\section{Introduction}
\label{Introduction}

Let $M^n$ be a flat Riemannian manifold of dimension $n$, i.e. a compact Riemannian manifold without boundary, with sectional curvature equal to zero.
From the Bieberbach theorem \cite{Sz12} it is known that the torsion free fundamental group $\Gamma = \pi_1(M^n)$ of the manifold $M^n$ defines a short
exact sequence
\begin{equation}\label{short}
0\to\Z^{n}\to\Gamma\stackrel{p}\to G\to 0,
\end{equation}
where $\Z^n$ is the unique maximal abelian subgroup of finite index in $\Gamma$. The finite group $G$ is the holonomy group of $M^n$.
By the \emph{holonomy representation} of the group $\Gamma$ we shall understand a homomorphism $\varphi_{\Gamma}\colon G\to \GL_n(\Z)$ given
by the formula
\[
\forall_{g\in G, z\in\Z^n} \varphi_{\Gamma}(g)(z) = \gamma z \gamma^{-1},
\]
where $\gamma \in \Gamma$ is any element such that $p(\gamma) = g$.
Since $\Z^n$ is a maximal abelian subgroup, $\varphi_{\Gamma}$ is a faithful representation.
Any short exact sequence (\ref{short}) corresponds to an element $\alpha_{\Gamma}\in H^2(G,\Z^n)$, where $G$ acts on $\Z^n$ via 
$\varphi_{\Gamma}.$
Moreover, an element $\alpha\in H^2(G,\Z^n)$ defines a flat manifold if and only if $\res^{G}_{C}\alpha\neq 0$ for any cyclic subgroup $C\subset G$.
Such an element $\alpha$ is called special, see \cite[p. 29, 36]{Sz12}.  
In this note we are interested in the question of the existence of flat manifolds for a special kind of holonomy representation.
We shall need some definitions.

Let $h\colon G\rightarrow \GL_m({\mathbb R})$ be a faithful representation.
Denote by $W$ a left ${\mathbb R}[G]$ module which is defined by the representation $h.$
We have a direct sum 
$W = V_1\oplus V_2\oplus \dots \oplus V_k$ 
of ${\mathbb R}[G]$-irreducible
modules $V_i$ for $i=1,2,...,k$.
It follows from \cite[Theorem (73.9)]{CR} that there are three kinds of summands: absolutely irreducible,
"complex" and "quaternionic" ones.
An irreducible ${\mathbb R}[G]$ module $V$ is "complex" if $\End_{{\mathbb R}[G]}(V) = {\mathbb C}$ (the complex numbers) and it is "quaternionic" if
$\End_{{\mathbb R}[G]}(V) = {\mathbb H}$ (the quaternionic field). If $\End_{{\mathbb R}[G]}(V) = {\mathbb R}$ then $V$ is absolutely irreducible.
Hence, we have a decomposition
\begin{equation}\label{decomposition} 
W = W_{{\mathbb R}}\oplus W_{{\mathbb C}}\oplus W_{{\mathbb H}} 
\end{equation}
into isotypic components of real, complex and quaternionic type.

There exists a similar characterization of ${\mathbb C}$-irreducible representations.
Let $\chi$ be character of $G$ and 
\[\nu_{2}(\chi) = \frac{1}{|G|}\sum_{g\in G}\chi(g^2).\]
By definition $\nu_2(\chi)$ is the Frobenius-Schur indicator. 

Let $V$ be as above an ${\mathbb R}[G]$-irreducible representation. Let $U$ be a simple submodule of the ${\mathbb C}[G]$-module ${\mathbb C} \otimes_{\mathbb R} V$. Then ${\mathbb C} \otimes_{\mathbb R} V = U$, which means that $V$ is absolutely irreducible or ${\mathbb C} \otimes_{\mathbb R} V = U \oplus \cconj{U}$, where $\cconj{U}$ denotes the conjugate module \cite[pages 106-109]{S77}. In the latter case, if $U\not\simeq\cconj{U}$ then $\End_{{\mathbb R}[G]}(V) = {\mathbb C}$ and if $U\simeq\cconj{U}$ then $\End_{{\mathbb R}[G]}(V) = {\mathbb H}$. Let $\chi_U$ denote the character of $U$. From \cite[p. 58]{Is76} we have
\[
\nu_2(\chi_{U}) = \left\{\begin{array}{rll}
1 & \mbox{, if $V$ is absolutely irreducible},\\
0 & \mbox{, if $V$ is of complex type},\\
-1 & \mbox{, if $V$ is of quaternionic type}.
\end{array}
\right.
\]

So knowing the irreducible representation on complex
spaces and their Frobenius-Schur indicators allows one to classify irreducible representations on real space.
  
