Frobenius-Schur indicators of characters in blocks with cyclic defect

TL;DR

This paper studies Frobenius-Schur indicators of characters in blocks with cyclic defect groups, showing uniform indicators for exceptional characters and relating indicators of non-exceptional characters to group elements.

Contribution

It establishes that all exceptional characters in such blocks share the same Frobenius-Schur indicator and provides a method to compute it, also relating indicators of non-exceptional characters to group structure.

Findings

Exceptional characters have identical Frobenius-Schur indicators.

The common indicator can be computed via the canonical character.

The number of characters with indicator -1 exceeds or equals the count of certain real elements.

Abstract

Let $p$ be an odd prime and let $B$ be a $p$-block of a finite group which has cyclic defect groups. We show that all exceptional characters in $B$ have the same Frobenius-Schur indicators. Moreover the common indicator can be computed, using the canonical character of $B$. We also investigate the Frobenius-Schur indicators of the non-exceptional characters in $B$. For a finite group which has cyclic Sylow $p$-subgroups, we show that the number of irreducible characters with Frobenius-Schur indicator $-1$ is greater than or equal to the number of conjugacy classes of weakly real $p$-elements in $G$.

Equations

$$ϵ(\chi ):=\frac{1}{|G|}\sum _{g\in G}\chi (g^2).$$

$$\begin{array}[]{lrrrrrrrrrrrrrrr}&2&4&4&3&3&4&4&2&3&3&3&3&2&2&2\\ &5&1&1&.&1&.&.&1&.&.&.&.&1&1&1\\ \\ &&1a&2a&2b&4a&4b&4c&5a&8a&8b&8c&8d&10a&20a&20b\\ \\ X.9&&2&-2&.&.&2i&-2i&2&.&.&.&.&-2&.&.\\ X.10&&2&-2&.&.&-2i&2i&2&.&.&.&.&-2&.&.\\ X.13&&4&-4&.&.&.&.&-1&.&.&.&.&1&\sqrt{-5}&-\sqrt{-5}\\ X.14&&4&-4&.&.&.&.&-1&.&.&.&.&1&-\sqrt{-5}&\sqrt{-5}\\ \end{array}$$

$$\begin{array}[]{lrrrrrrrrrr}&2&2&2&1&2&2&.&1&.&.\\ &3&1&1&1&.&.&1&1&1&1\\ &5&1&.&1&.&.&1&.&1&1\\ \\ &&1a&2a&3a&4a&4b&5a&6a&15a&15b\\ \\ X.5&&2&-2&-1&.&.&2&1&-1&-1\\ X.6&&2&2&-1&.&.&2&-1&-1&-1\\ X.8&&4&.&-2&.&.&-1&.&\alpha&\overline{\alpha}\\ X.9&&4&.&-2&.&.&-1&.&\overline{\alpha}&\alpha\\ \end{array}$$

$$\begin{array}[]{lrrrrrrrrrrrrrrrrrrr}&2&3&3&2&3&3&1&2&1&2&2&1&1&1&1\\ &3&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\ &5&1&1&1&.&.&1&1&1&.&.&1&1&1&1\\ \\ &&1a&2a&3a&4a&4b&5a&6a&10a&12a&12b&15a&15b&30a&30b\\ \\ X.11&&2&-2&-1&2i&-2i&2&1&-2&-i&i&-1&-1&1&1\\ X.12&&2&-2&-1&-2i&2i&2&1&-2&i&-i&-1&-1&1&1\\ X.15&&4&-4&-2&.&.&-1&2&1&.&.&\alpha&\overline{\alpha}&-\alpha&-\overline{\alpha}\\ X.16&&4&-4&-2&.&.&-1&2&1&.&.&\overline{\alpha}&\alpha&-\overline{\alpha}&-\alpha\\ \end{array}$$

$$\begin{array}[]{lrrrrrrrrr}&2&2&2&2&2&1&1&1&1\\ &5&1&1&.&.&1&1&1&1\\ \\ &&1a&2a&4a&4b&5a&5b&10a&10b\\ \\ X.3&&1&-1&i&-i&1&1&-1&-1\\ X.4&&1&-1&-i&i&1&1&-1&-1\\ X.5&&2&-2&.&.&\beta&{*\beta}&-\beta&-{*\beta}\\ X.6&&2&-2&.&.&{*\beta}&\beta&-{*\beta}&-\beta\\ \end{array}$$

$$\chi (xy)=\sum _{\phi }{d}_{\chi ,\phi }^{(x)}\phi (y),$$

$$\sum _{\chi }ϵ(\chi )d_{\chi ,\phi }^{(x)}=\sum _{\psi }ϵ(\psi )d_{\psi ,\phi }^{(x)},$$

$${ϵ}_{T/G}(\chi ):=\frac{1}{|G|}\sum _{t\in T\backslash G}\chi (t^2).$$

$${ϵ}_{G/G_{\varphi }}(\varphi {\uparrow }^{G_{\varphi }})=\frac{1}{|G_{\varphi }|}\sum _{g\in G\backslash G_{\varphi }}\varphi {\uparrow }^{G_{\varphi }}(g^2)=0.$$

$$\sum _{\chi \in \mathrm{Irr}(G)}⟨\varphi {\uparrow }^G,\chi ⟩ϵ(\chi )=0.$$

$$\sum_{x\in X\backslash W}\lambda(x)=\left\{\begin{array}[]{rl}|X|-|W|,&\quad\mbox{if\/ $X\subseteq\operatorname{ker}(\lambda)$.}\\ -|W|,&\quad\mbox{if\/ $W\subseteq\operatorname{ker}(\lambda)$ but $X\not\subseteq\operatorname{ker}(\lambda)$.}\\ 0,&\quad\mbox{if\/ $W\not\subseteq\operatorname{ker}(\lambda)$.}\\ \end{array}\right.$$

