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FROBENIUS–SCHUR INDICATORS FOR PROJECTIVE CHARACTERS WITH APPLICATIONS

Published online by Cambridge University Press:  08 October 2024

R. J. HIGGS*
Affiliation:
School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland
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Abstract

Let $\alpha $ be a complex valued $2$-cocycle of finite order of a finite group $G.$ The nth Frobenius–Schur indicator of an irreducible $\alpha $-character of G is defined and its properties are investigated. The indicator is interpreted in general for $n =2$ and it is shown that it can be used to determine whether an irreducible $\alpha $-character is real-valued under the assumption that the order of $\alpha $ and its cohomology class are both $2$. A formula, involving the real $\alpha $-regular conjugacy classes of $G,$ is found to count the number of real-valued irreducible $\alpha $-characters of G under the additional assumption that these characters are class functions.

Type
Research Article
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

1 Introduction

Throughout this paper, G will denote a finite group. Also, all the representations considered will be taken to be over the field of complex numbers. The set of all ordinary irreducible characters of G is denoted as usual by $\operatorname {\mathrm {Irr}}(G)$ , and $\operatorname {\mathrm {Lin}}(G)$ will denote the group of linear characters of $G.$

There are a number of results concerning $\operatorname {\mathrm {Irr}}(G)$ and Frobenius–Schur indicators, three of which are reviewed here. For the first, see [Reference Isaacs3, pages 49–50].

Theorem 1.1. Define $\theta _n: G\rightarrow \mathbb {Z}_{\geq 0}$ by $\theta _n(x) = \vert \{g\in G: g^n = x\}\vert $ for $n\in \mathbb {N}.$ Then

$$ \begin{align*}\theta_n = \sum_{\chi\in\operatorname{\mathrm{Irr}}(G)}v_n(\chi)\chi,\end{align*} $$

where

$$ \begin{align*}v_n(\chi) = \frac{1}{\vert G\vert}\sum_{x\in G}\chi(x^n)\end{align*} $$

is the nth Frobenius–Schur indicator of $\chi $ and $v_n(\chi )\in \mathbb {Z}.$

The second result is a consequence of this theorem (see [Reference Isaacs3, Corollary 4.6]).

Corollary 1.2. Let G have exactly t involutions. Then

$$ \begin{align*}1+t =\sum_{\chi\in\operatorname{\mathrm{Irr}}(G)}v_2(\chi)\chi(1).\end{align*} $$

An element of G is real if it is conjugate to its inverse and $\chi \in \operatorname {\mathrm {Irr}}(G)$ is real if $\chi (x)\in \mathbb {R}$ for all $x\in G.$ For the third result connecting these two concepts, see [Reference Isaacs3, Problem 6.13].

Theorem 1.3. The number of real conjugacy classes of G is equal to the number of real $\chi \in \operatorname {\mathrm {Irr}}(G).$

The purpose of this paper is to find generalisations of these three results if irreducible projective characters of G are considered instead of ordinary ones. To generalise Theorem 1.1 it will be necessary to define the nth Frobenius–Schur indicator of an irreducible projective character of $G.$ A number of remarks and examples were made and given in [Reference Humphreys2, pages 27–28] to show that this and Theorem 1.3 do not have a straightforward generalisation to the projective character situation, but our approach overcomes those difficulties.

In Section 2, basic facts about projective representations of G with $2$ -cocycle $\alpha $ will be stated. The nth Frobenius–Schur indicator of an irreducible projective character of G is then defined and interpreted for $n = 2.$ Using this, the generalisations sought of the three results will be found in Section 3, although for the last two restricted to the case when both $\alpha $ and its cohomology class have order $2$ .

2 Frobenius–Schur indicators for projective characters

All of the standard facts and concepts relating to projective representations below may be found in [Reference Karpilovsky4, Reference Karpilovsky5], or (albeit to a lesser extent) [Reference Isaacs3, Ch. 11] or [Reference Haggarty and Humphreys1].