In this paper we are interested in the question of the existence of flat manifolds whose ${\mathbb R}$-irreducible components of 
the holonomy representation
are either absolutely irreducible, or only of complex type or only of quaternionic type. Let us denote the
above classes of flat manifolds by ${\rm \R T}$, ${\rm \C T}$ and ${\rm \h T}$ respectively.

Equivalently, we are looking for a faithful $\Z G$ module $W$ such that $H^2(G,W)$ contains a special element and  ${\mathbb R} \otimes_\Z W = ({\mathbb R} \otimes_\Z W)_{F}$ for $F = {\mathbb R}, {\mathbb C}, {\mathbb H}$ respectively in the decomposition
(\ref{decomposition}).

In the first two cases (i.e. $F ={\mathbb R}$ and $F = {\mathbb C}$) such examples are well known and we shall present them in the next section. For example
any Hantzsche-Wendt manifold has holonomy representation whose ${\mathbb R}$-irreducible components are all absolutely irreducible.
Similarly, any flat manifold with holonomy group $(\Z_{3})^k$ and the first Betti number equal to zero and dimension
$2(2+k)$ (\cite[Th. 4.2]{Sz12}) has holonomy representation with ${\mathbb R}$-irreducible components only of complex type.

However in the case $F = {\mathbb H}$ such examples were unknown.
One of our main results (Theorem \ref{main}) is a construction of a flat manifold in the class ${\rm \h T}$. In the final section we shall
give a list of finite groups which are not holonomy groups of flat manifolds in the class ${\rm \h T}$.

An even dimensional flat manifold $M^{2n}$ is K\"ahler if and only if
all ${\mathbb R}$-irreducible summands of the holonomy representation $\varphi_{\pi_{1}(M^{2n})}$, 
which are also ${\mathbb C}$-irreducible occur with even multiplicity, see \cite[Prop. 7.2]{Sz12}. By the above definition
any flat manifold in the class ${\rm \C T}$ is K\"ahler. On the other side,
any K\"ahler flat manifold with non-zero first Betti number  does not belong to the class ${\rm \C T}.$

\begin{center}
\begin{tikzpicture}[rounded corners,line width=1]
\draw (-1,-.4) rectangle (3,.4);
\node at (-.5,.1) {${\mathbb C}$T};

\draw (-1.5,-.6) rectangle (3.5,1);
\node at (0.5,.7) {K\"ahler flat manifolds};
\end{tikzpicture}
\end{center}

Similarly, any flat manifold $M^{4n}$ of dimension $4n$ of the class ${\rm \h T}$
is hyperk\"ahler, i.e. $\varphi_{\pi_{1}(M^{4n})}(G)\subset \Sp(4n)$, see \cite{Hitchin}.
In fact, from (cf. \cite[p.58]{Is76}) it is well known that any irreducible complex representation $V$ with
$\nu_2(V) = -1$, has even dimension and carries a $G$-invariant, non-degenerate symplectic form.
However, any hyperk\"ahler flat manifold with the first Betti number different from zero is not ${\rm \h T}$. 
See also \cite{W82} and \cite{DM01}.

\begin{center}
\begin{tikzpicture}[rounded corners,line width=1]
\draw (-1,-.4) rectangle (3,.4);
\node at (-.5,.1) {${\rm \h T}$};

\draw (-1.5,-.6) rectangle (3.5,1);
\node at (1,.7) {hyperk\"ahler flat manifolds};
\end{tikzpicture}
\end{center}

For any finite group $G$, it has been observed (see \cite{John} and \cite[Prop. 3.2]{DM01}) that using "the double" 
construction it is possible to define  K\"ahler and hyperk\"ahler
manifolds with holonomy group $G$. 

