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arxiv: 2605.27280 · v1 · pith:2C7NCS7Bnew · submitted 2026-05-26 · 🧮 math.RT

On the Faithful Projective Representations of Finite Groups and their Minimal Dimension

Sumana Hatui , Poonam Nayak This is my paper

Pith reviewed 2026-06-29 14:22 UTC · model grok-4.3

classification 🧮 math.RT
keywords faithful projective representationsp-groupsinflation mapcohomology classesprojective embedding degreeextra-special groupsHeisenberg groupsdirect products
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The pith

A p-group admits a faithful irreducible projective representation precisely when its cocycle class is not inflated from any nontrivial central quotient.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper characterizes the cocycles that produce faithful projective representations of finite groups. For p-groups it gives an if and only if condition using the inflation map on cohomology. This matters because it tells exactly when the group can act faithfully on a projective space without a kernel. The authors also define the projective embedding degree as the smallest degree of a faithful projective representation and study it for direct products, abelian groups, extra-special p-groups, Heisenberg groups, and groups of small p-power order.

Core claim

A p-group G admits a faithful irreducible projective representation if and only if the cohomology class [α] does not lie in the image of the inflation map inf: H²(G/N, C×) → H²(G, C×) for any non-trivial central subgroup N of G. In the case where [α] lies in such an image, a criterion is given for when a direct sum of irreducible α-representations is faithful. The paper also describes the cocycles that yield faithful irreducible representations for direct products and defines the projective embedding degree of G as the smallest n such that G embeds into PGL_n(C).

What carries the argument

The inflation map inf: H²(G/N, C×) → H²(G, C×) from the second cohomology of a central quotient, which detects whether a given cocycle class yields a faithful projective representation.

If this is right

  • Faithfulness of a projective representation for a p-group can be decided by checking whether its class lies outside all inflation images from nontrivial central quotients.
  • When the class is inflated from a central quotient, a direct sum of several irreducible representations can still be faithful under an additional criterion.
  • The cocycles producing faithful irreducible representations for direct products of groups follow a specific described pattern.
  • The projective embedding degree and its irreducible variant can be computed for direct products, finite abelian groups, extra-special p-groups, Heisenberg groups, and all groups of order p^3, p^4, and p^5 for p at least 5.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The cohomological test may let researchers decide existence of faithful representations for additional families of p-groups beyond the ones already examined.
  • It cleanly separates the existence question for faithful representations from the separate task of finding their smallest possible degree.
  • The same inflation obstruction could be checked computationally for any concrete p-group by calculating its cohomology ring and the relevant maps from quotients.

Load-bearing premise

Projective representations correspond to cohomology classes in H²(G, C×) and the inflation map has the expected kernel and image properties with respect to central subgroups.

What would settle it

A counterexample would be a p-group that possesses a faithful irreducible projective representation whose cocycle class lies in the inflation image from some nontrivial central subgroup, or that lacks such a representation when the class avoids all such images.

Figures

Figures reproduced from arXiv: 2605.27280 by Poonam Nayak, Sumana Hatui.

Figure 1
Figure 1. Figure 1: Diagram 1 Proof. By Theorem 2.17, inf : H2 (Gj/Aj , C ×) → H2 (Gj , C ×) is surjective for j = 1, 2. Let ρi ∈ Irrα (G1) such that ρ = ⊕l i=1ρi be a faithful α-representation of G1 of degree τ (G1). By Theorem 3.3, there exists ˜ρi ∈ Irrβ tra(χi) (G1/A1) such that ϕ(ρi) = ˜ρi . Let ψi := η¯(χi) = χi ◦ η ∈ Hom(A2, C ×). Consider β ′ ∈ Z 2 (G2/A2, C ×) such that β ′ (gA2, hA2) = β(γ(gA2), γ(hA2)) for g, h ∈ G… view at source ↗
read the original abstract