$$(1,B)\subset (D_{a-1},b_{a-1})\subset \cdots \subset (D_1,b_1)\subset (D,b_0).$$

$$(1,{\chi }_1),\dots ,(1,{\chi }_e),(x_i^{{\sigma }_i},{\phi }_i^{n_i}),i=0,\dots ,a-1.$$

$$\mathrm{Irr}(B)=\{X_1,\dots ,X_e\}\bigcup \{X_{\lambda }\mid \lambda \in \Lambda \}.$$

$$\mathrm{Irr}(b_i)=\{X_{i,\lambda }^{\prime }\mid \lambda \in \mathrm{Irr}(D)\}.$$

$${ϵ}_u:=ϵ(X_{\lambda }).$$

$$\nu_{i}:=\epsilon(X_{i,1}^{\prime}),\quad\mbox{for $i=0,\dots,a-1$.}$$

$$\begin{array}[]{lcllcl}d_{X_{\lambda},\varphi_{i}^{\rho}}^{(x)}&=&\varepsilon_{0}\gamma_{i}\sum\limits_{\tau\in EC_{i}/C_{i}}\lambda({{}^{\rho\tau}}x),&\quad d_{X_{j},\varphi_{i}^{\rho}}^{(x)}&=&\varepsilon_{j}\gamma_{i},\quad\mbox{for $j=1,\dots,e$}\\ d_{X_{i,\lambda}^{\prime},\varphi_{i}^{\rho}}^{(x)}&=&\varepsilon_{0}^{\prime}\gamma_{i}\lambda({{}^{\rho}}x),&\quad d_{X_{i,1}^{\prime},\varphi_{i}^{\rho}}^{(x)}&=&1\end{array}$$

$$\sum _{j=1}^eϵ(X_j){\epsilon }_j{\gamma }_i+\sum _{\lambda \in \Lambda }ϵ(X_{\lambda }){\epsilon }_0{\gamma }_i\sum _{\tau \in E{C}_i/C_i}\lambda ({}^{\rho \tau }x)={\nu }_i.$$

$$\sum _{u=1}^a{ϵ}_u\sum _{\lambda \in {\Lambda }_u}\sum _{\tau \in E{C}_i}\lambda ({}^{\rho \tau }x)={\epsilon }_0{\gamma }_i{\nu }_i-\sigma ,$$

$$\sum _{u=1}^a{ϵ}_u\sum _{\lambda \in {\Lambda }_u}\sum _{x\in D_i-D_{i+1}}\lambda (x)=\left(\frac{p^{a-i}-p^{a-i-1}}{e}\right)\left({\epsilon }_0{\gamma }_i{\nu }_i-\sigma \right).$$

$$(p^{a-i}-p^{a-i-1})\sum _{u=1}^i\frac{p^u-p^{u-1}}{e}{ϵ}_u-p^{a-i-1}\frac{p^{i+1}-p^i}{e}{ϵ}_{i+1}=\frac{p^{a-i}-p^{a-i-1}}{e}({\epsilon }_0{\gamma }_i{\nu }_i-\sigma ).$$

$$\sum _{u=1}^i(p^u-p^{u-1}){ϵ}_u-p^i{ϵ}_{i+1}={\epsilon }_0{\gamma }_i{\nu }_i-\sigma .$$

$$-{ϵ}_1$$

$$(p-1){ϵ}_1$$

$$(p-1){ϵ}_1+(p^2-p){ϵ}_2$$

$$(p-1){ϵ}_1+(p^2-p){ϵ}_2+(p^3-p^2){ϵ}_3$$

$$⋮$$

$$(p-1){ϵ}_1+(p^2-p){ϵ}_2+\cdots +(p^{a-1}-p^{a-2}){ϵ}_{a-1}$$

$$p({ϵ}_1-{ϵ}_2)={\epsilon }_0({\gamma }_1{\nu }_1-{\gamma }_0{\nu }_0).$$

$$p^2({ϵ}_1-{ϵ}_3)={\epsilon }_0({\gamma }_2{\nu }_2-{\gamma }_0{\nu }_0).$$

$${ϵ}_1={ϵ}_2=\cdots ={ϵ}_a,{\gamma }_0{\nu }_0={\gamma }_1{\nu }_1=\cdots ={\gamma }_{a-1}{\nu }_{a-1}.$$

$$\epsilon(X_{i})=\left\{\begin{array}[]{ll}\epsilon_{-},&\quad\mbox{for $i=-\ell,\dots,-1$.}\\ \epsilon_{0},&\quad\mbox{for $i=0$.}\\ \epsilon_{+},&\quad\mbox{for $i=1,\dots,r$.}\end{array}\right.$$

Full text

For a finite group which has cyclic Sylow $p$-subgroups, we show that the number of irreducible characters with Frobenius-Schur indicator $-1$ is greater than or equal to the number of conjugacy classes of weakly real $p$-elements in $G$.

1. Introduction and preliminary results

The Frobenius-Schur (F-S) indicator of an ordinary character $\chi$ of a finite group $G$ is

[TABLE]

If $\chi$ is irreducible then $\epsilon(\chi)=0,\pm 1$. Moreover $\epsilon(\chi)\neq 0$ if and only if $\chi$ is real-valued.

R. Brauer showed how to partition the irreducible characters of $G$ into $p$-blocks, for each prime $p$. Each $p$-block has an associated defect group, which is a $p$-subgroup of $G$, unique up to $G$-conjugacy, which determines much of the structure of the block. If the defect group is trivial, the block contains a unique irreducible character. In the next most complicated case, E. Dade [D] determined the structure of a block which has a cyclic defect group and defined the Brauer tree of the block.