\section{Real and complex case}
To begin with, we shall present a class of examples of flat manifolds 
whose \emph{holonomy representations} have a decomposition (\ref{decomposition}) with only absolutely irreducible components.
Any flat manifold with holonomy group $(\Z_2)^k, k\geq 2$
has such property. The simplest ones are the Klein Bottle
and the $3$-dimensional Hantzsche-Wendt manifold whose fundamental group is defined as follows, see \cite{Sz12}: 
\[
\Gamma = \langle\gamma_1 = (A, (1/2, 1/2, 0)), 
\gamma_2 = (B, (0, 1/2, 1/2))\rangle \subset \SO(3)\ltimes{\mathbb R}^3,
\]
where
$A =\left[
\begin{array}{lll}
1 & 0 & 0\\
0 &-1 & 0\\
0 & 0 &-1
\end{array}
\right] \text{ and } B =\left[
\begin{array}{lll}
-1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 &-1
\end{array}
\right ].$

Let us present an example of a flat manifold $M_{k}$  where all components of the holonomy representation
are of "complex" type (have Frobenius-Schur indicator equal to $0$).
The manifold $M_k$ has holonomy group $(\Z_3)^k$, first Betti number zero and dimension equal to $2(k+2), k\geq 2$.


For example, to define the manifold $M_2$ we shall need matrices:
\[
D = \left[
\begin{array}{llll}
I_2 & 0 & 0 & 0\\
0 & C & 0 & 0\\
0 & 0 & C & 0\\
0 & 0 & 0 & C
\end{array}
\right], E = \left[
\begin{array}{llll}
C & 0 & 0 & 0\\
0 & I_2 & 0 & 0\\
0 & 0 & C & 0\\
0 & 0 & 0 & C^2
\end{array}
\right],
\text{ where }
C = \left[
\begin{array}{rr}
-1 & -1\\
1 & 0
\end{array}
\right ]
\]
and $I_k$ is the identity $k\times k$ matrix, for any $k \in {\mathbb N}$. Thus $D$ and $E$ are invertible integer matrices of degree $8$.
We define two elements
\[\gamma_{D} = (D,(-2/3,1/3,0,0,-2/3,1/3,-2/3,1/3)),\]
\[\gamma_{E} = (E, (0,0,-2/3,1/3,0,0,0,0))\]
of the group $SO(8)\ltimes{\mathbb R}^8$.
Let $\Gamma_{2} = \langle \gamma_{D},\gamma_{E},(I_8,x) : x\in \Z^8 \rangle$ and $M_2 = {\mathbb R}^8/\Gamma_2$. 
We have (see \cite{Sz12}) that $\pi_1(M_2) = \Gamma_2$.


\section{A flat manifold in the class ${\rm \h T}$}

In this section we present one of the main results of our article.
\begin{thm}\label{main}
There exists a flat manifold with holonomy representation of quaternionic type. 
\end{thm}
\begin{proof}

Let the presentation of a group $G$ be given as follows. $G$ fits into the central extension
\[
\ses{1}{C_2^2}{}{G}{}{C_2^4}{1},
\]
it is generated by the elements $a,b,c,d$ of order $4$, such that $a^2=c^2$ and $b^2=d^2$ generate the center $Z(G)$ and we have commutator relations:
\[
\begin{array}{lll}
[a,b] = a^2 & [a,c] = a^2b^2& [a,d] = b^2\\
            & [b,c] = a^2   & [b,d] = a^2b^2\\
            &               & [c,d] = 1
\end{array}
\]
This is an example of a skew group, i.e. a group for which we have
\[
\forall_{\chi \in \Irr(G)} \chi(1) = 1 \vee \nu_2(\chi) = -1,
\]
where $\Irr(G)$ denotes the set of complex irreducible characters of $G$. In the library of small groups of GAP \cite{GAP16} $G$ is labeled as [64,245].

\begin{table}[ht]
\[
\begin{array}{l|rrrr}
       & 1 & a^2 & b^2 & a^2b^2\\ \hline
\chi_1 & 4 &   4 &  -4 &     -4\\
\chi_2 & 4 &  -4 &   4 &     -4\\
\chi_3 & 4 &  -4 &  -4 &      4\\
\end{array}
\]
\caption{List of nonzero values of characters $\chi_1,\chi_2,\chi_3$.}
\label{tab:char}
\end{table}

$G$ has exactly $3$ characters $\chi_1, \chi_2, \chi_3$ with Frobenius-Schur indicator equal to $-1$. They are distinguished by their values on the center only, since on every element of the set $G \setminus Z(G)$ each of them is equal to zero. The non-zero values are presented in Table \ref{tab:char}. Moreover, the characters are conjugate one to another, which means that for every $1 \leq i < j \leq 3$ there exists an automorphism $f_{ij} \in \Aut(G)$ such that $\chi_j = \chi_i \circ f_{ij}$. Let us introduce a notation for two of those automorphisms:
\begin{equation}
f_2 := f_{12}, f_3 := f_{13}.
\end{equation}
From Table \ref{tab:char} and the fact that the center $Z(G)$ is a characteristic subgroup of $G$ we have that
\[
f_2(b^2) = f_3(a^2b^2) = a^2.
\]