Definition 2.1. 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 form 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 2.2. Let $\alpha $ be a $2$ -cocycle of $G.$ Then $x\in G$ is $\alpha $ -regular if $\alpha (x, y) = \alpha (y, x)$ for all $y\in C_G(x).$

Let $\beta \in [\alpha ]$ . Then $x\in G$ is $\alpha $ -regular if and only if it is $\beta $ -regular. If x is $\alpha $ -regular then so too are $x^{-1}$ and any conjugate of $x,$ so from the latter one may refer to the $\alpha $ -regular conjugacy classes of $G.$

Definition 2.3. 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 $P(x)P(y) = \alpha (x, y)P(xy)$ for all x, $y\in G.$

Observe that if P is an $\alpha $ -representation of $G,$ then $P(g)P(x)P(g)^{-1}\! =\! f_{\alpha }(g, x)P(gxg^{-1})$ and $P(x)^m = p_{\alpha }(x, m)P(x^m)$ for all $g, x\in G$ and $m\in \mathbb {N},$ where

$$ \begin{align*}f_{\alpha}(g, x) = \frac{\alpha(g, xg^{-1})\alpha(x, g^{-1})}{\alpha(g, g^{-1})} \quad \text{and} \quad p_{\alpha}(x, m) = \prod_{i=1}^{m-1}\alpha(x, x^i) \quad\text{for } m> 1.\end{align*} $$

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). Next $x\in G$ is $\alpha $ -regular if and only if $\xi (x)\not = 0$ 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 a class-function $2$ -cocycle, that is, the elements of $\operatorname {\mathrm {Proj}}(G, \alpha )$ are class functions. If $\alpha $ is a class-function $2$ -cocycle of G, then $x\in G$ is $\alpha $ -regular if and only if $f_{\alpha }(g, x) = 1$ for all $g\in G.$

The nth Frobenius–Schur indicator of $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha )$ can now be defined and agrees with the normal definition if $\alpha $ is trivial.

Definition 2.4. Let $\alpha $ be a $2$ -cocycle of G of finite order. Then the nth Frobenius–Schur indicator $v_n^{\alpha }(\xi )$ for $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha )$ and $n\in \mathbb {N}$ is given by

$$ \begin{align*}v_n^{\alpha}(\xi) = \left\{\! \begin{array}{@{}ll} \displaystyle{\frac{1}{\vert G\vert}\sum_{x\in G}p_{\alpha}(x, n)\xi(x^n)} & \text{if}\ n\equiv 0\pmod{o(\alpha)} \\ 0 & \text{otherwise.} \end{array} \right.\end{align*} $$

If $\alpha $ is a $2$ -cocycle of finite order of $G,$ then this allows the construction of the $\alpha $ -covering group H of G (see [Reference Karpilovsky4, Ch. 4, Section 1] or [Reference Haggarty and Humphreys1, page 191]). Let $\omega $ be a primitive $o(\alpha )$ th root of unity and let $A = \langle \omega \rangle .$ The set of elements of H may be taken to be $\{ar(x): a\in A, x\in G\}$ , and H is a group under the binary operation $ar(x)br(y) = ab\alpha (x, y)r(xy)$ for all $a, b\in A$ and all $x, y\in G.$ This is a central extension of G:

$$ \begin{align*}1\rightarrow A\rightarrow H \xrightarrow{\pi} G\rightarrow 1,\end{align*} $$

with $\pi (r(x) )= x$ for all $x\in G.$ It also has the following important property. Let P be an $\alpha ^i$ -representation of G for $i\in \mathbb {Z}$ . Then $R(ar(x)) = \lambda ^i(a)P(x)$ for all $a\in A$ and all $x\in G$ is an ordinary representation of $H,$ where $\lambda \in \operatorname {\mathrm {Lin}}(A)$ with $\lambda (\omega ) = \omega ;$ moreover, P is irreducible if and only if R is. Here R is said to linearise P (or to be the lift of P). Let $\operatorname {\mathrm {Irr}}(H\vert \lambda ^i) = \{\chi \in \operatorname {\mathrm {Irr}}(H): \chi _A = \chi (1)\lambda ^i\}$ for $i\in \mathbb {Z}.$ Then the linearisation process outlined means that for each such i there exists a bijection from $\operatorname {\mathrm {Irr}}(H\vert \lambda ^i)$ to $\operatorname {\mathrm {Proj}}(G, \alpha ^i)$ defined by $\chi \mapsto \xi ,$ where $\chi (r(x)) = \xi (x)$ for all $x\in G$ and it is convenient to say that $\chi $ linearises $\xi $ .

