PROJECTIVE CHARACTER VALUES ON REAL AND RATIONAL ELEMENTS

Abstract

Let $\alpha $ be a complex-valued $2$-cocycle of a finite group G with $\alpha $ chosen so that the $\alpha $-characters of G are class functions and analogues of the orthogonality relations for ordinary characters are valid. Then the real or rational elements of G that are also $\alpha $-regular are characterised by the values that the irreducible $\alpha $-characters of G take on those respective elements. These new results generalise two known facts concerning such elements and irreducible ordinary characters of $G;$ however, the initial choice of $\alpha $ from its cohomology class is not unique in general and it is shown the results can vary for a different choice.

1. Introduction

Throughout this paper G will denote a finite group.

Definition 1.1. A $2$ -cocycle of G over $\mathbb {C}$ is a function $\alpha : G\times G\rightarrow \mathbb {C}^*$ such that $\alpha (1, 1) = 1$ and $\alpha (x, y)\alpha (xy, z) = \alpha (x, yz)\alpha (y, z)$ for all x, y, $z\in G.$

The set of all such $2$ -cocycles of G forms a group $Z^2(G, \mathbb {C}^*)$ under multiplication. Let $\delta : G\rightarrow \mathbb {C}^*$ be any function with $\delta (1) = 1.$ Then $t(\delta )(x, y) = \delta (x)\delta (y)/\delta (xy)$ for all x, $y\in G$ is a $2$ -cocycle of G, which is called a coboundary. Two $2$ -cocycles $\alpha $ and $\beta $ are cohomologous if there exists a coboundary $t(\delta )$ such that $\beta = t(\delta )\alpha .$ This defines an equivalence relation on $Z^2(G, \mathbb {C}^*)$ and the cohomology classes $[\alpha ]$ form a finite abelian group, called the Schur multiplier $M(G).$

Definition 1.2. Let $\alpha $ be a $2$ -cocycle of $G.$ Then $g\in G$ is $\alpha $ -regular if $\alpha (g, h) = \alpha (h, g)$ for all $h\in C_G(g).$

Setting $y = z = 1$ in Definition 1.1 yields $\alpha (x, 1) = 1$ and similarly $\alpha (1, x) =1$ for all $x\in G,$ hence $1$ is $\alpha $ -regular. Let $\beta \in [\alpha ]$ . Then $g\in G$ is $\alpha $ -regular if and only if it is $\beta $ -regular. If g is $\alpha $ -regular then any conjugate of g is also $\alpha $ -regular, so one may refer to the $\alpha $ -regular conjugacy classes of G (see [3, Problem 11.4]). Finally, if $m\in \mathbb {N}$ is relatively prime to $o(g),$ then it is easy to show $g^m$ is $\alpha $ -regular.

Definition 1.3. Let $\alpha $ be a $2$ -cocycle of $G.$ Then an $\alpha $ -representation of G of dimension n is a function $P:G\rightarrow \mathrm {GL}(n, \mathbb {C})$ such that $P(g)P(h) = \alpha (g, h)P(gh)$ for all g, $h\in G.$