Now, for every $f \in \Aut(G)$, let $M^f$ denote the $G$-module twisted by $f$, i.e. $M^f = M$ and the action $\cdot_f$ comes from the original action of $G$ on $M$ as follows:
\[
g \cdot_f m = f(g) m,
\]
where $g \in G, m \in M^f$. Moreover, for every such automorphism $f$ and every subgroup $H$ of $G$ we have Diagram \ref{diag:commutativity}, where $(f_{|H})^*$ is an isomorphism. Hence, if we can find a $G$-lattice $M$ and an element $\alpha \in H^2(G,M)$ such that $\res_{C_1}\alpha \neq 0$, then
\[
\res_{C_i}f_i^*(\alpha)  = (f_{i|C_i})^*\res_{C_1}\alpha \neq 0,
\]
for $i=2,3$, where $C_i = \ker\chi_i$ is the group generated by $a^2,b^2$ and $a^2b^2$, for $i=1,2,3$ respectively. Hence the element
\begin{equation}
\label{eq:special}
\alpha + f_2^*(\alpha) + f_3^*(\alpha) \in H^2(G,M \oplus M^{f_2} \oplus M^{f_3})
\end{equation}
is special, since non-trivial elements of $Z(G)$ are the only elements of $G$ of order $2$.

\begin{fdiagram}
\[
\begin{tikzcd}
{H^{2}(G,M)} \arrow[r,"","f^*"] \arrow[d,"\res_{f(H)}"] 
& {H^2(G,M^f)} \arrow[d,"\res_{H}"] \\
{H^2(f(H),\res_{f(H)}M)} \arrow[r,"(f_{|H})^*"] 
& {H^2(H,\res_{H}M^f)}
\end{tikzcd}
\]
\caption{}
\label{diag:commutativity}
\end{fdiagram}

The construction of the lattice $M$ which works for the group $C_1$ in the above sense is as follows. Let $S_1 = \ind_{C_{1}}^{G} \Z$ 
be induced from the trivial $C_1$-module $\Z$. 
If we denote by $\chi$ the character of the $G$-module $S_1$, then direct calculations show that
\[
\langle \chi, \chi_1 \rangle = 4 \text{ and } \langle \chi, \chi_2 \rangle = \langle \chi, \chi_3 \rangle = 0.
\]
This means that there is a homogeneous component of the ${\mathbb C} G$-module ${\mathbb C}\otimes_\Z S_1$ of dimension $16$, which contains  quaternionic summands only.
Moreover, since $\chi_1$ is not the character of a real module, by \cite[Corollary 10.14]{Is76} we get that $2\chi_1$ is the character of some ${\mathbb Q} G$-module. If $\rho_1 \colon G \to \GL(32,\Z)$ is a permutation representation corresponding to the $G$-module $S_1$ (see below), then the procedure of finding homogeneous components in representations, presented in \cite[Section 2.6]{S77}, allows us to find a basis of the submodule of $S_1$ with character $4\chi_1$. To be more precise, 16 columns of the matrix
\[
B = \frac{2\chi_1(1)}{|G|} \sum_{g \in G} \cconj{2\chi_1(g)}\rho_1(g),
\]
up to multiplication by a rational constant, form a $\Z$-basis of a pure $\Z$-submodule of $S_1$, which admits a $G$ action. If we denote this module by $M$ then $M \cong \Z^{16}$ and computer calculations show that $H^2(G,M) = \langle \alpha \rangle \simeq \Z_2 $. Moreover,
$\res^G_{C_1} \alpha \neq 0$
and the formula \eqref{eq:special} defines a special element which corresponds to the torsion-free extension
\[
\ses{0}{M \oplus M^{f_2} \oplus M^{f_3}}{}{\Gamma}{}{G}{1}.
\]
Since the character of the $G$-module $M \oplus M^{f_2} \oplus M^{f_3}$ is equal to $4(\chi_1+\chi_2+\chi_3)$, it must be faithful, cf. Table \ref{tab:char}, and $\Gamma$ is a Bieberbach group.