Now x is $\alpha $ -regular if and only if $\omega ^ir(x)$ and $\omega ^jr(x)$ are not conjugate for all i and j with $0\leq i<j\leq o(\alpha )-1.$ So for counting purposes there are exactly $o(\alpha )$ conjugacy classes of H that map under $\pi $ to the conjugacy class of an $\alpha $ -regular element of G and fewer than this for an element that is not $\alpha $ -regular. If $o(\alpha ) = o([\alpha ]),$ then $A\leq H'$ and the mapping $\alpha ^i\mapsto [\alpha ^i] = [\alpha ]^i$ for $i =0,\ldots , o(\alpha )-1$ is a bijection.

Lemma 2.5. Let $\alpha $ be a $2$ -cocycle of G of finite order and let H be the $\alpha $ -covering group of $G.$ If $r(x)\in H$ is real, then so too is $x.$ Conversely if $x\in G$ is real, then $r(x)$ is real if and only if there exists $g\in G$ such that $gxg^{-1} = x^{-1}$ and $f_{\alpha }(g, x) = \alpha (x, x^{-1})^{-1}.$

Proof. If $r(x)$ is real with $r(g)r(x)r(g)^{-1} = r(x)^{-1},$ it follows that $f_{\alpha }(g, x)r(gxg^{-1}) = \alpha (x, x^{-1})^{-1}r(x^{-1}),$ so that in particular $gxg^{-1} = x^{-1}$ and x is real. The converse is now obvious.

Lemma 2.6. Let $\alpha $ be a $2$ -cocycle of G of finite order and let H be the $\alpha $ -covering group of $G.$ Let $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha ^i)$ for $i\in \mathbb {Z}$ and let $\chi \in \operatorname {\mathrm {Irr}}(H\vert \lambda ^i)$ linearise $\xi .$ Then $v_n^{\alpha ^i}(\xi ) = v_n(\chi ).$

Proof. Using the notation introduced, $r(x)^n = p_{\alpha }(x, n)r(x^n)$ for $n\in \mathbb {N}.$ So from Theorem 1.1,

$$ \begin{align*} v_n(\chi) & = \frac{1}{\vert H\vert}\sum_{a\in A, x\in G}\chi(a^np_{\alpha}(x, n)r(x^n)) \\ & = \frac{1}{\vert H\vert}\sum_{a\in A, x\in G}\lambda^i(a^n)p_{\alpha^i}(x, n)\xi(x^n) = v_n(\lambda^i)v_n^{\alpha^i}(\xi) = v_n^{\alpha^i}(\xi), \end{align*} $$

since $v_n(\lambda ^i) = v_1(\lambda ^{ni})$ from Theorem 1.1, so that $v_n(\lambda ^i) =1$ if $o(\lambda ^{ni}) = 1$ and is $0$ otherwise.

Let $\alpha $ be a $2$ -cocycle of G of finite order and let H be the $\alpha $ -covering group of $G.$ Consider another transversal of A in $H, \{s(x): x\in G\}$ with $s(1) = 1,$ where $s(x) = \delta (x)r(x)$ for $\delta (x)\in A.$ This gives rise to a new $2$ -cocycle $\beta \in [\alpha ]$ with $\beta = t(\delta )\alpha $ and for which $o(\beta )$ divides $o(\alpha ).$ Let $\chi \in \operatorname {\mathrm {Irr}}(H\vert \lambda ^i)$ . Then $\chi $ linearises $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha ^i)$ and $\xi '\in \operatorname {\mathrm {Proj}}(G, \beta ^i),$ where $\xi '(x) = \lambda ^i(\delta (x))\xi (x)$ for all $x\in G.$ Now $s(x)^n = r(x)^n$ for $n\equiv 0 \pmod {o(\alpha )}$ and so, from the proof of Lemma 2.6, $v_n^{\alpha ^i}(\xi ) = v_n^{\beta ^i}(\xi ')$ for $n\equiv 0 \pmod {o(\alpha )}.$ If $o(\alpha ) = o([\alpha ]),$ then $o(\beta ) = o(\alpha )$ and H is also the $\beta $ -covering group of $G.$