To avoid repetition all $\alpha $ -representations of G in this paper are defined over $\mathbb {C}.$ An $\alpha $ -representation is also called a projective representation of G with $2$ -cocycle $\alpha $ and its trace function is its $\alpha $ -character. Let $\operatorname {\mathrm {Proj}}(G, \alpha )$ denote the set of all irreducible $\alpha $ -characters of G. The relationship between $\operatorname {\mathrm {Proj}}(G, \alpha )$ and $\alpha $ -representations is much the same as that between $\operatorname {\mathrm {Irr}}(G)$ and ordinary representations of G (see [4, page 184] for details). The following known results concerning $\alpha $ -representations and characters may all be found in [3, Problems 11.7 and 11.8] and [1, Sections 1 and 4]. First, $\sum _{\xi \in \operatorname {\mathrm {Proj}}(G, \alpha )}\xi (1)^2 = \vert G\vert .$ Next $g\in G$ is $\alpha $ -regular if and only if $\xi (g)\not = 0$ for some ${\xi \in \operatorname {\mathrm {Proj}}(G, \alpha )}$ and $\vert \operatorname {\mathrm {Proj}}(G, \alpha )\vert $ is the number of $\alpha $ -regular conjugacy classes of $G.$ For $[\beta ]\in M(G)$ there exists $\alpha \in [\beta ]$ such that $o(\alpha ) = o([\beta ])$ and $\alpha $ is class-preserving, that is, the elements of $\operatorname {\mathrm {Proj}}(G, \alpha )$ are class functions. Henceforward it will be assumed that the initial choice of $2$ -cocycle $\alpha $ has these two properties, but the choice made within such $2$ -cocycles will affect the results obtained in Section 2. Under these assumptions the ‘standard’ inner product $\langle \phantom {\xi }, \phantom {\xi }\rangle $ may be defined on $\alpha $ -characters of G and the ‘normal’ orthogonality relations hold.

Definition 1.4. Let $g\in G.$ Then g is a real element if g is conjugate to $g^{-1}$ , and g is a rational element if g is conjugate to $g^m$ for all $m\in \mathbb {N}$ with m relatively prime to $o(g).$

Clearly every rational element of G is real; also G contains a nontrivial real element if and only if $\vert G\vert $ is even. The next two theorems are standard results in ordinary character theory concerning real and rational elements (see [3, Problems 2.11 and 2.12] and [6, Exercise XVIII.14]).

Theorem 1.5. Let $g\in G.$ Then $\chi (g)$ is real for all $\chi \in \operatorname {\mathrm {Irr}}(G)$ if and only if g is a real element.

Theorem 1.6. Let $g\in G.$ Then the following statements are equivalent:

1. (a) $\chi (g)$ is rational for all $\chi \in \operatorname {\mathrm {Irr}}(G);$

2. (b) g is conjugate to $g^m$ for all $m\in \mathbb {N}$ with m relatively prime to $\vert G\vert ;$

3. (c) g is a rational element.

In Section 2, these two results will be generalised to irreducible $\alpha $ -characters and an $\alpha $ -regular real or rational element of $G.$

2. Values of $\alpha $ -characters

Let P be an $\alpha $ -representation of G of dimension n with $\alpha $ -character $\xi .$ Then $P(g)P(g^{-1}) = \alpha (g, g^{-1})I_n$ for any $g\in G$ , and hence $P(g^{-1}) = \alpha (g, g^{-1})P(g)^{-1}.$ It follows that $\xi (g^{-1}) = \alpha (g, g^{-1})\overline {\xi (g)},$ where the bar denotes complex conjugation (see [5, Lemma 1.11.11]).

Theorem 2.1. Let $\alpha $ be a $2$ -cocycle of G and let $g\in G$ be $\alpha $ -regular. Then g is a real element if and only if $\xi (g) = \pm \vert \xi (g)\vert \omega $ for all $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha ),$ where $\omega ^2 = \alpha (g, g^{-1}).$

Proof. Suppose g is real and let $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha )$ such that $\xi (g)\neq 0.$ Then $\alpha (g, g^{-1})\overline {\xi (g)} = \xi (g)$ and the choice of $\alpha $ from Section 1 implies $\alpha (g, g^{-1})$ is a root of unity. Choose $\omega $ such that $\omega ^2 = \alpha (g, g^{-1}).$ Then $\xi (g)^2 = \vert \xi (g)\vert ^2\omega ^2$ and so $\xi (g) = \pm \vert \xi (g)\vert \omega .$

