Problem 8.9 — (G.

Definitions and mathematical statement

Let $H_k=H^0(\mathcal M_\Sigma,\mathcal L^{\otimes k})$, the space of global sections of the source's line-bundle power on its moduli space. G. Masbaum asks to study the specified action of the finite group $E(\Sigma)$ on $H_k$, determine its decomposition according to characters, and relate that decomposition to those of $V(\Sigma)$ in the TQFT derived from $U_q(\mathfrak{sl}_2)$ at a $(k+N)$-th root of unity.

For a genuine linear complex action $\rho_k$ on a finite-dimensional $H_k$, character decomposition means the isotypic splitting \[ H_k\cong\bigoplus_{\chi\in\operatorname{Irr}(E(\Sigma))} W_\chi\otimes{\mathbb{C}}^{m_\chi(k)},\qquad m_\chi(k)=\frac1{|E(\Sigma)|}\sum_{g\in E(\Sigma)} \overline{\chi(g)}\operatorname{tr}(\rho_k(g)). \] The task is to identify the actual action, character multiplicities and comparison of summands; this general character identity alone is not their computation. No assumption that $E(\Sigma)$ is abelian or has only one-dimensional characters is made.

The excerpt's "this action" lacks its defining antecedent. The group, moduli space, line bundle, action/lift, level range and parameter $N$ are not defined. If the action is initially projective, the linear or central-extension convention must be specified before applying the displayed splitting. Nor is an identification between $H_k$ and the TQFT vector space supplied. The exact $(k+N)$-th-root condition is retained without substituting a guessed level shift or root order.

Short English statement

Compute the character decomposition of the specified finite-group action on the holomorphic section space, then compare it with decompositions in the stated $\mathfrak{sl}_2$ TQFT. The action and geometric/level conventions are missing.

Sources

• Local UnsolvedMath registry, permanent ID 10400155, source code catalogue entry; statement attribution G. Masbaum; proposer field and proposal year absent; updated 2026-08-06.
• Ohtsuki, Problems on invariants of knots and 3-manifolds (2002), Problem 8.9, PDF page 125: https://msp.org/gtm/2002/04/gtm-2002-04-024s.pdf. Registry access date 2026-07-29; PDF font-structure extraction. Provenance distinguishes 231 numbered bold headings. Present primary-literature status and redistribution terms require review.
• The supplied summary repeats the finite-group action question without precise additional definitions, references, or a verified resolution.