Using this notation, $\{s(x): x\in G\}$ can be chosen to be conjugacy-preserving, that is, $s(x)$ and $s(y)$ are conjugate in H whenever x and y are conjugate in G (see [Reference Karpilovsky5, Lemma 4.1.1] or [Reference Haggarty and Humphreys1, Proposition 1.1]) and this choice makes $\beta $ a class-function $2$ -cocycle.

The next result is an immediate corollary of Lemma 2.6 from [Reference Isaacs3, page 58].

Corollary 2.7. Let $\alpha $ be a $2$ -cocycle of G with $o(\alpha ) = o([\alpha ]) = 2.$ Let $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha ).$ Then $v_2^{\alpha }(\xi ) = 0$ or $\pm 1.$ Moreover, $v_2^{\alpha }(\xi ) = 0$ if and only if $\xi $ is nonreal, $v_2^{\alpha }(\xi ) =1$ if and only if $\xi $ is afforded by a real $\alpha $ -representation, and $v_2^{\alpha }(\xi ) = -1$ if and only if $\xi $ is real but is not afforded by any real $\alpha $ -representation of $G.$

Lemma 2.6 also explains why the second Frobenius–Schur indicator is defined to be $0$ when $o(\alpha )> 2,$ but another rationale follows. If $\alpha (x, y)\not \in \mathbb {R}$ and P is an $\alpha $ -representation of $G,$ then at least one of the three matrices $P(x), P(y)$ and $P(xy)$ must contain a nonreal entry.

Example 2.8. Consider the elementary abelian group $G = C_p\times C_p$ for p a prime number, which has $M(G)\cong C_p$ (see [Reference Karpilovsky4, Proposition 10.7.1]). Let $\alpha $ be any $2$ -cocycle of G with $o([\alpha ]) = p.$ Then the only $\alpha $ -regular element of G is the identity element and consequently the only element $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha )$ has $\xi (1) = p$ and $\xi (x) = 0$ for $x\ne 1$ (see [Reference Karpilovsky5, Theorem 8.2.21]). So $\xi $ is integer-valued, but is not afforded by any real $\alpha $ -representation for $p\geq 3$ from the remark preceding this example. If $o(\alpha )\geq 3$ and is finite, let H be the $\alpha $ -covering group of G and let $\chi \in \operatorname {\mathrm {Irr}}(H\vert \lambda )$ linearise $\xi .$ Then $\chi $ is nonreal since $\lambda $ is nonreal.

It can be concluded from Example 2.8 that the results of Corollary 2.7 do not hold in general for any group G with a $2$ -cocycle of finite order greater than $2$ and in this case $v_2^{\alpha }(\xi ) = 0$ for all $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha )$ can only be interpreted as meaning that each $\xi $ is not afforded by any real $\alpha $ -representation of $G.$

It should be noted that in general the value of $v_n^{\alpha }(\xi )$ for $n\equiv 0\pmod {o(\alpha )}$ depends upon the choice of $\alpha $ , even if $o(\alpha ) = o([\alpha ]) =2$ , as the next example illustrates.