Recall that a $p$-block is said to be real if it contains the complex conjugates of its characters. We wish to determine the F-S indicators of the irreducible characters in a real $p$-block which has a cyclic defect group. In [M2, Theorem 1.6] we dealt with the case $p=2$; there are six possible indicator patterns, and the extended defect group of the block determines which occurs. In this paper we consider the case $p\neq 2$.

R. Gow showed [G, 5.1] that a real $p$-block has a real irreducible character, if $p=2$. This is false for $p\neq 2$, as was first noticed by H. Blau in the early 1980’s, in response to a question posed by Gow. His example was for $p=5$ and $G=6.S_{6}$ (Atlas notation). G. Navarro has recently found a solvable example with $p=3$ and $G=\operatorname{SmallGroup}(144,131)$ (GAP notation). We give examples for blocks with cyclic defect below.

Now let $B$ be a real $p$-block which has a cyclic defect group $D$. The inertial index of $B$ is a certain divisor $e$ of $p-1$. Dade showed that $B$ has $e$ irreducible Brauer characters and $e+\frac{|D|-1}{e}$ ordinary irreducible characters. The latter he divided into $\frac{|D|-1}{e}$ exceptional characters and $e$ non-exceptional characters.

Suppose that $\frac{|D|-1}{e}=1$ (which can only occur when $|D|=p$). Then the choice of exceptional character is arbitrary, and the convention in [F] is to regard $B$ as having no exceptional characters. However, we will see that in this event $B$ has real irreducible characters, all of which have the same F-S indicators. So our convention is to assume that $B$ has a real exceptional character.

The Brauer tree of $B$ is a planar graph which describes the decomposition matrix of $B$. There is one exceptional vertex, representating all the exceptional characters, and one vertex for each of the non-exceptional characters. Two vertices are connected by an edge if their characters share a modular constituent.

J. Green [Gr] showed that all real objects in the Brauer tree lie on a line segment, now called the real-stem of $B$. The exceptional vertex belongs to the real-stem (see Lemma 6 below). So it divides the real non-exceptional vertices into two, possibly empty, subsets. We find it convenient to refer to the corresponding real non-exceptional characters as being on the left or the right of the exceptional vertex. Here is our main theorem:

Theorem 1**.**

Let $p$ be an odd prime and let $B$ be a real $p$-block which has a cyclic defect group. Then

• (i)

All exceptional characters in $B$ have the same F-S indicators.

• (ii)

On each side of the exceptional vertex, the real non-exceptional characters have the same F-S indicators.

• (iii)

If $B$ has a real exceptional character then all real irreducible characters in $B$ have the same F-S indicators.

• (iv)

Suppose that $B$ has no real exceptional characters, and that there are an odd number of non-exceptional vertices on each side of the exceptional vertex. Then the real non-exceptional characters have F-S indicator $+1$ on one side of the exceptional vertex and $-1$ on the the other side.

Note that (i) is not a consequence of Galois conjugacy, as there are at least two Galois conjugacy classes of exceptional characters, when $|D|>p$.

In Proposition 15 we show that the F-S indicators of the exceptional characters in $B$ agree with those of the Brauer corresponding block in the normalizer of a defect group. In Theorem 16 we compute this common indicator using the ‘canonical character’ of $B$.

Next recall that an element of $G$ is said to be weakly real if it is conjugate to its inverse in $G$, but it is not inverted by any involution in $G$. Here is an application of Theorem 1 whose statement does not refer to blocks or to modular representation theory:

Theorem 2**.**

Let $p$ be an odd prime and let $G$ be a finite group which has cyclic Sylow $p$-subgroups. Then the number of irreducible characters of $G$ with F-S indicator $-1$ is greater than or equal to the number of conjugacy classes of weakly real $p$-elements in $G$.

We use the notation and results of [NT] for group representation theory, and use [D] and [F, VII] for notation specific to blocks with cyclic defect. When referring to the character tables of a finite simple group we use the conventions of the ATLAS [A]. For other character tables, we use the notation of the computer algebra system GAP [GAP].

2. Examples

We begin with a number of examples which illustrate the possible patterns of F-S indicators in a block which has a cyclic defect group. Throughout $G$ is a finite group and $B$ is a real $p$-block of $G$ which has a cyclic defect group $D$. Also $N_{0}$ is the normalizer in $G$ of the unique order $p$ subgroup of $D$ and $B_{0}$ is the Brauer correspondent of $B$ in $N_{0}$.

Example 1: There are many blocks with cyclic defect group whose irreducible characters all have the same F-S indicators. For blocks with all indicators $+1$, choose $n\geq 2$, a prime $p$ with $n/2\leq p\leq n$ and any $p$-block of the symmetric group $S_{n}$. There are numerous blocks with all indicators $-1$ among the faithful $p$-blocks of the double cover $2.A_{n}$ of an alternating group, with $n/2\leq p\leq n$ e.g. the four faithful irreducible characters of $2.A_{5}$ have F-S indicator $-1$ and constitute a $5$-block with a cyclic defect group.

Example 2: If $e$ is odd then $B$ has a real non-exceptional character. Now it follows from [D, Part 2 of Theorem 1 & Corollary 1.9] that $B$ has a Galois conjugacy class consisting of $\frac{p-1}{e}$ exceptional characters. So $B$ has a real exceptional character if $\frac{p-1}{e}$ is odd. Thus $B$ always has a real irreducible character if $p\equiv 3\ (\mathrm{mod}\ 4)$.