The representation $\rho_1$ may be defined as follows:
\[
\setlength{\arraycolsep}{.1em}
\begin{array}{lll}
a & \mapsto & P_{(1\,13)(2\,14)(3\,9)(4\,10)(5\,21)(6\,22)(7\,11)(8\,12)(15\,24)(16\,23)(17\,26)(18\,25)(19\,27)(20\,28)(29\,31)(30\,32)},\\
b & \mapsto & P_{(1\,3\,2\,4)(5\,18\,6\,17)(7\,15\,8\,16)(9\,14\,10\,13)(11\,24\,12\,23)(19\,30\,20\,29)(21\,25\,22\,26)(27\,32\,28\,31)},\\
c & \mapsto & P_{(1\,5)(2\,6)(3\,18)(4\,17)(7\,20)(8\,19)(9\,26)(10\,25)(11\,27)(12\,28)(13\,22)(14\,21)(15\,29)(16\,30)(23\,31)(24\,32)},\\
d & \mapsto & P_{(1\,8\,2\,7)(3\,15\,4\,16)(5\,19\,6\,20)(9\,23\,10\,24)(11\,14\,12\,13)(17\,30\,18\,29)(21\,28\,22\,27)(25\,32\,26\,31)},\\
\end{array}
\]
where $P_s$ denotes the permutation matrix of degree $32$ of an element $s$ of $S_{32}$, the symmetric group on $32$ letters.
 With this specific representation a basis of the module $M$ consists of the elements
\[
e_1-e_2,e_3-e_4,\ldots,e_{31}-e_{32},
\]
where $e_i$ denotes the $i$-th column of the identity matrix of degree $32$. In this case the structure of the $G$-module $M=\Z^{16}$ is given by the integral representation $\rho_M \colon G \to \GL(16,\Z)$ defined as follows
\[
\setlength{\arraycolsep}{.1em}
\def1.3{1.3}
\begin{array}{lll}
a & \mapsto & P^{8,9,12,13}_{( 1\, 7)( 2\, 5)( 3\,11)( 4\, 6)( 8\,12)( 9\,13)(10\,14)(15\,16)},\\
b & \mapsto & P^{2,3,5,6,8,10,13,14}_{( 1\, 2)( 3\, 9)( 4\, 8)( 5\, 7)( 6\,12)(10\,15)(11\,13)(14\,16)},\\
c & \mapsto & P^{2,4,5,7,9,10,11,13}_{( 1\, 3)( 2\, 9)( 4\,10)( 5\,13)( 6\,14)( 7\,11)( 8\,15)(12\,16)},\\
d & \mapsto & P^{1,6,8,9,10,11,12,13}_{( 1\, 4)( 2\, 8)( 3\,10)( 5\,12)( 6\, 7)( 9\,15)(11\,14)(13\,16)},\\
\end{array}
\]
where the superscripts define rows of a permutation matrix which should be multiplied by $-1$.
The cocycle $\hat{\alpha}$ which defines non-zero cohomology class $\alpha$ in the group $H^1(G,{\mathbb Q}^{16}/\Z^{16}) \cong H^2(G,\Z^{16})$ is given by:
\[
\setlength{\arraycolsep}{.1em}
\renewcommand{1.3}{1.3}
\begin{array}{lll}
a & \mapsto & \left( 0, \tfrac{1}{2}, 0, 0, 0, \tfrac{1}{2}, \tfrac{1}{2}, \tfrac{1}{2}, \tfrac{1}{2}, 0, \tfrac{1}{2}, 0, 0, \tfrac{1}{2}, \tfrac{1}{2}, 0 \right) + \Z^{16},\\
b & \mapsto & \left( 0, 0, 0, \tfrac{1}{2}, 0, \tfrac{1}{2}, 0, \tfrac{1}{2}, 0, \tfrac{1}{2}, 0, \tfrac{1}{2}, 0, \tfrac{1}{2}, \tfrac{1}{2}, \tfrac{1}{2} \right) + \Z^{16},\\
c & \mapsto & \left( 0, \tfrac{1}{2}, \tfrac{1}{2}, 0, 0, \tfrac{1}{2}, \tfrac{1}{2}, \tfrac{1}{2}, 0, \tfrac{1}{2}, 0, 0, \tfrac{1}{2}, 0, 0, \tfrac{1}{2} \right) + \Z^{16},\\
d & \mapsto & \left( 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 \right) + \Z^{16}.\\
\end{array}
\]
One gets that $\rho_M(a^2) = I_{16}$, the identity matrix of degree $16$, and that
\[
\hat{\alpha}(a^2) = \left(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2}\right) + \Z^{16},
\]
hence $\res_{C_1} \alpha \neq 0$.