Example 2.9. Let $G = C_2\times C_2.$ It is well known that G has two Schur representation groups (also known as covering groups) up to isomorphism, namely D and Q, the dihedral and quaternion groups of order $8$ , respectively. The character tables of these two groups are identical, and the irreducible characters $\chi $ and $\chi '$ of degree $2$ of each linearise $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha )$ and $\xi '\in \operatorname {\mathrm {Proj}}(G, \alpha ')$ respectively, where $\alpha $ and $\alpha '$ are the $2$ -cocycles of G constructed from D and Q of order $2$ with $o([\alpha ]) = o([\alpha ']) = 2.$ Now $\xi $ and $\xi '$ are identical and integer-valued from Example 2.8; however, $v_2^{\alpha }(\xi ) = v_2(\chi ) = 1,$ whereas $v_2^{\alpha '}(\xi ') = v_2(\chi ') = -1.$

Using Lemma 2.6 other results concerning $v_n$ carry over to $v_n^{\alpha },$ as in the next lemma.

Lemma 2.10. Let $\alpha $ be a $2$ -cocycle of G of finite order. Let $\xi \in \operatorname {\mathrm {Proj}}(G, \alpha )$ and let $\mu \in \operatorname {\mathrm {Lin}}(G)$ with $\mu ^n$ trivial for $n\in \mathbb {N}.$ Then $v_n^{\alpha }(\xi )\in \mathbb {Z}$ and $v_n^{\alpha }(\mu \xi ) = v_n^{\alpha }(\xi ).$

Proof. Let H be the $\alpha $ -covering group of G and $\chi \in \operatorname {\mathrm {Irr}}(H\vert \lambda )$ linearise $\xi $ . Then $v_n^{\alpha }(\xi )\in \mathbb {Z}$ from Lemma 2.6 and Theorem 1.1. Now let $\nu \in \operatorname {\mathrm {Lin}}(H)$ linearise $\mu .$ Then $\nu \chi $ linearises $\mu \xi $ and $\nu ^n$ is trivial, so $v_n^{\alpha }(\mu \xi ) = v_n(\nu \chi ) = v_n(\chi ) = v_n^{\alpha }(\xi )$ using [Reference Isaacs3, Lemma 4.8] and Lemma 2.6.

3 Frobenius–Schur indicator applications

Let $\alpha $ be a $2$ -cocycle of G of finite order and define

$$ \begin{align*} \theta_n^{\alpha} = \sum_{\xi\in\operatorname{\mathrm{Proj}}(G, \alpha)}v_n^{\alpha}(\xi)\xi. \end{align*} $$

From Lemma 2.10, $\theta _n^{\alpha }$ is an integral linear combination of $\alpha $ -characters of G and so $\theta _n^{\alpha }(x) = 0$ if x is not $\alpha $ -regular. If, in addition, $\alpha $ is a class-function $2$ -cocycle, then $\theta _n^{\alpha }$ is a class function. If $o(\alpha ) = 1,$ then $\theta _n^{\alpha } = \theta _n$ as in Theorem 1.1.

By analogy with the definition in Theorem 1.1, define $\theta _n^+: G\rightarrow \mathbb {Z}_{\geq 0}$ by

$$ \begin{align*} \theta_n^+(x) = \vert\{g\in G: p_{\alpha}(g, n) = 1\text{ and } g^n = x\}\vert \end{align*} $$

for $n\in \mathbb {N}$ . This function is used in the generalisation of Theorem 1.1.

Theorem 3.1. Let $\alpha $ be a $2$ -cocycle of G with $o(\alpha ) = o([\alpha ])$ of finite order m and let $n\in \mathbb {N}$ with $n\equiv 0\pmod {m}.$ Then

$$ \begin{align*}\sum_{i=1}^{m-1}\theta_n^{\alpha^i} = m\theta_n^+ - \theta_n.\end{align*} $$