When $e$ is even and $p\equiv 1\ (\mathrm{mod}\ 4)$, $B$ may have no real irreducible characters. For example SmallGroup$(80,29)=\langle a,b\mid a^{20},a^{10}=b^{4},a^{b}=a^{7}\rangle$ has such a block, for $p=5$. It consists of the four irreducible characters lying over the non-trivial irreducible character of $\langle a^{10}\rangle$. Here is its character table. The first two rows indicate the 2 and 5 parts of the class centralizers. The third row labels the classes by their element orders:

[TABLE]

Note that SmallGroup$(80,29)$ has Sylow $2$-subgroups isomorphic to SmallGroup$(16,6)=\langle s,t\mid s^{8},t^{2},s^{t}=s^{5}\rangle$. This 2-group is sometimes denoted $M_{4}(2)$.

Example 3: $B$ may have a real non-exceptional character but no real exceptional characters. For example SmallGroup$(60,7)=\langle a,b\mid a^{15},b^{4},a^{b}=a^{2}\rangle$ has such a block, for $p=5$. It consists of the four irreducible characters lying over a non-trivial irreducible character of $\langle a^{5}\rangle$. This is also an example of part (iv) of Theorem 1; the non-exceptional characters $X.5$ and $X.6$ have F-S indicators $-1$ and $+1$, respectively. Here is the table of character values, with $\alpha=(1+\sqrt{-15})/2$:

[TABLE]

Example 4: There is no apparent relationship between the F-S indicators of the non-exceptional characters in $B$ and in $B_{0}$. For example, let $B$ be the $5$-block $2.A_{8}$ with $\operatorname{Irr}(B)=\{\chi_{15},\chi_{19},\chi_{21},\chi_{22}\}$. Then the two non-exceptional characters $\chi_{15}$ and $\chi_{19}$ have F-S indicator $+1$ and $-1$, respectively. However $B_{0}$ is a real block which has no real irreducible characters.

The character table of $B$ can be found on p22 of The Atlas. Now $N_{0}$ is isomorphic to SmallGroup$(120,7)=\langle a,b\mid a^{15},b^{8},a^{b}=a^{2}\rangle$. Here is the table of character values of its $5$-block $B_{0}$. Again $\alpha=(1+\sqrt{-15})/2$. In order to save space, we have omitted $4$ columns of zero values for the four classes of elements of order $8$:

[TABLE]

We note that $B$ has $2$ irreducible modules and $2$ weights, in conformity with Alperin’s weight conjecture [Al]. However the irreducible modules are self-dual and the weights are duals of each other. This shows that there is no obvious ‘real’ version of the weight conjecture for $p$-blocks, when $p\neq 2$.

Consider the inclusion of groups $N_{0}<\operatorname{PSL}_{2}(11)<M_{11}$, where $N_{0}\cong{11:5}$. The principal $11$-blocks each have $5$ non-exceptional characters. It is somewhat surprising that the number of real non-exceptional characters in these blocks is $1$, $5$ and $3$, respectively.

Example 5: Finally $B$ may have a real exceptional character but no real non-exceptional characters. For example let $B$ be the $5$-block containing the four faithful irreducible characters of SmallGroup$(20,1)=\langle a,b\mid a^{5},b^{4},a^{b}=a^{-1}\rangle$. The two exceptional characters have F-S indicators $-1$, but neither of the two non-exceptional characters is real. Here is the character table of $B$, with $\beta=(-1+\sqrt{5})/2$ and $*\beta=(-1-\sqrt{5})/2$:

[TABLE]

3. Miscellaneous results

We need general results from representation theory, some of which are not so well-known. So in this section $p$ is a prime and $B$ is a $p$-block of a finite group $G$.

Let $\chi$ be an irreducible character in $B$, let $x$ be a $p$-element of $G$ and let $y$ be a $p$-regular element of $\operatorname{C}_{G}(x)$. Then

[TABLE]

where $\varphi$ ranges over the irreducible Brauer characters in blocks of $\operatorname{C}_{G}(x)$ which Brauer induce to $B$, and each $d_{\chi,\varphi}^{(x)}$ is an algebraic integer, called a generalized decomposition number; if $x=1$, $\varphi$ is an irreducible Brauer character in $B$ and $d_{\chi,\varphi}^{(x)}$ is simplified to $d_{\chi,\varphi}$. It is an integer called an ordinary decomposition number of $B$.

Brauer [B, Theorem (4A)] used his Second Main Theorem to prove the following remarkable ‘local-to-global’ formula for F-S indicators:

[TABLE]

where $\chi$ ranges over the irreducible characters in $B$ and $\psi$ ranges over the irreducible characters in blocks of $\operatorname{C}_{G}(x)$ which Brauer induce to $B$. We have previously used this formula to determine the F-S indicators of the irreducible characters in 2-blocks with a cyclic, Klein-four or dihedral defect group.

Our next result relies on Clifford theory. However it was inspired by (and can be proved using) the notion of a weakly real $2$-block, as introduced in [M1]. Suppose that $N$ is a normal subgroup of $G$ and $\phi\in\operatorname{Irr}(N)$, with stabilizer $G_{\phi}$ in $G$. If $G_{\phi}\subseteq H\subseteq G$, the Clifford correspondence is a bijection $\operatorname{Irr}(G\mid\phi)\leftrightarrow\operatorname{Irr}(H\mid\phi)$ such that $\chi\leftrightarrow\psi$ if and only if $\langle\chi{\downarrow_{H}},\phi\rangle\neq 0$ or $\chi=\psi{\uparrow^{G}}$. The stabilizer of $\{\phi,\overline{\phi}\}$ in $G$ is called the extended stabilizer of $\phi$, here denoted by $G_{\phi}^{*}$. So $|G_{\phi}^{*}:G_{\phi}|\leq 2$, with equality if and only if $\phi\neq\overline{\phi}$ but $\phi$ is $G$-conjugate to $\overline{\phi}$. If $G_{\phi}^{*}\subseteq H$ it is easy to see that $\chi$ is real if and only if $\psi$ is real. Moreover in this case $\epsilon(\chi)=\epsilon(\psi)$.