The calculations are presented in more detail on the web page \cite{Lu19}.
\end{proof}

\section{Holonomy groups of flat manifolds from the class ${\rm \h T}$}

Let $\ensuremath{\mathcal HT}$ denote the class of holonomy groups of manifolds from the class ${\rm \h T}$. This section is motivated by the search for further examples in  \ensuremath{\mathcal HT}. We will present some necessary conditions for a group $G$ to belong to \ensuremath{\mathcal HT}. Then we will exclude certain simple groups from the class \ensuremath{\mathcal HT}. Finally, we comment on $2$-groups.

\subsection{Some necessary conditions}

\begin{dfn}[{\cite[Definition 4.2]{Sz12}}]
We say that a finite group is primitive if it is a holonomy group of a flat manifold with first Betti number equal to zero.
\end{dfn}

Since the holonomy representation of any flat manifold with non-zero first Betti is of non-quaternionic type,
a group in $\ensuremath{\mathcal HT}$ is primitive.
Moreover by \cite[Theorem 4.1]{Sz12} a finite group is primitive if and only if
no non-trivial cyclic Sylow $p$-subgroup of $G$ has a normal complement.

\begin{exl}
By \cite[Proposition 4.2]{Sz12} the following groups are non-primitive:
\begin{enumerate}[label=(\roman*)]
\item metacyclic groups $\Z_n\rtimes\Z_m, (m,n) = 1$;
\item the Borel subgroup $\Z_q\rtimes\Z_{q-1}\subset \SL_2(q)$, where $q$ is power of a prime $p$;
\item the group $\SL_2(3)$ (it has a normal $3$-complement).
\end{enumerate}
\end{exl}

\begin{prop}
\label{prop:htproperties}
Let $G \in \ensuremath{\mathcal HT}$. Then all of the following hold:
\begin{enumerate}[label=\arabic*.,ref=\arabic*]
\item \label{ht:even} 
	$G$ has even order.
\item \label{ht:non-abelian}
	$G$ is non-abelian.
\item \label{ht:center} 
	$Z(G)$ is an elementary abelian $2$-group.
\item \label{ht:primitive}
	If $p$ is a prime such that $p \mid G$ and a Sylow $p$-subgroup of $G$ is cyclic, then $G$ does not have a normal $p$-complement.
\item \label{ht:primitive2}
	The Sylow $2$-subgroup of $G$ is non-cyclic.
\item \label{ht:squareroots}
	Put $I(G):=|\{ g \in G \mid g^2=1 \}|$. Then \[I(G) < \sum_{\chi\in \Irr(G)}\chi(1) \text{ and } I(G) \leq |G|/2.\]
\item \label{ht:nonhomogeneous}
	$|\{ \chi \in \Irr(G) \mid \nu_2(\chi) = -1 \}| \geq 2$.
\item \label{ht:centerrep}
	Suppose $z \in Z(G)$ with $|z|=2$. Then there exist $\chi,\psi \in \Irr(G)$ with $\nu_2(\chi) = \nu_2(\psi) = -1$ and $\chi(z) = \chi(1), \psi(z)=-\psi(1)$. 
\end{enumerate}
\end{prop}

\begin{proof}
\begin{description}[before={\renewcommand\makelabel[1]{\bfseries Ad.\,##1:}}]
\item[\ref{ht:even}] By assumption the map  $l:G\to G, g\mapsto g^2$ is a bijection. Hence $\nu_2(\chi)\neq -1$ for any $\chi\in \Irr(G)$.
In fact $\nu_2(\chi) = \frac{1}{|G|}\Sigma_{g\in G}\chi(g^2) = \frac{1}{|G|}\Sigma_{g\in G}\chi(g) = \langle\chi, 1\rangle.$
Here $\langle,\rangle$ denotes the scalar product of characters, and $1$ the trivial character.

\item[\ref{ht:non-abelian}] The Frobenius-Schur indicator of every character of every abelian group is equal to $0$ or $1$.

\item[\ref{ht:center}] Suppose that $Z(G)$ contains an element $z$ with $|z|=4$ or $|z|=p$ for some odd prime $p$. Let $\psi$ be a faithful complex character of $G$. Then there is some irreducible constituent $\chi$ of $\psi$ such that $z^2$ is not in the kernel of $\chi$. As $\res_{Z(G)}\chi = \chi(1)\lambda$ for some $\lambda \in \Irr(Z(G))$, we obtain $\chi(z) \in {\mathbb C} \setminus {\mathbb R}$. Hence $\chi \neq \cconj{\chi}$ and so $G$ cannot have a faithful quaternionic representation.

