Theorem. Compact character expansions and surface integrals [fdip-007C]
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  • November 1, 2026

Let \(G\) be a compact Lie group with normalized Haar measure. Choose unitary representatives of its complex irreducibles, indexed by \(\widehat G\), with dimensions \(d_\rho \). The Peter–Weyl theorem says that the functions \(\sqrt {d_\rho }\rho (\cdot )_{ij}\) form an orthonormal basis of \(L^2(G)\). Its central subspace has orthonormal basis \(\chi _\rho \). Thus a central \(f\in L^2(G)\) has the expansion \[f=\sum _{\rho \in \widehat G}c_\rho \chi _\rho , \qquad c_\rho =\int _G f(h)\overline {\chi _\rho (h)}\,d\mu _G(h),\] with convergence in \(L^2\). This is a cited statement, namely Woit's Section 5, Theorem 1 and Corollary 1 [woit2012peterweyl, p. 6]. It does not assert pointwise or absolute convergence for an arbitrary \(L^2\) function.

For an inequivalent exhaustive family of finite-dimensional unitary continuous models, TauCeti's TauCeti.peterWeylBasis constructs the normalized matrix-coefficient Hilbert basis, with its inner-product convention for matrix coefficients. Over \(\mathbb C\), its coefficient \(\langle \rho (g)e_a,e_b\rangle \) is the conjugate of the matrix entry \(\rho (g)_{ba}\); conjugation, or reindexing by dual representations, gives the equivalent entry-coordinate basis. The separate TauCeti.characterBasis constructs the actual character basis of the central \(L^2\) subspace. Compact Hausdorff groups over \(\mathbb C\), hence the compact Lie groups here, satisfy the stated hypotheses. These correspondences support the two \(L^2\) basis statements, not the uniform absolute convergence or surface integral established next.

Suppose now that \(f\) is continuous and central and that \[\sum _{\rho \in \widehat G}|c_\rho |d_\rho <\infty .\] Then its character expansion converges uniformly and absolutely. For the closed oriented polygonal surface, with \(P\) faces and genus \(g\), define the edge integral \[Z_\Sigma (f)=\int _{G^E}\prod _p f(U_p)\,d\mu (U).\] Under the displayed summability condition, \[Z_\Sigma (f)=\sum _{\rho \in \widehat G} d_\rho ^{2-2g}\left (\frac {c_\rho }{d_\rho }\right )^P,\] and this numerical series is absolutely convergent.

Proof. 

  • November 1, 2026

Unitarity gives \(|\chi _\rho (h)|\le d_\rho \). The sum of the uniform bounds is finite, so the character series has a continuous uniform limit. Peter–Weyl identifies that limit with \(f\) in \(L^2\); continuity and full support of Haar measure make the equality pointwise. Expanding a product over the finite face set gives a sum whose total absolute value is bounded uniformly by \[\left (\sum _\rho |c_\rho |d_\rho \right )^P.\] This bound is integrable against probability measure, justifying interchange of the full face-label sum and the edge integrals.

Haar averaging gives the same matrix orthogonality as finite averaging: for \(A:V_\tau \to V_\rho \), the integral \(\int _G\rho (h)A\tau (h)^{-1}\,d\mu _G(h)\) is an intertwiner. Schur's lemma and its trace make it zero for inequivalent representatives and \((\operatorname {tr}A/d_\rho )I\) for equal ones. The edge contractions and circular vertex links in the finite surface proof therefore leave one common face label and one free index per vertex. Their contribution is \(c_\rho ^P d_\rho ^{|V|-|E|}\), the displayed formula. Absolute convergence follows as well: the absolute values of these integrated terms are bounded by the integrals of the absolute face terms, whose total has just been bounded. In particular \(f=1\) still gives \(Z_\Sigma (1)=1\).