---
tag: 442JP05SRX
kind: concept
name: Projective representation
aliases: [projective-rep, unit-ray-representation]
sources:
  - wikipedia: en:348860@r1350652513
    locator: Projective representation
    note: adopted
    mention: wikipedia://en/348860
    sections: [Summary]
same_as:
  wikidata: Q7249476
  wikipedia: Projective representation
inventory: [6DR3QSG4G5]
---

## Summary

In the field of representation theory in mathematics, a **projective representation** of a group $G$ on a [[vector space|000AYPA9AJ]] $V$ over a [[field|04C5ZBG1WT]] $F$ is a [[group homomorphism|0HV9Y7WG5R]] from $G$ to the projective linear group

$$\mathrm{PGL}(V) = \mathrm{GL}(V) / F^*,$$

where GL(*V*) is the [[general linear group|0FCHE902D7]] of invertible linear transformations of $V$ over $F$, and $F^{\ast}$ is the [[normal subgroup|0CB0DX5BVV]] of $\mathrm{GL}(V)$ consisting of nonzero scalar multiples of the identity transformation (see Scalar transformation).