Theorem. Handle convolution computes the homomorphism count [fdip-00A8]
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  • November 3, 2026

Let \(G\) be a finite group, and choose one unitary representative of each irreducible complex representation, as in the surface data. Write \(d_\rho \) for its dimension and \(\chi _\rho \) for its character. For an integer \(g\ge 0\), let \(\Pi _g\) be the group presented by generators \(x_1,y_1,\ldots ,x_g,y_g\) and the single relation that the product of the commutators \(x_iy_ix_i^{-1}y_i^{-1}\) is the identity. The case \(g=0\) is the trivial group. Write a commutator as \([a,b]=aba^{-1}b^{-1}\), and write \((\psi *\phi )(h)\) for the convolution \(\sum _{x\in G}\psi (x)\phi (x^{-1}h)\). Let \(\psi _g(h)\) be the number of \(2g\)-tuples in \(G\) whose product of commutators equals \(h\), with the empty product equal to the identity. Then each \(\psi _g\) is a class function, \(\psi _{g+1}=\psi _g*\psi _1\), and \[\psi _g(h)=\sum _{\rho \in \widehat G}|G|^{2g-1}d_\rho ^{1-2g}\chi _\rho (h).\] The number of homomorphisms satisfies \[\frac {|\operatorname {Hom}(\Pi _g,G)|}{|G|} =\sum _{\rho \in \widehat G} \left (\frac {|G|}{d_\rho }\right )^{2g-2}.\]

Let \(f\) be the class function with \(f(1)=|G|\) and with value \(0\) at every other group element. For every surface admitted by the finite surface character formula, including the exceptional one-vertex sphere, \[Z_\Sigma (f)=\frac {\psi _g(1)}{|G|^{2g-1}}.\] Thus the normalized flat partition function depends on the cellulation only through the genus, and it is the identity value of the \(g\)-fold handle convolution divided by \(|G|^{2g-1}\).

Attaching one handle is convolution with \(\psi _1\), and closing the remaining boundary is evaluation at the identity. The papers that identify this composition with the two-dimensional theory of the center of the group algebra have not been read, and this note does not state their result. Nothing proved here identifies the pure-gauge transfer operator with this convolution.

Proof. 

  • November 3, 2026

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.