\item[\ref{ht:primitive}] This is in fact a definition of a primitive group (see above).

\item[\ref{ht:primitive2}] This follows from \ref{ht:primitive} and \cite[Theorem 1]{T}.

\item[\ref{ht:squareroots}] By \cite[Lemma 2 on p. 252]{Wall} and \cite[p. 155]{Gruyter}, if any of given conditions does not hold then for every $\chi \in \Irr(G)$ we have $\nu_2(\chi)=1$.

\item[\ref{ht:nonhomogeneous}] The fact that the holonomy representation of a flat manifold with non-trivial holonomy group is non-homogeneous is the main result of \cite{Lu18}.

\item[\ref{ht:centerrep}] By assumption there exists a faithful $G$ lattice $L$ and a special element $\alpha\in H^2(G,L)$.
If $\chi(z) = \chi(1)$ for all $\chi \in \Irr(G)$, then $z$ is in the kernel of ${\mathbb C}\otimes_{\Z} L$, i.e. $z$ acts trivially on $L$,
and we have a contradiction. Suppose that $\chi(z) = -\chi(1)$ for all  $\chi \in \Irr(G)$. Then $z$ acts as $-1$ on  ${\mathbb C}\otimes_{\Z} L$,
hence $z$ acts as $-1$ on $L$. But then $H^2(G,L)$ cannot contain a special element, since $H^2(\langle z\rangle, \res_{\langle z\rangle}L) = 0$.
\end{description}
\end{proof}

\subsection{Some simple groups and their covering groups}

\begin{lem}\label{Remark2}
Let $G$ be a finite group and $p$ a prime number. 
Let $O_{p'}(G)$ denote the maximal normal subgroup of order prime to $p$.
Then $O_{p'}(G)$ is contained in the kernel of every $\chi\in\Irr(G)$
in the principal $p$-block. 
\end{lem}
\begin{proof}
Follows from \cite[Lemma (4.12)(ii)]{Feit}.
\end{proof}

\begin{lem}
\label{lem:principalpblock}
Let $G$ be a finite group and $L$ be a $G$-lattice. If $H^2(G,L)$ contains a special element, then for every prime divisor $p$ of $|G|$ there exists a constituent of ${\mathbb C} \otimes_\Z L$ which lies in the principal $p$-block of $G$.
\end{lem}
\begin{proof}
Follows from proofs of Lemmas 2.1 and 2.2.a of \cite{HS}.
\end{proof}

\begin{prop}
The following groups do not belong to \ensuremath{\mathcal HT}:
\begin{enumerate}[label=(\roman*),ref=\roman*]
\item \label{ht:s:sl}  $\SL_2({\mathbb F}_q), \PSL_2({\mathbb F}_q)$, where $q$ is a power of a prime;
\item \label{ht:s:sn}  $A_n, 2.A_n, S_n, 2.S_{n}, n \geq 5$;
\item \label{ht:s:pce} a perfect central extension of a sporadic simple group.
\end{enumerate}
\end{prop}

\begin{proof}
\begin{description}
\item[Ad \ref{ht:s:sl}] These groups are excluded by Lemma \ref{Remark2}, as all characters $\chi$ with $\nu_2(\chi) = -1$ satisfy $\chi(z) = -\chi(1)$
for $z\in Z(G), |z| = 2$ and hence $O_{p'}(G) \not \subset \ker \chi$ for any prime $p \neq 2$.

\item[Ad. \ref{ht:s:sn}] Let $n \geq 5$. Then $S_n$ and $A_n$ do not have irreducible representations of quaternionic type.
For example all ${\mathbb C}$-irreducible representations of $S_n$ are afforded by real representations, \cite[page 56, Corollary 4.15]{Is76}.
Similar properties have irreducible ${\mathbb C}$-representations of a dihedral group.
For the alternating group we shall use properties of complex irreducible representations of symmetric groups and Clifford theory, see \cite[page 79]{Is76}. Let $W$ be an ${\mathbb R} S_n$-irreducible module. We shall consider two cases.