Conversely, suppose $\xi (g) = \pm \vert \xi (g)\vert \omega $ for all $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha ),$ where $\omega ^2 = \alpha (g, g^{-1}).$ Then

$$ \begin{align*} \sum_{\xi\in\operatorname{\mathrm{Proj}}(G, \alpha)} \xi(g)\overline{\xi(g^{-1})} = \overline{\alpha(g, g^{-1})}\omega^2\sum_{\xi\in\operatorname{\mathrm{Proj}}(G, \alpha)} \vert\xi(g)\vert^2 = \vert C_G(g)\vert, \end{align*} $$

and hence by the second orthogonality relation for $\alpha $ -characters g is conjugate to $g^{-1}.$

Let $g\in G$ be $\alpha $ -regular. From Theorem 2.1, if $\alpha (g, g^{-1}) = 1$ or $-1,$ then g is a real element if and only if $\xi (g)$ is real or purely imaginary, respectively, for all $\xi \in \operatorname {\mathrm {Proj}}(G,\alpha ).$ It should be noted that the root of unity $\omega $ that occurs in Theorem 2.1 depends upon the choice of $\alpha $ , as the next example illustrates.

Example 2.2. Every element of the symmetric group $S_4$ is rational and $M(S_4)$ is cyclic of order 2. Also $S_4$ has two Schur representation groups (also known as covering groups) up to isomorphism (see [4, Theorem 12.2.2]). One is the binary octahedral group and an $\alpha $ -character table of $S_4$ for $o(\alpha ) = 2$ constructed from this group is given in Table 1 (see [5, Theorem 5.6.4]). We deduce that $\alpha (g, g^{-1}) = 1$ for all $\alpha $ -regular $g\in S_4.$ The other Schur representation group is $\mathrm {GL}(2, 3)$ and it is easy to check that a $\beta $ -character table of $S_4$ for $o(\beta ) =2$ constructed from this group is identical to Table 1 except that the three entries in the last column are multiplied by $i,$ so $\beta ((1~2~3~4), (1~2~3~4)^{-1}) = -1.$

Table 1 $\alpha $ -character table of $S_4$ .

Two variations of Theorem 2.1 are discussed next, the first of which is easy to see.

Corollary 2.3. Let $\alpha $ be a $2$ -cocycle of G and let $g\in G$ be $\alpha $ -regular. Then g is a real element if and only if $\xi (g)^2\alpha ^{-1}(g, g^{-1})\in \mathbb {R}_{\geq 0}$ for all $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha ).$

Proof. Let $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha )$ and suppose $\xi (g)^2\alpha ^{-1}(g, g^{-1})= r$ for $r\in \mathbb {R}_{\geq 0}.$ Then $r~=~\vert \xi (g)\vert ^2$ and the result follows from Theorem 2.1.

Suppose g is an $\alpha $ -regular real element of $G.$ Then it was shown in Theorem 2.1 that $\xi (g)$ lies on a line in the complex plane of the form $\{r\omega : r\in \mathbb {R}\}$ for all $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha ),$ where $\vert \omega \vert = 1$ . Conversely, this latter condition is sufficient to guarantee that an $\alpha $ -regular element g of G is a real element.

Corollary 2.4. Let $\alpha $ be a $2$ -cocycle of G and let $g\in G$ be $\alpha $ -regular. Then g is a real element if and only if there exists an $\omega \in \mathbb {C}$ such that $\xi (g) = \pm \vert \xi (g)\vert \omega $ for all $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha ).$

Proof. Suppose the second condition holds. Then, using the same argument as that at the end of the proof of Theorem 2.1, it must be the case that the product of $\omega ^2$ and the root of unity $\overline {\alpha (g, g^{-1})}$ is $1$ and so g is a real element from Theorem 2.1. The converse obviously holds from Theorem 2.1.

Note that $\omega ^2 = \alpha (g, g^{-1})$ from Theorem 2.1 or the proof of Corollary 2.4. So $\omega $ is a $\vert G\vert $ th root of unity if $\vert G\vert $ is even (see [4, Theorem 10.11.1]). If $\vert G\vert $ is odd, then just one of $\omega $ and $-\omega $ is a $\vert G\vert $ th root of unity.