We need one other idea. Suppose that $T$ is a degree $2$ extension of $G$. Then the Gow indicator [G, 2.1] of a character $\chi$ of $G$ with respect to $T$ is defined to be

[TABLE]

Clearly $\epsilon(\chi{\uparrow^{T}})=\epsilon(\chi)+\epsilon_{T/G}(\chi)$. Just like the F-S indicator, $\epsilon_{T/G}(\chi)=0,\pm 1$, for each $\chi\in\operatorname{Irr}(G)$. Moreover $\epsilon_{T/G}(\chi)\neq 0$ if and only if $\chi$ is $T$-conjugate to $\overline{\chi}$.

Lemma 3**.**

Let $N$ be a normal odd order subgroup of $G$ and let $\phi\in\operatorname{Irr}(N)$. Suppose that $G_{\phi}^{*}$ does not split over $G_{\phi}$. Then there exists $\chi\in\operatorname{Irr}(G\mid\phi)$ such that $\epsilon(\chi)=-1$.

Proof.

We first show that there exists $\psi\in\operatorname{Irr}(G\mid\phi)$ such that $\epsilon(\psi)=+1$. For let $S$ be a Sylow $2$-subgroup of $G$. As $\phi{\uparrow^{G}}$ vanishes on the $2$-singular elements of $G$, we have $(\phi{\uparrow^{G}}){\downarrow_{S}}=\frac{\phi(1)|G|}{|N||S|}\rho_{S}$, where $\rho_{S}$ is the regular character of $S$. Now $\frac{\phi(1)|G|}{|N||S|}$ is an odd integer. So $\langle(\phi{\uparrow^{G}}){\downarrow_{S}},1_{S}\rangle$ is odd. Moreover $\phi{\uparrow^{G}}$ is a real character of $G$. So $\langle(\phi{\uparrow^{G}}),\psi\rangle=\langle(\phi{\uparrow^{G}}),\overline{\chi}\rangle$, for each $\psi\in\operatorname{Irr}(G)$. Pairing each irreducible character of $G$ with its complex conjugate, we see that there exists a real-valued $\psi\in\operatorname{Irr}(G\mid\phi)$ such that $\langle\psi{\downarrow_{S}},1_{S}\rangle$ is odd. Then $\epsilon(\psi)=\epsilon(1_{S})=+1$.

Following the discussion before the lemma, we may assume that $G=G_{\phi}^{*}$. So $|G:G_{\phi}|=2$. Next suppose that $g\in G$ and $\phi{\uparrow^{G_{\phi}}}(g^{2})\neq 0$. Write $g=xy=yx$, where $x$ is a $2$-element and $y$ is a $2$-regular element. Then $g^{2}=x^{2}y^{2}$. As $\phi{\uparrow^{G_{\phi}}}$ vanishes off $N$, we have $x^{2}=1$ and $y^{2}\in N$. So $x\in G_{\phi}$, as $G_{\phi}$ contains all involutions in $G$. Moreover $y\in N$, as $y$ has odd order. Thus $g\in G_{\phi}$, whence

[TABLE]

Now $\operatorname{Irr}({G_{\phi}\mid\phi})$ contains no real characters, as $\phi\neq\overline{\phi}$. So $\epsilon(\phi{\uparrow^{G}})=\epsilon_{G/G_{\phi}}(\phi{\uparrow^{G_{\phi}}})+\epsilon(\phi{\uparrow^{G_{\phi}}})=0$. Equivalently

[TABLE]

Together with the fact that $\langle\phi{\uparrow^{G}},\psi\rangle\epsilon(\psi)>0$, this implies that $\langle\phi{\uparrow^{G}},\chi\rangle\epsilon(\chi)<0$, for some $\chi\in\operatorname{Irr}(G)$. Thus $\chi\in\operatorname{Irr}(G\mid\phi)$ and $\epsilon(\chi)=-1$, which completes the proof. ∎

It is well-known that each $G$-invariant irreducible character of a normal subgroup of $G$ extends to $G$, when the quotient group is cyclic.

Lemma 4**.**

Suppose that $N$ is a normal subgroup of $G$ such that $G/N$ is cyclic and of even order. Let $\varphi\in\operatorname{Irr}(N)$ be real and $G$-invariant. Then $\varphi$ has a real extension to $G$ if and only if $\varphi$ has a real extension to $T$, where $N\subset T\subseteq G$ and $T/N$ has order $2$.

Proof.

The ‘only if’ part is obvious. So assume that $\varphi$ has a real extension to $T$. Then both extensions of $\varphi$ to $T$ are real. Let $\omega$ be a generator of the abelian group $\operatorname{Irr}(G/N)$ and let $\chi$ be any extension of $\varphi$ to $G$. Then $\omega^{i}\chi$, $i\geq 0$ give all extensions of $\varphi$ to $G$. Here $\omega^{i}=\omega^{j}$ if and only if $i\equiv j\ (\mathrm{mod}\ |G/N|)$.

As $\overline{\chi}$ lies over $\varphi$, we have $\overline{\chi}=\omega^{i}\chi$, for some $i\geq 0$. Now $\chi{\downarrow_{T}}$ is an extension of $\varphi$ to $T$ and $\overline{\chi}{\downarrow_{T}}=(\omega^{i}{\downarrow_{T}})(\chi{\downarrow_{T}})$. As $\chi{\downarrow_{T}}$ is real, it follows that $\omega^{i}{\downarrow_{T}}$ is trivial. So $T\subseteq\operatorname{ker}(\omega^{i})$, whence $i\equiv 2j\ (\mathrm{mod}\ |G/N|)$, for some $j\geq 0$. Now $\overline{\omega^{j}\chi}=\omega^{i-j}\chi=\omega^{j}\chi$. So $\omega^{j}\chi$ is a real extension of $\varphi$ to $G$. ∎

Notice that in this context $\varphi$ has a real extension to $T$ if and only if $\epsilon(\varphi)=\epsilon_{T/N}(\varphi)$. When $G/N$ has even order, but is not cyclic, and $\varphi$ is a real irreducible character of $N$ which extends to $G$, it is not clear whether there is a sensible sufficient criteria for $\varphi$ to have a real extension to $G$.