Proof. Let H be the $\alpha $ -covering group of $G.$ Then, using Theorem 1.1 and Lemma 2.6,

$$ \begin{align*} \theta_n(r(x)) &= m\vert\{g\in G: p_{\alpha}(g, n) = 1\ \text{and}\ g^n = x\}\vert \\ &= \sum_{\chi\in\operatorname{\mathrm{Irr}}(H)} v_n(\chi)\chi(r(x)) = \sum_{\psi\in\operatorname{\mathrm{Irr}}(G)} v_n(\psi)\psi(x) + \sum_{i= 1}^{m-1}\sum_{\xi\in\operatorname{\mathrm{Proj}}(G, \alpha^i)} v_n^{\alpha^i}(\xi)\xi(x) \\ &= \theta_n(x) + \sum_{i=1}^{m-1}\theta_n^{\alpha^i}(x) \end{align*} $$

for all $x\in G.$

Continuing with the notation and hypotheses in Theorem 3.1, suppose $g\in G$ with $g^n = x$ and let $y\in C_G(x).$ Then

$$ \begin{align*} f_{\alpha}(y, x)p_{\alpha}(g, n)r(x) = (r(y)r(g)r(y)^{-1})^n = p_{\alpha}(ygy^{-1}, n)r(x). \end{align*} $$

Now if m is a prime number and x is not $\alpha $ -regular, then $r(x)$ is conjugate to $ar(x)$ for all $a\in A.$ So if $r(y)r(x)r(y)^{-1} = ar(x),$ then the mapping $g\mapsto ygy^{-1}$ defines a bijection from $\{g\in G: p_{\alpha }(g, n) = 1~ \text {and}~g^n = x\}$ to $\{g\in G: p_{\alpha }(g, n) = a~ \text {and}~g^n = x\},$ which explains why $m\theta _n^+(x) = \theta _n(x)$ in this scenario.

The next result is a special case of Theorem 3.1 that generalises Corollary 1.2.

Corollary 3.2. Let $\alpha $ be a $2$ -cocycle of G with $o(\alpha ) = o([\alpha ]) = 2.$ Let H be the $\alpha $ -covering group of G and let H and G have exactly t and s involutions, respectively. Then

$$ \begin{align*}t-s = \sum_{\xi\in\operatorname{\mathrm{Proj}}(G, \alpha)} v_2^{\alpha}(\xi)\xi(1).\end{align*} $$

Proof. Using Corollary 1.2 and the proof of Theorem 3.1,

$$ \begin{align*}\sum_{\xi\in\operatorname{\mathrm{Proj}}(G, \alpha)} v_2^{\alpha}(\xi)\xi(1) = \theta_2(r(1)) - \theta_2(1) = t -s.\\[-45pt] \end{align*} $$

The final aim is to generalise Theorem 1.3, which involves an analysis of the real conjugacy classes of $G.$

Lemma 3.3. Let $\alpha $ be a class-function $2$ -cocycle of G with $o(\alpha ) = o([\alpha ]) = 2.$ Let H be the $\alpha $ -covering group of G with its associated central subgroup $A = \langle -1\rangle $ and transversal $\{r(x): x\in G\}.$ Let $x\in G$ be real. Then $r(x)$ is nonreal if and only if x is $\alpha $ -regular and $\alpha (x, x^{-1}) = -1.$

Proof. If x is $\alpha $ -regular, then $r(x)$ is real if and only if $\alpha (x, x^{-1}) = 1$ from Lemma 2.5. On the other hand, if x is not $\alpha $ -regular, then there exists $y\in C_G(x)$ such that $r(y)r(x^{-1})r(y)^{-1} = -r(x^{-1}).$ Now if $gxg^{-1} = x^{-1},$ then either $f_{\alpha }(g, x)$ or $f_{\alpha }(yg, x)$ equals $\alpha (x, x^{-1})^{-1}$ and so $r(x)$ is real from Lemma 2.5.