Rational elements are now considered. Continuing with the notation at the start of this section, an easy proof by induction shows $P(g)^m = f_{\alpha }(g, m)P(g^m)$ for any $g\in G$ and any $m\in \mathbb {N},$ where $f_{\alpha }(g, 1) = 1$ and

$$ \begin{align*}f_{\alpha}(g, m) = \alpha(g, g)\cdots \alpha(g, g^{m-1}) \quad\text{for}~m>1.\end{align*} $$

Let $\zeta $ be a primitive $\vert G\vert $ th root of unity. Then $\xi (g)\in \mathbb {Q}[\zeta ]$ and is an algebraic integer for any $g\in G$ (see [5, Corollary 1.2.7]). If $(m, \vert G\vert ) = 1$ then, as shown in the proof of [2, Theorem 2],

$$ \begin{align*}\xi(g^m) = f_{\alpha}^{-1}(g , m)\sigma_m(\xi(g)),\end{align*} $$

where $\sigma _m$ is the automorphism of $\mathbb {Q}[\zeta ]$ over $\mathbb {Q}$ that maps $\zeta $ to $\zeta ^m.$ The Galois group of $\mathbb {Q}[\zeta ]$ over $\mathbb {Q}$ is abelian and $\sigma _{-1}$ represents the restriction of complex conjugation to $\mathbb {Q}[\zeta ].$ Thus for all $z\in \mathbb {Q}[\zeta ]$ , $\sigma _m(\overline {z}) = \overline {\sigma _m(z)}$ and $\sigma _m(\vert z\vert ^2) = \vert \sigma _m(z)\vert ^2.$ So $\vert \xi (g^m)\vert ^2 = \sigma _m(\vert \xi (g)\vert ^2).$

Theorem 2.5. Let $\alpha $ be a $2$ -cocycle of G and let $g\in G$ be $\alpha $ -regular. Then g is conjugate to $g^m$ for all $m\in \mathbb {N}$ that are relatively prime to $\vert G\vert $ if and only, if for all $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha )$ ,

1. (a) there exists a $\vert G\vert $ th root of unity $\omega $ with $\omega ^2 = \alpha (g, g^{-1})$ such that $\xi (g) = \pm \vert \xi (g)\vert \omega $ and

2. (b) either $\sigma _m(\vert \xi (g)\vert ) = \vert \xi (g)\vert $ and $f_\alpha (g, m) = \omega ^{m-1}$ , or $\sigma _m(\vert \xi (g)\vert ) = -\vert \xi (g)\vert $ and $f_\alpha (g, m) = -\omega ^{m-1}.$

Proof. Suppose g is conjugate to $g^m$ for all $m\in \mathbb {N}$ with $(m, \vert G\vert ) = 1.$ Then, in particular, g is a real element of G from Theorem 1.6. Thus $\xi (g) = \pm \vert \xi (g)\vert \omega $ for all $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha ),$ where $\omega ^2 = \alpha (g, g^{-1})$ by Theorem 2.1. If $g =1,$ then (a) and (b) hold with $\omega = 1$ and so, as previously noted, in all cases $\omega $ is a $\vert G\vert $ th root of unity. By supposition $\xi (g) = \xi (g^m)$ and so $\vert \xi (g)\vert ^2 = \sigma _m(\vert \xi (g)\vert ^2)$ for all such $m.$ Thus $\vert \xi (g)\vert ^2\in \mathbb {Q}_{\geq 0}.$ Also

$$ \begin{align*}\pm\vert\xi(g)\vert\omega = f_{\alpha}^{-1}(g , m)\sigma_m(\pm\vert\xi(g)\vert\omega) = \pm f_{\alpha}^{-1}(g , m)\sigma_m(\vert\xi(g)\vert)\omega^m,\end{align*} $$

and consequently

$$ \begin{align*}\vert\xi(g)\vert = f_{\alpha}^{-1}(g , m)\sigma_m(\vert\xi(g)\vert)\omega^{m-1}.\end{align*} $$

Now $\sigma _m(\vert \xi (g)\vert ) = \pm \vert \xi (g)\vert $ . For the positive sign the conclusion is $f_{\alpha }(g, m) = \omega ^{m-1},$ since $\xi (g)\ne 0$ for some $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha ),$ and similarly for the negative sign.

