LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 2026-09-27

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The cocycle central extension linearizes a projective representation

Statement

Let $Q$ be a group, let $α : Q \times Q \to C^{\times}$ be a normalized two-cocycle, and let $E_{α} = Q \times C^{\times}$ be the group with $(q, z) (r, w) = (q r, α (q, r) z w)$ of The twisted product of a normalized cocycle is a central extension. Then the normalized projective $Q$ -representations with factor set $α$ correspond to the ordinary representations $D$ of the group $E_{α}$ satisfying $D (1, z) = z id$ for all $z \in C^{\times}$ , on the same space $V$ , by $D (q, z) = z P (q) and P (q) = D (q, 1) .$ The two constructions are mutually inverse on maps, and a linear map intertwines two projective representations exactly when it intertwines the corresponding $E_{α}$ -representations.

Facts & Assumptions

Given: A group $Q$ , a normalized two-cocycle $α : Q \times Q \to C^{\times}$ , the group $E_{α} = Q \times C^{\times}$ with product $(q, z) (r, w) = (q r, α (q, r) z w)$ and identity $(1, 1)$ , and a nonzero finite-dimensional complex vector space $V$ .

[F1]

$E_{α}$ is a group with the displayed product and identity $(1, 1)$ , and $α (1, q) = α (q, 1) = 1$ . (The twisted product of a normalized cocycle is a central extension).

[F2]

For finite $Q$ , Projective representations and normalized factor sets defines a normalized projective representation by $P : Q \to GL (V)$ , $P (1) = {id}_{V}$ and $P (q) P (r) = α (q, r) P (q r)$ . In this lemma, for arbitrary $Q$ we use these same equations as the definition, on the given nonzero finite-dimensional complex space $V$ . The constructions below verify the correspondence directly in this convention; no finiteness of $Q$ is assumed.

[F3]

A representation of a group $G$ over a field $k$ is a group homomorphism $ρ : G \to GL (V)$ on a finite-dimensional $k$ -space $V$ . (A finite-dimensional representation $ρ : G \to GL (V)$ over a field, and its degree).

[A1]

For $T \in End (V)$ and scalars $z, w \in C$ one has $(z T) (w S) = z w T S$ and $z T = T z$ , and $z T$ is invertible when $z \neq 0$ and $T$ is invertible.

Proof

technique · direct

1.1

Let $P$ be a normalized projective representation of $Q$ with factor set $α$ and define $D (q, z) : = z P (q)$ for $(q, z) \in E_{α}$ . Each $D (q, z)$ is invertible by [A1], and $D (1, z) = z {id}_{V}$ . For $(q, z), (r, w) \in E_{α}$ , $D (q, z) D (r, w) = z P (q) w P (r) = z w P (q) P (r) = z w α (q, r) P (q r) = D (q r, α (q, r) z w) = D ((q, z) (r, w))$ by [F2] and [A1], so $D$ is a homomorphism, that is, a representation of the group $E_{α}$ in the sense of [F3].

A1F1F2F3

1.2

Conversely let $D : E_{α} \to GL (V)$ be a representation of the group $E_{α}$ with $D (1, z) = z {id}_{V}$ for every $z \in C^{\times}$ , and define $P (q) : = D (q, 1)$ . Then $P (q)$ is invertible, $P (1) = D (1, 1) = {id}_{V}$ , and for $q, r \in Q$ one has $P (q) P (r) = D (q, 1) D (r, 1) = D ((q, 1) (r, 1)) = D (q r, α (q, r))$ by [F1], while $(q r, 1) (1, α (q, r)) = (q r, α (q r, 1) α (q, r)) = (q r, α (q, r))$ and hence $D (q r, α (q, r)) = D (q r, 1) D (1, α (q, r)) = α (q, r) P (q r)$ ; thus $P$ is a normalized projective representation of $Q$ with factor set $α$ in the sense of [F2].

A1F1F2F3

2.1

The two constructions are inverse. If $P$ is given and $D (q, z) = z P (q)$ , then the projective representation reconstructed from $D$ is $q \mapsto D (q, 1) = P (q)$ . Conversely, if $D$ is given and $P (q) = D (q, 1)$ , then $z P (q) = D (1, z) D (q, 1) = D ((1, z) (q, 1)) = D (q, α (1, q) z) = D (q, z)$ for all $q, z$ , because $(1, z) (q, 1) = (q, z)$ by [F1]; this also shows that the condition $D (1, z) = z {id}_{V}$ is exactly the requirement that the reconstruction be consistent, and it is automatic for the representations produced in step 1.1.

F1step 1.1step 1.2algebra

2.2

A linear map $T : V \to V^{'}$ intertwines a projective representation $P$ with a projective representation $P^{'}$ of $Q$ , that is $T P (q) = P^{'} (q) T$ for all $q$ , if and only if it intertwines the corresponding representations $D, D^{'}$ of $E_{α}$ : indeed $T D (q, z) = z T P (q)$ and $D^{'} (q, z) T = z P^{'} (q) T$ by [A1], so $T D (q, z) = D^{'} (q, z) T$ for all $(q, z)$ is equivalent to $T P (q) = P^{'} (q) T$ for all $q$ , since $z \neq 0$ may be cancelled.

A1step 1.1step 1.2algebra

3.1

Steps 1.1 and 1.2 give mutually inverse constructions between normalized projective $Q$ -representations with factor set $α$ and representations $D$ of the group $E_{α}$ with $D (1, z) = z {id}_{V}$ , on a fixed space $V$ , by the formulas $D (q, z) = z P (q)$ and $P (q) = D (q, 1)$ , and step 2.2 shows that they match intertwiners; consequently the projective representation theory of $Q$ with factor set $α$ is the ordinary representation theory of the central extension $E_{α}$ restricted to the representations with the prescribed central character $z \mapsto z$ .

step 1.1step 1.2step 2.1step 2.2∎

Depends on

Used by

The quaternion group as a cocycle central extension of C2 x C2 Example

Dependency tree · two levels

15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Britta Späth, Reduction theorems for some global-local conjectures — Theorem 1.12(b), printed pp. 4–5 (standard reference, not scraped)
Tammo tom Dieck, Representation Theory — §4.2, printed pp. 54–57 (standard reference, not scraped)