The first part of this article is devoted to characterizing the cocycles $\alpha$ of a finite group $G$ that give rise to faithful projective representations of $G$. We prove that a $p$-group $G$ admits a faithful irreducible projective representation if and only if the cohomology class $[\alpha]$ does not lie in the image of the inflation map $\operatorname{inf}: \mathrm{H}^2\!\left(G / N, \mathbb{C}^{\times}\right) \longrightarrow \mathrm{H}^2\!\left(G, \mathbb{C}^{\times}\right)$ for any non-trivial central subgroup $N$ of $G$. In the case where $[\alpha] \in \operatorname{Im}(\operatorname{inf})$, we determine a criterion such that a direct sum of irreducible $\alpha$-representations is faithful. We conclude this part by describing the behaviour of cocycles $\alpha$ that yield faithful irreducible representations for direct products of groups. In the second part, we introduce the notion of the projective embedding degree of a finite group $G$, defined as the smallest integer $n$ such that $G$ embeds into $\mathrm{PGL}_n(\mathbb{C})$; equivalently, it is the smallest $n$ such that $G$ has a faithful complex projective representation of degree $n$. We also define the analogous notion of the irreducible projective embedding degree of $G$. These invariants have been investigated for several classes of groups, including direct products of groups, finite abelian groups, extra-special $p$-groups, Heisenberg groups, and groups of order $p^3$, $p^4$ (for primes $p$), and $p^5$ (for $p \geq 5$).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript characterizes the cohomology classes [α] ∈ H²(G, ℂ×) for which a finite group G admits a faithful irreducible α-projective representation. For p-groups the criterion is that [α] lies outside the image of every inflation map inf: H²(G/N, ℂ×) → H²(G, ℂ×) with N a nontrivial central subgroup. When [α] lies in such an image the paper gives a criterion for a direct sum of irreducibles to be faithful. The second part defines the projective embedding degree of G (minimal n such that G embeds in PGL_n(ℂ)) and its irreducible variant, then computes or describes these invariants for direct products, finite abelian groups, extraspecial p-groups, Heisenberg groups, and groups of order p³, p⁴ (all p) and p⁵ (p ≥ 5).

Significance. If the stated characterizations hold, the work supplies an explicit, checkable cohomological test for faithfulness of irreducible projective representations of p-groups together with new numerical invariants that measure the smallest dimension of a faithful projective representation. The explicit computations for several families of p-groups furnish concrete data that can be used to test conjectures or to guide further calculations in group cohomology. The arguments rest on standard identifications between projective representations and 2-cocycles and on the exactness properties of inflation, which are already in the literature.

minor comments (3)
  1. [§1] §1, paragraph after Definition 1.2: the phrase 'projective embedding degree' is introduced without an immediate comparison to the ordinary minimal faithful representation degree; a single sentence relating the two would clarify the novelty of the invariant.
  2. [Theorem 2.4] Theorem 2.4 (p-group criterion): the statement assumes the reader recalls that inflation corresponds precisely to representations with N in the kernel; a one-line reminder of this bijection immediately before the theorem would improve readability for non-specialists.
  3. [Table 1] Table 1 (computations for groups of order p³): the column headings use 'irred. proj. emb. deg.' without defining the abbreviation on first use in the table caption; expand or define the abbreviation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the manuscript and for the positive recommendation of minor revision. The report correctly captures the cohomology criteria for faithful irreducible projective representations of p-groups and the introduction of the projective embedding degree together with its computations for various families of p-groups.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central result is a characterization of cocycles yielding faithful irreducible projective representations for p-groups, stated in terms of the cohomology class lying outside the image of inflation maps from quotients by nontrivial central subgroups. This rests on the standard bijection between projective representations and H²(G, C×) classes together with the exact correspondence between inflation and representations factoring through G/N (hence having N in the kernel). Both facts are external textbook results in group cohomology; the paper invokes them without deriving them internally or via load-bearing self-citations. No equations, definitions, or fitted parameters reduce the claimed characterization to its own inputs by construction. The work is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper rests on the standard axioms of group cohomology and the inflation map; the only invented entity is the newly defined embedding degree, which has no independent evidence supplied in the abstract.

axioms (1)
  • standard math Standard properties of the second cohomology group H²(G, C×) and the inflation homomorphism from quotients by central subgroups
    Invoked directly in the p-group characterization and the criterion for direct sums.
invented entities (1)
  • projective embedding degree no independent evidence
    purpose: Smallest n such that G embeds into PGL_n(C)
    Newly introduced invariant whose value is studied for listed families of groups.

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