The number of a set of irreducible projective characters vs the number of the ordinary characters of a finite group G.

Question

I need valid references to show that the number of a set of irreducible projective characters with non-trivial factor set is always strictly less than the number of the ordinary characters of a finite group G.

Answer 1

This long answer mostly just explains Geoff Robinson's answer. The short version is "read and deeply understand chapter 11, and carefully work exercise 3.10 in Isaacs's Character Theory of Finite Groups. This answer lacks: (a) a proof or reference to a proof of Burnside's non-negativity result, and (b) a reference to this entire claim in a published work (I just learned it from Geoff Robinson's nice answer).

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Let $G$ be a finite group, $M(G)$ its Schur multiplier, and $X$ a Schur cover. Let $\lambda:M(G) \to \operatorname{GL}_1(\mathbb{C})$ be a group homomorphism. Define $k_\lambda(G)$ to be the number of projective irreducible characters of $G$ above $\lambda$. Then we have:

$$k_\lambda(G) = \sum_{\chi \in \operatorname{Irr}(X)} \frac{ \langle \chi{\downarrow_{M(G)}}, \lambda \rangle }{ \chi(1) }$$ and $$k_\lambda(G) \leq k_1(G) = k(G)$$

The equality is by definition, but the definition is fairly complicated, so I'll explain. The second equality depends on a non-negativity result of Burnside, but otherwise is simple.

The equality: (Part 1: projective representations.) Let $\alpha:G \times G \to \mathbb{C}^\times$ be any (set-theoretic) function. Then an $\alpha$-projective representation is a (set-theoretic) function $\hat\pi:G \to \operatorname{GL}_n(\mathbb{C})$ satisfying $\hat\pi(gh) = \hat\pi(g) \cdot \hat\pi(h) \cdot \alpha(g,h)$. A projective representation is a group homomorphism $\pi:G \to \operatorname{PGL}_n(\mathbb{C})$. These two notions are basically equivalent: Let $\mu:\operatorname{PGL}_n(\mathbb{C}) \to \operatorname{GL}_n(\mathbb{C})$ be any set theoretic function so that the composition $\operatorname{PGL}_n(\mathbb{C}) \xrightarrow{\mu} \operatorname{GL}_n(\mathbb{C}) \to \operatorname{PGL}_n(\mathbb{C})$ is the identity. Given an $\alpha$-projective representation, compose it with the surjection $\operatorname{GL}_n(\mathbb{C}) \to \operatorname{PGL}_n(\mathbb{C})$ and the $\alpha$ term drops out, leaving a group homomorphism, that is, a projective representation. Given a projective representation $\pi$, define $\hat\pi$ as the composition $G \xrightarrow{\pi} \operatorname{PGL}_n(\mathbb{C}) \xrightarrow{\mu} \operatorname{GL}_n(\mathbb{C})$. Since the kernel of the natural surjection $\operatorname{GL}_n(\mathbb{C}) \to \operatorname{PGL}_n(\mathbb{C})$ consists only of scalar matrices, we can define $\alpha(g,h)$ as the scalar of those scalar matrices so that $\hat\pi$ is an $\alpha$-projective representation.

(Part 2: cohomology of $\alpha$.) For a given $\pi$ there are many $\alpha$ that work to form $\hat\pi$. Each one is given by a different choice $\mu$. If $\pi=1:G\to \operatorname{PGL}_n(\mathbb{C}) : g \mapsto \bar 1$ is the trivial projective representation, then set of all $\alpha$ that work are closed under pointwise multiplication, so form an abelian group called the second coboundaries, $B^2(G,\mathbb{C}^\times)$. For any other $\pi$, the set of all $\alpha$ forms a coset of $B^2(G,\mathbb{C}^\times)$. If we let $\pi$ vary, then the set of all $\alpha$ is closed under pointwise multiplication, so forms an abelian group called the second cocycles, $Z^2(G,\mathbb{C}^\times)$. If we only want one $\alpha$ per $\pi$, then we need to stick to cosets, so we work in the second cohomology $H^2(G,\mathbb{C}^\times) = Z^2(G,\mathbb{C}^\times)/B^2(G,\mathbb{C}^\times)$, also called the Schur multiplier, $M(G)$.

