Problem 2.2 — Decompose the representation on H0 into irreducible representations of ModΣ.

Definitions and mathematical statement

Problem 2.2. Decompose the source's representation of $\operatorname{Mod}_\Sigma$ on $\mathcal H_0$ into irreducible representations. The excerpt does not define $\mathcal H_0$, its relation to $\mathcal H$, the group $G$ in general, the measure, or the surface and mapping-class convention. A conventional setting would use a Hilbert space of square-integrable functions on $\operatorname{Hom}(\pi_1\Sigma,G)/G$, with a measure-preserving mapping-class action; that setup is conditional, not supplied in full. Nor is a direct sum of irreducibles guaranteed for an arbitrary unitary representation; a direct-integral formulation may require separate justification.

The source gives the example $G=U(1)$, $\Sigma=T^2$. Its representation space identifies with $T^2$, with coordinate functions $\alpha,\beta$ corresponding to a basis of $\pi_1(\Sigma)$. The Fourier functions \[ \varphi_{m,n}=\alpha^m\beta^n,\qquad(m,n)\in{\mathbb{Z}}^2, \] form the stated Hilbert basis of $\mathcal H$. The source uses the linear $\operatorname{GL}_2({\mathbb{Z}})$-action on its index set; if only orientation-preserving mapping classes are intended, the torus group is instead $\operatorname{SL}_2({\mathbb{Z}})$, so this convention needs checking.

The index orbits are labeled by $d\geq0$, with \[ O_d=\{(m,n):\gcd(m,n)=d\},\qquad C_d=\overline{\operatorname{span}}\{\varphi_{m,n}:(m,n)\in O_d\}, \] using $O_0=\{(0,0)\}$. The source calls the $C_d$ irreducible constituents. Orbit transitivity alone does not prove irreducibility of a permutation Hilbert representation. Indeed, inversion of Fourier indices $(m,n)\mapsto(-m,-n)$ commutes with the linear action and provides a nontrivial symmetry on $C_d$ for $d>0$; the claimed irreducibility requires correction or additional hypotheses before use. If $\mathcal H_0$ means the orthogonal complement of constants, that also needs explicit confirmation.

The source further considers the power homomorphism $\Phi_d:U(1)\to U(1)$, inducing a covering of the torus representation space for positive $d$, and defines $L_d$ as the closure of the image of an induced operator on $\mathcal H$. It records \[ L_d=\widehat{\bigoplus}_{d'\mid d}C_{d'},\qquad C_d=L_d\ominus\overline{\sum_{d'\mid d,\ d'\neq d}L_{d'}}. \] These displayed source formulas are preserved as unresolved, not asserted identities: the induced operator's direction is unspecified, and at $d=1$ the identity power map should have image all of $\mathcal H$, while the written divisor formula gives only $C_1$. Also $d=0$ labels a Fourier orbit but is not a covering power map. These inconsistencies must be resolved before the example supports an irreducible decomposition.

Short English statement

Decompose the mapping-class representation on the source's Hilbert space $\mathcal H_0$. The torus example supplies Fourier orbit subspaces, but its irreducibility assertion and divisor formula for covering-induced images have inconsistencies that must be reviewed.

Sources

• Local UnsolvedMath registry, permanent ID 11000188, source code catalogue entry, Problem 2.2, PDF page 216. No individual proposer is supplied; page distinguishes repeated labels.
• Farb, Problems on Mapping Class Groups and Related Topics. Supplied URL: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf; recorded access 2026-07-29; extraction pdf-font-structure. The edited volume contains 329 explicitly typeset headings across chapters.
• The Fourier, orbit, and covering formulas are supplied explanatory text with no separate authoritative citation or missing-definition references in this record.
• Supplied status open and OPEN-TRIAGE. The research summary says only "Cannot verify." Current primary-literature status and redistribution terms for the publicly accessible author/publisher copy remain NEEDS_REVIEW.