrm{\Psi }\circ \mathrm{\Phi }=\mathrm{id}$.

For $F\in {\mathrm{Ind}}_H^G({\mathrm{Ind}}_K^{H}W)$ and $h\in H$, one has $\mathrm{\Phi }(\mathrm{\Psi }(F))(g)(h)=\mathrm{\Psi }(F)(gh)=F(gh)(e)=(h^{-1}\cdot F(g))(e)=F(g)(h)$, so $\mathrm{\Phi }(\mathrm{\Psi }(F))=F$ as functions $G\to {\mathrm{Ind}}_K^{H}W$.

Steps 3.1 and 3.2 show that $\mathrm{\Phi }$ and $\mathrm{\Psi }$ are inverse $G$-equivariant $R$-module isomorphisms. Therefore induction is transitive along $K\le H\le G$.

For an honest pair of characters $\chi$ and $\psi$ and any $g\in G$, [F1] gives ${\mathrm{Ind}}_H^G(\chi {\mathrm{Res}}_H^G\psi )(g)=\frac{1}{\mathrm{\mid }H\mathrm{\mid }}{\sum }_{x\in G,x^{-1}gx\in H}\chi (x^{-1}gx)\psi (x^{-1}gx)$.

Since $\psi$ is a class function by [F3], $\psi (x^{-1}gx)=\psi (g)$ for each summand of step 1.1. Factoring that constant out of the finite sum and applying [F1] again yields ${\mathrm{Ind}}_H^G(\chi {\mathrm{Res}}_H^G\psi )(g)=\psi (g){\mathrm{Ind}}_H^G\chi (g)=(({\mathrm{Ind}}_H^G\chi )\psi )(g)$.

So the identity holds for ordinary characters. By [F4], both induction and multiplication extend $\mathbb{Z}$-bilinearly to virtual characters, so the same pointwise identity holds for all $\chi \in R(H)$ and $\psi \in R(G)$.

This pointwise equality is the projection formula in $R(G)$, and [F2] identifies the pointwise product on the right with the character-ring product coming from tensor products. The adjoint viewpoint from [F5] is compatible with it, but step 3.1 already proves the formula.

Let $V$ be an irreducible representation of $G$ affording $\psi$. Its restriction to $H$ is completely reducible by [F1], so ${\mathrm{Res}}_H^{G}V\cong {⨁}_{i=1}^r{m}_i{W}_i$ for irreducible $H$-representations $W_i$ with characters ${\phi }_i$.

Because ${\mathrm{Res}}_H^{G}V$ is nonzero, some multiplicity $m_i$ is positive. By [F2], this means $m_i=⟨{\phi }_i,{\mathrm{Res}}_H^G\psi {⟩}_H>0$.

Applying [F3] to $\phi ={\phi }_i$ gives $⟨{\mathrm{Ind}}_H^G\phi ,\psi {⟩}_G=⟨\phi ,{\mathrm{Res}}_H^G\psi {⟩}_H=m_i>0$.

The strict positivity in step 3.1 means that $\psi$ occurs in the induced character ${\mathrm{Ind}}_H^G\phi$, by the multiplicity interpretation of [F2].

The chosen irreducible constituent $\phi :={\phi }_i$ therefore has the required property.