Invertible Projective Representations

• Invertible projective representations are algebraic and categorical structures defined by twisted cocycle data that enable lifting to linear representations via central extensions.
• They are classified using cohomological invariants like the second cohomology group, distinguishing finite-dimensional cases from infinite-dimensional ones in quantum settings.
• Computational methods adapt classical algorithms to handle twisted character tables, facilitating applications in harmonic analysis and condensed matter physics.

An invertible projective representation is a representation—generally defined up to scalar multiples—of a group, quantum group, or category, whose “twisting datum” (commonly a 2-cocycle, Galois co-object, or anomaly functor) is itself invertible in the relevant algebraic or categorical sense. The concept subsumes classical projective group representations, quantum group corepresentations with twist, and higher-categorical analogues arising in TQFTs, and has significant applications in harmonic analysis, quantum algebra, and condensed matter physics. The defining feature is the existence of a cohomological or extension-theoretic structure that allows the projective representation to be “lifted,” “untwisted,” or “linearized,” often by passing to a central extension or appropriate module category.

1. Algebraic and Operator-Algebraic Characterizations

Invertible projective representations may be described by a twisted multiplication law parameterized by a 2-cocycle or more generally a categorical extension. For a group $G$ and multiplier $\alpha$, the classical formula is: $$U(g) U(g') = \alpha(g, g') U(gg'), \quad (\text{with } U(g) \text{ unitary/automorphism})$$ where $\alpha(g, g')$ is a cohomological factor satisfying the cocycle condition: $$\alpha(g, h) \alpha(gh, k) = \alpha(h, k) \alpha(g, hk)$$ Such structures generalize to compact quantum groups via twisted coactions. For $(M, \Delta_M)$ a compact quantum group and $(N, \Delta_N)$ a Galois co-object, a corepresentation $G \in N \otimes B(H)$ satisfies: $(\Delta_N \otimes \mathrm{id})(G) = G_{13} G_{23}$ Here, invertibility requires the Galois co-object to be cleft: there exists a unitary $u$ such that the twisting cocycle can be absorbed as a coboundary, i.e. $\Omega = (u^* \otimes u^*)\Delta_M(u)$, or for categorical settings an anomaly functor factors through invertible objects in the Picard group (Commer, 2010, Fiorenza et al., 2 Jun 2025, Constantinescu, 2011).

2. Dimensionality and Classification

A striking phenomenon in the quantum (non-Kac) case is the appearance of infinite-dimensional irreducible invertible projective corepresentations, while for Kac-type algebras and classical settings all irreducible invertible projective representations are finite-dimensional. The dichotomy reflects the role of modularity: Kac algebras admit only algebraic (finite-dimensional) objects, whereas more general quantum groups support analytic deformations resulting in infinite-dimensional corepresentations (Commer, 2010). In categorical and group-theoretical contexts, the classification of projective (and hence invertible) representations is governed by the second cohomology group, $H^2(G, \mathbb{C}^*)$ or its analogues (Schur and Bogomolov multipliers), which organize all possible twistings and the equivalence classes of invertible projective representations (Kobayashi et al., 16 Jul 2025).

3. Computational Methods and Algorithms

Algorithms for computing irreducible invertible projective representations have been generalized from classical (linear) to projective settings. The Burnside algorithm is adapted for twisted character tables: let $C^{(i)}$ be $\alpha$-regular conjugacy classes, then for each such class $A$,

$$(M_A)_{BC} = \frac{1}{\sqrt{|B||C|}} \sum_{a \in A,\, b \in B,\, ab \in C} \frac{\beta(c_0, ab)}{\beta(a_0, a)\beta(b_0, b)} \alpha(a, b)$$

with class factors $\beta$ defined via the cocycle. Dixon's algorithm, transposed to finite fields, yields exact arithmetic solutions and supports floating-point implementations when the multiplier is only known approximately (Szabó, 20 May 2025). Both approaches bypass the need to construct the representation group of the Schur multiplier directly, relying instead on explicit cocycle data.

