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

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 $\inf :{\mathrm{H}}^2\left(G/N,{\mathbb{C}}^{\times }\right)⟶{\mathrm{H}}^2\left(G,{\mathbb{C}}^{\times }\right)$ for any non-trivial central subgroup $N$ of $G$. In the case where $[\alpha ]\in \mathrm{Im}(\inf )$, 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 $p\ge 5$).