\begin{description}
\item[Case 1] $\Res^{S_n}_{A_n}(W)\simeq V_1\oplus V_2$ where the $V_i$ are irreducible. 
That means $({\mathbb C}\otimes V_1)\oplus ({\mathbb C}\otimes V_2)\simeq \Res^{S_n}_{A_{n}}({\mathbb C}\otimes W).$
By Clifford's Theorem the ${\mathbb C}\otimes V_i, i = 1,2$ are
irreducible i.e. the $V_i$ are absolutely irreducible

\item[Case 2] $V:= \Res^{S_n}_{A_n}(W)$ is an irreducible ${\mathbb R} G$-module.Then:

\begin{enumerate}[label=(\roman*)]
\item ${\mathbb C}\otimes V$ is irreducible. This means that $V$ is absolutely irreducible.
\item ${\mathbb C}\otimes V = U \oplus \cconj{U}$. Since the index of $A_n$ in $S_n$ is equal $2$, then by \cite[Corollary 6.19]{Is76} $U \not\simeq \cconj{U}$.
By definition, we can conclude that $V$ is of "complex type". 
\end{enumerate}
\end{description}

For the groups $2.A_n$ and $2.S_n$ we argue as in the proof of \ref{ht:s:sl}.

\item[Ad. \ref{ht:s:pce}] Assume first that $G$ is simple. Using the Atlas \cite{ATL} we find that $\nu_2(\chi) \in \{ 0,1 \}$ for all $\chi \in \Irr(G)$ except if $G = \ensuremath{\text{McL}}$
and $\chi \in \{ \chi_{11}, \chi_{13} \}$. A calculation in GAP \cite{GAP16} shows that $\chi_{11}$ and $\chi_{13}$ do not belong to the principal $11$-block of $G$. Hence $\ensuremath{\text{McL}} \notin \ensuremath{\mathcal HT}$ by Lemma \ref{lem:principalpblock}. If $G$ is non-simple, by Proposition \ref{prop:htproperties}.\ref{ht:center} we get that $G = 2.S$ for a simple group $S \ncong \ensuremath{\text{McL}}$ (Schur multiplier of $\ensuremath{\text{McL}}$ is of order $3$). Then $G \notin \ensuremath{\mathcal HT}$ by Proposition \ref{prop:htproperties}.\ref{ht:centerrep}. This concludes the proof.
\end{description}
\end{proof}

\subsection{$2$-groups}

The following proposition shows the importance of the family of $2$-groups in our investigation.

\begin{prop}[{\cite[Theorem 1]{Mann} and \cite[Satz]{WW}}]
If a finite group $G$ is non-abelian and all its non-linear characters have Frobenius-Schur indicator equal to $-1$ then $G$ is a $2$-group.
\end{prop}


\begin{proof}
We follow \cite[p.354]{Mann}.
Let $G$ be a minimal counter example to the theorem.
First, since all non-linear irreducible characters have even degree, \cite[Theorem 1]{T} shows that $G$ has a normal $2$-complement $K$. Let $T$ be a Sylow $2$-subgroup of $G$. Then $T\simeq G/K$, therefore $G$ and $T$
have the same number of linear characters of order $2$ and all irreducible characters
of $T$ can be considered as characters of $G$. The Frobenius-Schur formula
\begin{equation}\label{invo}
I(G) = \Sigma_{\chi\in \Irr(G)}\nu_{2}(\chi)\chi(1),
\end{equation}
shows that $T$ has at least as many involutions as $G$ has. This is possible only
if $G$ and $T$ have the same number of involutions, and then (\ref{invo})
implies that all non-linear irreducible characters of $G$ are characters of $T,$
which means that $K$ is contained in the kernels of all these characters.
But then $K = 1$ and $G$ is a $2$-group.
\end{proof}

\begin{exl}
Let $E$ be an extraspecial $2$-group of order $2^{2m+1}, m \geq 1$. Then $E/E'$ is elementary abelian and there is a unique irreducible character of degree $2^m$ (see \cite[Problem 2.13]{Is76}).

There are two non-isomorphic such groups and $\nu_2(\chi)=-1$ in one of these copies (if $E$ is a central product of the quaternion group of order $8$ and several dihedral groups of order $8$).

In particular, $E$ does not belong to ${\mathcal HT}$ by the main result of \cite{Lu18}.
\end{exl}

\begin{rem}
Proposition \ref{prop:htproperties} may be used to show that the group $[64,245]$, defined in the proof of Theorem \ref{main}, is the only group of order less than or equal to $64$, which is the holonomy group of a flat manifold from the class ${\rm \h T}$.
\end{rem}