4. Peter–Weyl Theory and Decomposition

The extension of the Peter–Weyl theorem to projective representations via Galois co-objects shows that every projective corepresentation admits a decomposition into (possibly infinite) direct sums of indecomposable, and in many cases irreducible, projective representations. Orthogonality and matrix coefficient relations generalize the classical results: $$\sum_{k} \langle G^{(r)}_{ij}, G^{(s)}_{kl} \rangle = \delta_{rs}\delta_{ik} \cdot \text{factor}$$ for indecomposable components $E_{p_r}G$ parametrized by central projections $p_r$. This forms the foundation for harmonic analysis and fusion rules on the twisted quantum group (Commer, 2010).

5. Representation Group, Central Extensions, and Linearization

Invertible projective representations are closely tied to the existence and structure of central extensions and representation groups; this is both a technical and conceptual tool for lifting projective systems to linear ones. For a group $G$ and multiplier $\alpha$, one forms a central extension $G^\alpha$ (or a twisted group algebra), and there is an equivalence: $$\text{projective representations of } G\ (\alpha)\ \longleftrightarrow\ \text{linear representations of } G^\alpha \text{ with central multiplicities}$$ In categorical settings, extension categories $\mathcal{C}^J$ and specialized subcategories $\mathcal{C}^J_{\mathrm{ST}}$ encode this passage, with anomalies recording the twisting data. The practical effect is the ability to "linearize" a projective (or anomalous) representation by extending the base category with appropriate module structures (Fiorenza et al., 2 Jun 2025).

6. Physical Applications and Invertibility

Invertible projective representations play a central role in condensed matter and quantum field theory. In symmetry-protected topological (SPT) phases, edge states carry irreducible invertible projective representations classified by $H^2(G, U(1))$. In symmetry-enriched topological (SET) phases, fractionalized anyonic excitations transform projectively, and the Bogomolov multiplier identifies those cocycles undetectable by string order but physically distinct upon gauging: interfaces host novel ground state degeneracies, and fusion rules of local order parameters acquire nontrivial phase twists (Yang et al., 2016, Kobayashi et al., 16 Jul 2025). In quantum simulation, enforcing Hamiltonians with invertible projective symmetry can remove the Monte Carlo sign problem by ensuring eigenvalue degeneracies via nontrivial projective antiunitary symmetries.

7. Higher-Categorical Generalizations

Invertible projective 2-representations, e.g., in the Morita 2-category of super vector spaces, arise naturally in extended TQFTs with defects. The “freeness property” asserts that invertible assignments of objects, morphisms, and, where relevant, 2-morphisms extend to full projective 2-representations whose cocycle data is encoded in canonical trace formulas on higher-categorical structures: $$\alpha_{f,g} = \mathrm{tr}\left(\rho_{f \circ g}^{-1}(\rho_f \circ \rho_g)\right),\quad l_\Xi = \mathrm{tr}(M_{f \circ g}^{-1}(M_f \otimes M_g))$$ The Clifford/Fock construction provides a key example, where invertible Clifford algebras and Fock bimodules yield projective 2-representations of the category of Lagrangian correspondences, and the twisting by Pfaffian lines measures the failure of strict composition, thus recovering and explaining results such as those of Ludewig–Roos (Fiorenza et al., 20 Sep 2025).

Summary Table: Key Formalisms

Setting	Invertibility Criterion	Lifting Mechanism
Classical groups	$\alpha$ cohomologically trivial or coboundary-equivalent	Central extension, twisted group algebra
Compact quantum groups	Cleft Galois co-object, untwist via unitary conjugation	Cohomologous cocycle, ergodic coaction
Operator algebra/Clifford/Fock	Unitary generators, precise bimodule structure	Morita equivalence, Pfaffian line twist
Categorical / TQFT	Anomaly factors through invertible Picard elements	Extension $\mathcal{C}^J$, scalar action

Invertible projective representations unify and generalize twisting phenomena in algebra, analysis, geometry, and physics. Their classification, decomposition, and computational construction depend critically on cohomological, extension, and anomaly-theoretic data, while their applications range from harmonic analysis and operator algebras to topological phases and higher-categorical field theories.