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 [Reference Isaacs3, 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 [Reference Karpilovsky4, page 184] for details). The following known results concerning
$\alpha $
-representations and characters may all be found in [Reference Isaacs3, Problems 11.7 and 11.8] and [Reference Haggarty and Humphreys1, 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 [Reference Isaacs3, Problems 2.11 and 2.12] and [Reference Lang6, 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:
-
(a)
$\chi (g)$
is rational for all
$\chi \in \operatorname {\mathrm {Irr}}(G);$
-
(b) g is conjugate to
$g^m$
for all
$m\in \mathbb {N}$
with m relatively prime to
$\vert G\vert ;$
-
(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 [Reference Karpilovsky5, 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 [Reference Karpilovsky4, 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 [Reference Karpilovsky5, 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 [Reference Karpilovsky4, 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 [Reference Karpilovsky5, Corollary 1.2.7]). If
$(m, \vert G\vert ) = 1$
then, as shown in the proof of [Reference Humphreys2, 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
$\xi \in \operatorname {\mathrm {Proj}}(G, \alpha )$
,
-
(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 -
(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
$\xi \in \operatorname {\mathrm {Proj}}(G, \alpha ).$
Acknowledgement
The author would like to thank the diligent referee for making suggestions to shorten the proof of Theorem 2.1, use alternative references and improve the clarity of this paper.
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