Finally we need the following consequence of the first orthogonality relation:

Lemma 5**.**

Let $W\subseteq X\subseteq Y$ be finite abelian groups. Then for $\lambda\in\operatorname{Irr}(Y)$ we have

[TABLE]

4. The Brauer tree and its real-stem

From now on $G$ is a finite group, $p$ is an odd prime and $B$ is a real $p$-block of $G$ which has a cyclic defect group. To avoid trivialities we assume that the defect group is non-trivial.

Dade asserts [D, Theorem 1, Part 2] that each decomposition number in $B$ is either [math] or $1$. The Brauer tree of $B$ is a planar graph with edges labelled by the irreducible Brauer character in $B$ and with vertices labelled by the irreducible characters in $B$ (the exceptional characters in $B$ label a single ‘exceptional’ vertex). The edge labelled by an irreducible Brauer character $\theta$ meets the vertex labelled by an irreducible character $\chi$ if and only if the decomposition number $d_{\chi,\theta}$ is not [math].

When $B$ is real, complex conjugation acts on the Brauer tree of $B$, and in particular fixes the exceptional vertex. However, as we have seen in Examples 2,3 and 4 above, $B$ may have no real exceptional characters. So we restate [F, VII,9.2] in the following more precise fashion:

Lemma 6**.**

The subgraph of the Brauer tree of $B$ consisting of the exceptional vertex and those vertices and edges which correspond to real characters and Brauer characters is a straight line segment.

Feit calls this line segment the real-stem of $B$. An easy consequence is:

Corollary 7**.**

The number of real non-exceptional characters in $B$ equals the number of real irreducible Brauer characters in $B$.

Proof.

Suppose that $B$ has $r$ real irreducible Brauer characters. Then the real-stem of the Brauer tree has $r$ edges and $r+1$ vertices. One of these is the exceptional vertex. So $B$ has $r$ real non-exceptional characters. ∎

Let $\theta$ be a real irreducible $p$-Brauer character of a finite group $G$. As $p$ is odd, the $G$-representation space of $\theta$ affords a non-degenerate $G$-invariant bilinear form which is either symmetric or skew-symmetric. Given the symmetry groups of such forms, we refer to $\theta$ as being of orthogonal or symplectic type. Thompson and Willems [W, 2.8] proved that there is a real irreducible character $\chi$ of $G$ such that $d_{\chi,\theta}$ is odd. Moreover $\theta$ has orthogonal type if $\epsilon(\chi)=+1$ or symplectic type if $\epsilon(\chi)=-1$. This implies that $\epsilon(\psi)=\epsilon(\chi)$, for all real irreducible characters $\psi$ such that $d_{\psi,\theta}$ is odd.

Proof of part (ii) of Theorem 1.

Let $X$ and $Y$ be real non-exceptional characters which lie on the same side of the exceptional vertex in the real-stem of $B$. Then by Lemma 6 there is a sequence $X=X_{0},X_{1},\dots,X_{n}=Y$ of real non-exceptional characters and a sequence $\theta_{1},\dots,\theta_{n}$ of real irreducible Brauer characters such that $d_{X_{i-1},\theta_{i}}=1=d_{X_{i},\theta_{i}}$, for $i=1,\dots,n$. The Thompson-Willems result implies that $\epsilon(X_{i-1})=\epsilon(X_{i})$, for $i=1,\dots,n$. So $\epsilon(X)=\epsilon(Y)$. This gives part (ii) of Theorem 1. ∎

A similar argument gives the following weak form of parts (i) and (iii) of Theorem 1:

Lemma 8**.**

If $B$ has a real exceptional character and a real non-exceptional character, then all real irreducible characters in $B$ have the same F-S indicators.

Notice that if $B$ is the principal $p$-block of a group with a cyclic Sylow $p$-subgroup, and $B$ has an irreducible character with F-S indicator $-1$ (e.g. the principal $7$-block of $\operatorname{U}(3,3)$) then the lemma implies that $B$ has no real exceptional characters.

5. The exceptional characters

We outline some results from [D] using the language of subpairs. See [NT, Chapter 5.9] for a full description of the theory. We then prove results about the local blocks in $B$, in Proposition 10, and the exceptional characters in $B$, in Proposition 11. This allows us to prove parts (i), (iii) and (iv) of Theorem 1.

Recall that $B$ is a $p$-block with a non-trivial cyclic defect group $D$. Write $|D|=p^{a}$, where $a>0$, and let $1\subset D_{a-1}\subset D_{a-2}\subset\dots\subset D_{1}\subset D_{0}=D$ be the complete list of subgroups of $D$. So $[D:D_{i}]=p^{i}$, for $i=0,\dots,a-1$. Set $C_{i}=C_{G}(D_{i})$ and $N_{i}=N_{G}(D_{i})$. So $C_{0}\subseteq C_{1}\subseteq\ldots\subseteq C_{a-1}$, and $N_{0}\subseteq N_{1}\subseteq\ldots\subseteq N_{a-1}$.

