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
where
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
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
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
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:
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,
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
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
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
Proof. Let H be the $\alpha $ -covering group of $G.$ Then, using Theorem 1.1 and Lemma 2.6,
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
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
Proof. Using Corollary 1.2 and the proof of Theorem 3.1,
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 $k^- = 1.$