# Is each representation induced by a simpler representation

Let $$G$$ be a finite group, $$V$$ be a real vector space of dimension $$N$$, and $$N > |G|$$.

Can you find a representation $$\sigma: G \to GL(\text{End}(V))$$ such that $$\sigma$$ is not induced by a representation $$\rho: G \to GL(V)$$ via $$\sigma_g(X)=\rho_g^{-1}X\rho_g$$; $$g \in G$$ and $$X \in GL(V)$$?

We know each automorphism of $$GL(\text{End}(V))$$ is inner. That is, for each $$g \in G$$, there exists $$U_g \in GL(V)$$ such that $$\sigma_g(X)=U_g^{-1}XU_g$$. Does this necessitate the existant of $$\rho: G \to GL(V)$$ such that $$\rho_g =U_g$$ when $$N> |G|$$?

• $GL(GL(V))$ does not make sense as far as I know. Commented Aug 9 at 20:36
• @CaptainLama I believe it should really be $GL(\operatorname{End}(V))$ Commented Aug 9 at 20:41
• Were the answer to the question in your title yes, then there would have to be an infinite sequence of ever-simpler representations. Commented Aug 9 at 21:30
• You don't even mean $GL(\text{End}(V))$ but $\text{Aut}(\text{End}(V))$, right? That is I assume you're not interested in representations which don't preserve the multiplication (which clearly cannot be induced by conjugation but for simpler reasons). Commented Aug 9 at 23:06
• Would you explain what the differences between GL(algebra) and Aut(algebra) is? I thought they were the same. I once asked which notation is better between $\sigma: G \to GL(S^N)$ and $\sigma: G \to Aut(S^N)$, where $S^N$ is the space of $N \times N$ symmetric matrices, and you said the later is awkward. Commented Aug 10 at 1:45

I assume you mean $$\text{Aut}(\text{End}(V))$$. By the Skolem-Noether theorem this is $$PGL(V)$$, so your question is whether every map $$G \to PGL(V)$$ lifts to a map $$G \to GL(V)$$.

This is well-understood. Maps $$G \to PGL(V)$$ are called projective representations, and whether they lift to $$GL(V)$$ is controlled by whether a corresponding cohomology class in $$H^2(G, K^{\times})$$ vanishes. This class is sometimes called the Schur multiplier but that name also refers to several other closely related objects; it should really be called something like the Schur class. It is the cohomology class you get when you try to lift each individual element of $$G$$ and compare the difference between the lift of $$gh$$ and the product of the lifts of $$g$$ and $$h$$.

The condition on the dimension of $$V$$ is irrelevant because the Schur class doesn't change when you add a genuine representation.

A typical way to produce nontrivial examples of projective representations is to take genuine representations of a central extension

$$1 \to Z \to H \to G \to 1$$

of $$G$$ on which the center $$Z$$ acts by scalars (it's a classical fact that every projective representation of a finite group arises in this way), which means the induced map $$H \to PGL(V)$$ has kernel $$Z$$ and hence gives a map $$G \to PGL(V)$$ which usually won't lift to $$GL(V)$$.

The smallest nontrivial central extension is the quaternion group $$Q_8$$, which is a central extension of $$V_4$$ by $$C_2$$. It has a unique $$4$$-dimensional irreducible representation $$\mathbb{R}^4 \cong \mathbb{H}$$ induced by the inclusion $$Q_8 \hookrightarrow \mathbb{H}$$, on which the center acts by $$-1$$. This gives a projective representation $$V_4 \to PGL_4(\mathbb{R})$$ which does not lift to $$GL_4(\mathbb{R})$$.

To see this explicitly we can think of $$Q_8$$ as $$\{ \pm 1, \pm i, \pm j, \pm k \}$$ as a subgroup of $$\mathbb{H}$$, with center $$\pm 1$$, so the quotient by the center is $$V_4 = \{ 1, i, j, k \}$$ with $$i, j, k$$ all of order $$2$$. Any lift of the corresponding projective representation $$V_4 \to PGL_4(\mathbb{R})$$ to $$GL_4(\mathbb{R})$$ must lift $$i, j, k$$ to some nonzero real multiples $$pi, rj, qk$$, but none of these commute.

• Can you also provide a counterexample when we work over reals? Commented Aug 9 at 20:54
• @khashayar: $Q_8$ is also a counterexample over the reals, I've just edited. Commented Aug 9 at 20:56
• I see, so helpful. Commented Aug 9 at 20:57