As $p$ is odd, $\operatorname{Aut}(D_{i})$ is a cyclic group of order $p^{a-i-1}(p-1)$. So $N_{i}/C_{i}$ is a cyclic group whose order divides $p^{a-i-1}(p-1)$. Moroever the centralizer of $D_{i}$ in $\operatorname{Aut}(D)$ has order $p^{i}$. So $C_{i}\cap N_{0}/C_{0}$ is a cyclic $p$-group. We note that the unique involution in $\operatorname{Aut}(D)$ inverts every element of $D$.

Fix a Sylow $B$-subpair $(D,b_{0})$. So $b_{0}$ is a $p$-block of $C_{0}$ such that $b_{0}^{G}=B$ and the pair $(D,b_{0})$ is uniquely determined up to $G$-conjugacy. Set $b_{i}:=b_{0}^{C_{i}}$, for $i=1,\dots,a-1$. Then by [NT, 5.9.3] the lattice of $B$-subpairs contained in $(D,b_{0})$ is

[TABLE]

Set ${E:=\operatorname{N}(D,b_{0})}$, the stabilizer of $b_{0}$ in $N_{0}$. Then ${e:=|E:C_{0}|}$ is called the inertial index of $B$. Now ${p\not\hskip 2.84544pt\mid e}$, by Brauer’s extended first main theorem. So $e\mid(p-1)$. Let $x\in E$. Then $D_{i}^{x}=D_{i}$. As $(D_{i},b_{i}),(D_{i},b_{i}^{x})\subseteq(D,b_{0})$, it follows from (2) that $b_{i}^{x}=b_{i}$. So $EC_{i}\subseteq\operatorname{N}(D_{i},b_{i})$. Conversely let $n\in\operatorname{N}(D_{i},b_{i})$. As $(D,b_{0})$ and $(D,b_{0})^{n}$ are Sylow $b_{i}$-subpairs (in the group $C_{i}$), there is $c\in C_{i}$ such that $nc_{i}\in E$. This shows that $\operatorname{N}(D_{i},b_{i})\subseteq EC_{i}$. This recovers Dade’s observation that $\operatorname{N}(D_{i},b_{i})=EC_{i}$.

Now $E\cap C_{i}/C_{0}$ is a subgroup of $C_{i}\cap N_{0}/C_{0}$ and a quotient of the group $E/C_{0}$. As $C_{i}\cap N_{0}/C_{0}$ is a $p$-group and $E/C_{0}$ has $p^{\prime}$-order, we deduce that $E\cap C_{i}=C_{0}$. It follows from this $EC_{i}/C_{i}\cong E/C_{0}$, and in particular $|EC_{i}:C_{i}|=e$.

By [D, Theorem 1, Part 1] $B$ has $e$ irreducible Brauer characters, listed as $\chi_{1},\dots,\chi_{e}$. Each $b_{i}$ has inertial index $1$. So $b_{i}$ has a unique irreducible Brauer character, denoted $\varphi_{i}$.

From the above discussion there are $|N_{i}:EC_{i}|=\frac{|N_{i}:C_{i}|}{e}$ distinct blocks of $C_{i}$ which induce to $B$, namely $b_{i}^{\tau}$ as $\tau$ ranges over $N_{i}/EC_{i}$. Also there are $\frac{p^{a-i}-p^{a-i-1}}{|N_{i}:C_{i}|}$ conjugacy classes of $G$ which contain a generator of $D_{i}$. So $B$ has $\frac{p^{a-i}-p^{a-i-1}}{e}$ subsections $(x,b)$, with $D_{i}=\langle x\rangle$. A consequence of Brauer’s second main theorem [NT, 5.4.13(ii)] is that the number of irreducible characters in a block equals the number of columns in the block.

Lemma 9**.**

A complete set of columns of $B$ is

[TABLE]

Here $x_{i}$ is a fixed generator of $D_{i}$, $\sigma_{i}$ ranges over a set of representatives for the cosets of the image of $N_{i}/C_{i}$ in $\operatorname{Aut}(D_{a-i})$ and $n_{i}$ ranges over a set of representatives for the cosets of $EC_{i}$ in $N_{i}$. In particular $k(B)=e+\frac{p^{a}-1}{e}$.

Let $\Lambda$ be a set of representatives for the $\frac{p^{a}-1}{e}$ orbits of $E$ on $\operatorname{Irr}(D)^{\times}$. Then

[TABLE]

Also set $X_{0}:=\sum_{\lambda\in\Lambda}X_{\lambda}$. Dade refers to the $X_{\lambda}$ as the exceptional characters of $B$.

Notice that as $\ell(b_{i})=1$, $b_{i}$ is real if and only if $\varphi_{i}$ is real. The next two propositions are relatively elementary.

Proposition 10**.**

All the blocks $b_{0},b_{1},\dots,b_{a-1}$ are real or none of them are real.

Proof.

We have $(b_{i}^{o})^{G}=B^{o}=B$. So $(D,b_{0})$ and $(D,b_{o}^{o})$ are Sylow $B$-subpairs, and there is $n\in\operatorname{N}_{0}$ such that $b_{0}^{o}=b_{0}^{n}$.

Suppose that $b_{j}$ is real, for some $j=0,\dots,a-1$. As $(D_{j},b_{j}^{n}),(D_{j},b_{j}^{o})\subset(D_{0},b_{0}^{o})$, it follows from (2) that $b_{j}^{n}=b_{j}^{o}=b_{j}$. So $n\in\operatorname{N}(D_{j},b_{j})=EC_{j}$. Write $n=ec$, where $e\in E$ and $c\in C_{j}$. Then $c=e^{-1}n\in C_{j}\cap N_{0}$ and $b_{0}^{c}=b_{0}^{n}=b_{0}^{o}$. So $c^{2}\in C_{j}\cap E=C_{0}$. But $C_{j}\cap N_{0}/C_{0}$ has odd order, as it is a $p$-group. So $c\in C_{0}$, which shows that $n\in E$. As $b_{0}^{n}=b_{0}$, it follows that $b_{0}$ is real.