Let P be an $\alpha $ -representation of G of dimension $n.$ Then for all $g, x\in G$ , $P(g)P(x)P(x^{-1})P(g)^{-1}$ equals $f_{\alpha }(g, x)f_{\alpha }(g, x^{-1})\alpha (gxg^{-1}, gx^{-1}g^{-1})I_n,$ but it also equals $\alpha (x, x^{-1})I_n.$ Thus if $\alpha $ is a class-function $2$ -cocycle of G and x is $\alpha $ -regular, then $\alpha (x, x^{-1}) = \alpha (gxg^{-1}, gx^{-1}g^{-1})$ for all $g\in G.$ In the context of Lemma 3.3 and using this result, let $k_0, k^+$ and $k^-$ denote the number of conjugacy classes $\mathcal {C}$ of G that are respectively (a) real and not $\alpha $ -regular, (b) real and $\alpha $ -regular with $\alpha (x, x^{-1}) = 1$ for all $x\in \mathcal {C},$ and (c) real and $\alpha $ -regular with $\alpha (x, x^{-1}) = -1$ for all $x\in \mathcal {C}.$

Theorem 3.4. Let $\alpha $ be a class-function $2$ -cocycle of G with $o(\alpha ) = o([\alpha ]) = 2.$ Then the number of real elements of $\operatorname {\mathrm {Proj}}(G, \alpha )$ is $k^+ - k^-.$

Proof. Let H be the $\alpha $ -covering group of $G.$ The number of real conjugacy classes of G and H is $k_0 + k^+ + k^-$ and $k_0 + 2k^+$ , respectively, from Lemma 3.3 and previous remarks. Thus from Theorem 1.3 the number of real elements of $\operatorname {\mathrm {Proj}}(G, \alpha )$ is the second number minus the first.

If $\alpha '$ is a $2$ -cocycle of G with $o(\alpha ') = o([\alpha ']) = 2,$ then we may let H be the $\alpha '$ -covering group of $G.$ As explained after Lemma 2.6: (a) there exists a change of transversal so that the resultant $2$ -cocycle $\alpha $ of G is a class-function $2$ -cocycle with $o(\alpha ) = 2$ and $\alpha \in [\alpha '];$ (b) the numbers of real elements of $\operatorname {\mathrm {Proj}}(G, \alpha )$ and $\operatorname {\mathrm {Proj}}(G, \alpha ')$ are equal, with this number given by Theorem 3.4.

Example 3.5. Every element of the symmetric group $S_4$ is real, $M(S_4)\cong C_2$ and $S_4$ has two Schur representation groups up to isomorphism (see [Reference Morris6, Theorem 1]). One is the binary octahedral group, and the three elements of $\operatorname {\mathrm {Proj}}(S_4, \alpha )$ constructed from this group, for a class-function $2$ -cocycle $\alpha $ with $o(\alpha ) = o([\alpha ]) = 2,$ are all real (see [Reference Morris6, page 70]), so $k^+ = 3$ and $k^- = 0.$ The other Schur representation group is $\mathrm {GL}(2, 3)$ , and only one element of $\operatorname {\mathrm {Proj}}(S_4, \alpha ')$ constructed from this group, for a class-function $2$ -cocycle $\alpha '$ with $o(\alpha ') = 2$ and $\alpha '\in [\alpha ],$ is real (see [Reference Humphreys2, Remark (ii), pages 27–28] or [Reference Morris6, page 56]), so here $k^+ = 2$ and $k^- = 1.$

References

Haggarty, R. J. and Humphreys, J. F., ‘Projective characters of finite groups’, Proc. Lond. Math. Soc. (3) 36(1) (1978), 176192.CrossRefGoogle Scholar
Humphreys, J. F., ‘Rational valued and real valued projective characters of finite groups’, Glasg. Math. J. 21(1) (1980), 2328.CrossRefGoogle Scholar
Isaacs, I. M., Character Theory of Finite Groups, Pure and Applied Mathematics, 69 (Academic Press, New York, 1976).Google Scholar
Karpilovsky, G., Group Representations. Volume 2, North-Holland Mathematics Studies, 177 (North-Holland Publishing Co., Amsterdam, 1993).Google Scholar
Karpilovsky, G., Group Representations. Volume 3, North-Holland Mathematics Studies, 180 (North-Holland Publishing Co., Amsterdam, 1994).Google Scholar
Morris, A. O., ‘The spin representation of the symmetric group’, Proc. Lond. Math. Soc. (3) 12 (1962), 5576.CrossRefGoogle Scholar