(Part 3: Schur cover and lifting.) Schur (1904) noticed that the way we lift projective representations, $\pi$, to $\alpha$-projective representations, $\hat\pi$, can be used to lift the group $G$ to a larger group, $X$, called the Schur cover that has the following great properties: there is a natural surjection $X \to G$ whose kernel is $M(G) \leq Z(X) \cap [X,X]$, and every projective representation $\pi:G \to \operatorname{PGL}_n(\mathbb{C})$ has an associated group homomorphism $\rho:X \to \operatorname{GL}_n(\mathbb{C})$ such that compositions $X \xrightarrow{\rho} \operatorname{GL}_n(\mathbb{C}) \to \operatorname{PGL}_n(\mathbb{C})$ and $X \to G \xrightarrow{\pi} \operatorname{PGL}_n(\mathbb{C})$ are equal. The definition for $\pi$ to be irreducible is basically that $\rho$ is irreducible. To find $\alpha$ from $\rho$ requires finding a second function $\mu:G \to X$ so that the composition $G \xrightarrow{\mu} X \to G$ is the identity. Then $X$ itself is defined (by Schreier) via the function $\zeta:G \times G \to M(G): (g,h) \mapsto \mu(gh)\mu(g)^{-1}\mu(h)^{-1}$ and $\alpha(g,h) = \rho( \zeta(g,h) )$. If $\rho$ is irreducible, then $\rho$ restricted to $M(G) \leq Z(X)$ consists only of scalar matrices, so $\chi{\downarrow_{M(G)}} = \chi(1) \lambda$ where $\chi$ is the character of $\rho$ and $\lambda\in \operatorname{Irr}(M(G))$ is a (irreducible) linear character of $M(G)$. While $\alpha$ and $\zeta$ depend on $\mu$, $\pi$ and $\lambda$ do not, so we can collect the various $\pi$ by which $\lambda$ they have. Since $\langle \chi{\downarrow_{M(G)}}, \lambda \rangle = \langle \chi(1)\lambda,\lambda\rangle = \chi(1)$, we get that $$\frac{\langle\chi{\downarrow_{M(G)}},\lambda\rangle}{\chi(1)}=\begin{cases} 1 & \text{if $\chi$ lies over $\lambda$}\\ 0 & \text{otherwise} \end{cases}$$ Hence $k_\lambda(G)$ literally adds up this 1/0 indicator function for all irreducible characters $\chi$ of $X$.

The inequality: This follows from a lemma of Burnside, which says in part that $$\sum_{\chi\in \operatorname{Irr}(X)} \frac{\chi(z)}{\chi(1)} \geq 0 \quad\forall z \in X$$ Given Burnside's lemma, we just expand and do a standard regrouping: $$\begin{array}{rcl} k_\lambda(G) &=&\sum_{\chi \in \operatorname{Irr}(X)} \frac{ \langle \chi{\downarrow_{M(G)}}, \lambda \rangle }{ \chi(1) } \\ &=&\sum_{\chi \in \operatorname{Irr}(X)} \frac{1}{|M(G)|\chi(1)} \sum_{z\in M(G)} \chi(z) \lambda(z^{-1}) \\ &=&\sum_{z\in M(G)} \frac{\lambda(z^{-1})}{|M(G)|}\left( \sum_{\chi \in \operatorname{Irr}(X)} \frac{\chi(z)}{\chi(1)} \right) \\ &\leq&\sum_{z\in M(G)} \frac{|\lambda(z^{-1})|}{|M(G)|}\left| \sum_{\chi \in \operatorname{Irr}(X)} \frac{\chi(z)}{\chi(1)} \right| \\ &\leq&\sum_{z\in M(G)} \frac{1}{|M(G)|}\left( \sum_{\chi \in \operatorname{Irr}(X)} \frac{\chi(z)}{\chi(1)} \right) \\ &=&\sum_{\chi \in \operatorname{Irr}(X)}\frac{1}{|M(G)|\chi(1)} \sum_{z\in M(G)} \chi(z) \mathbb{1}(z) \\ &=&\sum_{\chi \in \operatorname{Irr}(X)}\frac{\langle \chi{\downarrow_{M(G)}}, \mathbb{1} \rangle}{\chi(1)} \\ &=& k_1(G) = k(G) \end{array}$$

Burnside's lemma is theoretically proven in Gallagher (1962) (I think this is the existence of Fejer kernels), but probably that reference is not useful here. Burnside himself only gives it as an exercise on page 319, exercise 7, section 288 of the second edition of textbook on the Theory of Groups of Finite Order.

Bibliography

• Burnside, W. Theory of groups of finite order. 2d ed. Dover Publications, Inc., New York, 1955. xxiv+512 pp. MR69818
• Gallagher, P. X. “Group characters and commutators.” Math. Z. 79 (1962) 122–126. MR140590 DOI:10.1007/BF01193110