If \(\prod _i[a_i,b_i]=h\), replacing every entry by its
conjugate by a fixed element replaces the product by the conjugate
of \(h\). That replacement is a bijection of tuples, so \(\psi _g\) is
a class function. Splitting off the last commutator gives
\(\psi _{g+1}(h)=\sum _x\psi _g(x)\psi _1(x^{-1}h)\) for \(g\ge 0\). The
identity \(\psi _0(h)=1\) when \(h=1\), and \(0\) otherwise, is the
empty tuple.
The matrix orthogonality in Theorem [fdip-006W] says
\[\frac 1{|G|}\sum _{h\in G}
\rho (h)_{ij}\tau (h^{-1})_{ab}
=\begin {cases}
d_\rho ^{-1}\delta _{ib}\delta _{ja},&\rho =\tau ,\\
0,&\rho \ne \tau .
\end {cases}\]
Unitarity gives \(\tau (h^{-1})_{ab}=\overline {\tau (h)_{ba}}\).
Relabelling the indices therefore gives
\[\frac 1{|G|}\sum _{h\in G}
\rho (h)_{ij}\overline {\tau (h)_{kl}}
=\begin {cases}
d_\rho ^{-1}\delta _{ik}\delta _{jl},&\rho =\tau ,\\
0,&\rho \ne \tau .
\end {cases}\]
Sum \(\rho (h)_{ii}\overline {\tau (h)_{kk}}\) over the diagonal
indices \(i\) and \(k\). The identity supplies a factor
\(d_\rho ^{-1}\) only when \(i=k\), and there are \(d_\rho \) such
indices, so
\[\frac 1{|G|}\sum _{h\in G}
\chi _\rho (h)\overline {\chi _\tau (h)}=\begin {cases}
1,&\rho =\tau ,\\
0,&\rho \ne \tau .
\end {cases}\]
In particular \(\sum _h|\chi _\rho (h)|^2=|G|\). The trace of
the unitary identity \(\tau (h^{-1})=\tau (h)^*\) gives
\(\chi _\tau (h^{-1})=\overline {\chi _\tau (h)}\).
Let \(f(h)=\sum _\rho d_\rho \chi _\rho (h)\). The
character expansion gives \(f\) the coefficient
\[c_\tau =\frac 1{|G|}\sum _h f(h)\overline {\chi _\tau (h)}=d_\tau \]
against \(\chi _\tau \). The class function \(|G|\) at the identity
and \(0\) elsewhere has the same coefficients, so
\(\sum _\rho d_\rho \chi _\rho (h)\) equals \(|G|\) at the identity and
equals \(0\) elsewhere. Thus
\(\psi _0=\sum _\rho (d_\rho \,|G|^{-1})\chi _\rho \), and evaluating at the
identity gives \(\sum _\rho d_\rho ^2=|G|\).
For a linear map \(A\) on the space of \(\rho \), the average
\[S(A)=\frac 1{|G|}\sum _{g\in G}\rho (g)A\rho (g)^{-1}\]
commutes with every \(\rho (h)\). Over \(\mathbb C\) the
endomorphism \(S(A)\) has an eigenvalue. Its eigenspace is
invariant, and irreducibility forces the eigenspace to be the
whole space, so \(S(A)\) is scalar. Its trace equals
\(\operatorname {tr}A\), hence
\(S(A)=(\operatorname {tr}A/d_\rho )I\). Taking \(A=\rho (b)\) gives
\[\sum _{a\in G}\rho (a)\rho (b)\rho (a)^{-1}
=\frac {|G|}{d_\rho }\chi _\rho (b)\,I.\]
Multiply by \(\rho (b)^{-1}\) and sum over \(b\). The left side is
\(\sum _{a,b}\rho ([a,b])\). The right side is
\((|G|/d_\rho )\sum _b\chi _\rho (b)\rho (b)^{-1}\).
Set \(C=\sum _b\overline {\chi _\rho (b)}\rho (b)\). Then
\(\rho (h)C\rho (h)^{-1}=C\), because \(\chi _\rho \) is a class
function, so the same scalar argument gives
\(C=(|G|/d_\rho )I\). The trace computation uses
\(\sum _b|\chi _\rho (b)|^2=|G|\). The change of variable
\(b\mapsto b^{-1}\) and
\(\chi _\rho (b)=\overline {\chi _\rho (b^{-1})}\) identify
\(\sum _b\chi _\rho (b)\rho (b)^{-1}\) with \(C\). Therefore
\[\sum _{a,b\in G}\rho ([a,b])
=\left (\frac {|G|}{d_\rho }\right )^2 I.\]
Taking traces gives
\(\sum _{a,b}\chi _\rho ([a,b])=|G|^2/d_\rho \). Inverting a
commutator swaps the two summation variables, so the same sum
with \(\overline {\chi _\rho }\) is equal. The coefficient in the
character expansion is that sum divided by \(|G|\),
hence \(|G|/d_\rho \):
\[\psi _1=\sum _{\rho \in \widehat G}
\frac {|G|}{d_\rho }\chi _\rho .\]
Expand the character convolution and insert a matrix product
for the second character:
\[(\chi _\rho *\chi _\tau )(z)
=\sum _x\sum _{i,k,l}
\rho (x)_{ii}\tau (x^{-1})_{kl}\tau (z)_{lk}.\]
The matrix identity already displayed sums the variable \(x\):
the sum over \(x\) of \(\rho (x)_{ii}\tau (x^{-1})_{kl}\) equals
\(|G|d_\rho ^{-1}\) when \(\rho =\tau \), \(i=l\) and \(i=k\), and
equals zero otherwise. Multiplying by \(\tau (z)_{lk}\) and
summing those indices leaves the trace of \(\tau (z)\), so
\[(\chi _\rho *\chi _\tau )(z)
=\begin {cases}
(|G|/d_\rho )\chi _\rho (z),&\rho =\tau ,\\
0,&\rho \ne \tau .
\end {cases}\]
Consequently, if \(\alpha _\rho \) and \(\beta _\rho \) are the
character coefficients of two class functions, the coefficient of
\(\chi _\rho \) in their convolution is
\(\alpha _\rho \beta _\rho (|G|/d_\rho )\). Convolution against
\(\psi _1\) multiplies the coefficient \(\alpha _\rho \) by
\(|G|^2/d_\rho ^2\). Starting from \(\psi _0\) and applying this
multiplication \(g\) times gives the displayed formula for
\(\psi _g\).
A homomorphism \({\Pi _g}\to G\) is exactly a \(2g\)-tuple whose
product of commutators is the identity, so
\(|\operatorname {Hom}(\Pi _g,G)|=\psi _g(1)\). The displayed
expansion at the identity is
\(|G|^{2g-1}\sum _\rho d_\rho ^{2-2g}\). Dividing by \(|G|\) gives
the homomorphism formula.
For the stated face weight, the character expansion in
Definition [fdip-006V] has \(c_\rho =d_\rho \). Every ratio \(c_\rho/d_\rho \)
equals \(1\), so Theorem [fdip-006W] gives
\(Z_\Sigma (f)=\sum _\rho d_\rho ^{2-2g}\). The evaluation of
\(\psi _g\) at the identity is \(|G|^{2g-1}\) times that sum.