Conversely, suppose (a) and (b) are true for all $m\in \mathbb {N}$ with $(m, \vert G\vert ) = 1.$ Then

$$ \begin{align*} \xi(g^m) = \pm f_{\alpha}^{-1}(g , m)\sigma_m(\vert\xi(g)\vert)\omega^m, \end{align*} $$

with the sign corresponding to that of $\xi (g) = \pm \vert \xi (g)\vert \omega .$ In either case, using (b),

$$ \begin{align*} \sum_{\xi\in\operatorname{\mathrm{Proj}}(G, \alpha)} \xi(g)\overline{\xi(g^m)} = f_{\alpha}(g, m)\omega^{1-m}\sum_{\xi\in\operatorname{\mathrm{Proj}}(G, \alpha)} \vert\xi(g)\vert^2 = \vert C_G(g)\vert, \end{align*} $$

and hence by the second orthogonality relation g is conjugate to $g^m.$

Suppose $\alpha $ is trivial and g is conjugate to $g^m$ for all $m\in \mathbb {N}$ with $(m, \vert G\vert ) = 1.$ Then with $\omega = 1,$ (a) in Theorem 2.5 implies that $\chi (g)$ is real for all $\chi \in \operatorname {\mathrm {Irr}}(G)$ . In addition, $f_{\alpha }(g, m) = 1$ for all such m, and so from (b), $\vert \chi (g)\vert \in \mathbb {Q}$ . Thus $\chi (g)\in \mathbb {Q}.$ Conversely, if $\chi (g)\in \mathbb {Q}$ for all $\chi \in \operatorname {\mathrm {Irr}}(G),$ then (a) and (b) in Theorem 2.5 obviously hold with $\omega =1.$ So Theorem 2.5 reduces to Theorem 1.6 in this case.

It is possible to replace (a) in Theorem 2.5 by: ‘(a) $'$ there exists an $\omega \in \mathbb {C}$ such that $\xi (g) = \pm \vert \xi (g)\vert \omega $ and’. Suppose (a) $'$ and (b) hold. Then $\omega ^2 = \alpha (g, g^{-1})$ from the proof of Corollary 2.4. Theorem 2.5 will then still hold using this variation provided $\omega $ is a $\vert G\vert $ th root of unity, which is the case if $\vert G\vert $ is even, using the remarks after Corollary 2.4. Suppose $\vert G\vert $ is odd and let $\gamma $ denote the unique $\vert G\vert $ th root of unity with $\gamma ^2 = \alpha (g, g^{-1})$ . Now $f_{\alpha }(g, m)$ is a $\vert G\vert $ th root of unity, and from (b), $f_{\alpha }(g, m) = \pm \gamma ^{m-1}$ or $\pm (-\gamma )^{m-1}.$ Setting $m = 1$ and then $2$ shows that $f_{\alpha }(g, m) = \gamma ^{m-1},$ so $\omega $ must equal $\gamma $ in this situation.

Of course, using Theorem 1.6, the conditions in Theorem 2.5 are necessary and sufficient for an $\alpha $ -regular element of G to be a rational element. Also $\mathbb {Q}$ can be replaced by $\mathbb {Z}$ in either formulation of Theorem 2.5, since as previously noted $\xi (g)$ is an algebraic integer for all $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha )$ and any $g\in G.$ This yields the following useful consequence of Theorem 2.5.

Corollary 2.6. Let $\alpha $ be a $2$ -cocycle of G and let $g\in G$ be $\alpha $ -regular. If g is a rational element, then $\xi (g)^2\alpha ^{-1}(g, g^{-1})\in \mathbb {Z}_{\geq 0}$ for all $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha ).$