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].

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 $M(G).$

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 $G.$

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.

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].

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.

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.

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

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 $G.$

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 $x\in \mathcal {C}.$

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.