Now let $i=0,\dots,a-1$. Then $(D_{i},b_{i}),(D_{i},b_{i}^{o})\subset(D_{0},b_{0})=(D_{0},b_{0}^{o})$. So $b_{i}=b_{i}^{o}$, for $i=0,\dots,a-1$, using (2). This shows that all $b_{0},\dots,b_{a-1}$ are real. ∎

We showed in [M3, 1.1] that the number of real irreducible characters in a block equals the number of real columns in the block. Here $(x,\varphi)$ is real if $x^{g}=x^{-1}$ and $\varphi^{g}=\overline{\varphi}$, for some $g\in G$.

Let $i=0,\dots,a-1$. As $b_{i}$ has inertial index $1$, it has $|D|$ irreducible characters. Modifying [D, p26] we use the notation

[TABLE]

Here $X_{i,1}^{\prime}$ is the unique non-exceptional character in $b_{i}$, and all characters $X_{i,\lambda}^{\prime}$ with $\lambda\neq 1$ are exceptional. Suppose that $b_{i}$ is real. The columns of $b_{i}$ are $(d,\varphi_{i})$, for $d\in D$. As $C_{i}$ acts trivially on the columns, the only real column is $(1,\varphi_{i})$. So $X_{i,1}^{\prime}$ is the only real irreducible character in $b_{i}$.

We will refine the next result in part (i) of Theorem 1:

Proposition 11**.**

All exceptional characters in $B$ are real or none are real.

Proof.

It follows from Corollary 7 and Lemma 9 that the number of real exceptional characters in $B$ equals the number of real columns $(x,\varphi)$ with $x\in D^{\times}$ and $\varphi\in\operatorname{IBr}(\operatorname{C}_{G}(x))$.

Suppose that $B$ has a real exceptional character, and let $(x,\varphi)$ be a real column of $B$, with $x\in D^{\times}$. Then $\langle x\rangle=D_{i}$, for some $i=0,\dots,a-1$. As $N_{i}/C_{i}$ is abelian, the columns $(x^{\prime},\varphi_{i}^{n_{i}})$ are real, for all generators $x^{\prime}$ of $D_{i}$ and all $n_{i}\in N_{i}$. In particular $(x_{i},\varphi_{i})$ is a real column. Choose $n\in N_{i}$ such that $x_{i}^{n}=x_{i}^{-1}$ and $\varphi_{i}^{n}=\overline{\varphi}_{i}$. We may suppose that $n^{2}\in C_{i}$.

Suppose first that $b_{i}$ is real. As $\varphi_{i}=\overline{\varphi}_{i}$, $n$ fixes $\varphi_{i}$ and inverts $D_{i}$. So $nC_{i}$ is an involution in $EC_{i}/C_{i}$. As $EC_{i}/C_{i}\cong E/C_{0}$, we may assume without loss that $nC_{0}$ is an involution in $E/C_{0}$. Now all the blocks $b_{0},\dots,b_{a-1}$ are real. Hence all $\varphi_{0},\dots,\varphi_{a-1}$ are real. As $n$ inverts $D_{j}$ and fixes $\varphi_{j}$, all columns $(x_{j},\varphi_{j})$ are real. Thus all columns $(x,\varphi)$, with $x\in D^{\times}$, are real. So all exceptional characters in $B$ are real in this case.

Conversely, suppose that $b_{i}$ is not real. As $nC_{i}$ is the unique involution in $N_{i}/C_{i}$, but $n\not\in EC_{i}$, it follows that $|EC_{i}:C_{i}|=e$ is odd. Now $(D,b_{0})$ and $(D,b_{0}^{o})$ are Sylow $B$-subpairs, but $b_{0}\neq b_{0}^{o}$. So there is $m\in N_{0}\backslash E$ such that $b_{0}^{m}=b_{0}^{o}$. As $m^{2}\in E$ and $|E:C_{0}|$ is odd, we may choose $m$ so that $m^{2}\in C_{0}$. Then $mC_{0}$ is the unique involution in $N_{0}/C_{0}$. In particular $m$ inverts every element of $D$. Let $j=0,\dots,a-1$. Then $(D_{j},b_{j}^{m})$ and $(D_{j},b_{j}^{o})$ are $B$-subpairs contained in $(D,b_{0}^{o})$. So $b_{j}^{m}=b_{j}^{o}$ and thus $(d_{j},\varphi_{j})^{m}=(d_{j}^{-1},\overline{\varphi}_{j})$. It follows that all exceptional characters in $B$ are real in this case also. ∎

Examination of the proof shows that:

Corollary 12**.**

All exceptional characters in $B$ are real if and only if $b_{0}$ is real and $e$ is even, or $b_{0}$ is not real and $e$ is odd.

We need some additional notation. Set $\Lambda_{u}:=\{\lambda\in\Lambda\mid\operatorname{ker}(\lambda)=D_{u}\}$, for $u=1,\dots,a$. So $|\Lambda_{u}|=\frac{p^{u}-p^{u-1}}{e}$. Now choose $\lambda\in\Lambda_{u}$ and set

[TABLE]

Note that $X_{\lambda}$ and $X_{\mu}$ are Galois conjugates, for all $\lambda,\mu\in\Lambda_{u}$ (this follows from [D, part 2 of Theorem 1 and Corollary 1.9]). So $\epsilon_{u}$ does not depend on $\lambda$.

Recall our notation (4) for the irreducible characters $X_{i,\lambda}^{\prime}$ in $b_{i}$. As already noted, $X_{i,1}^{\prime}$ is the only possible real irreducible character in $b_{i}$. We set

[TABLE]

Now let $i=0,\dots,a-1$ and choose $x\in D_{i}-D_{i+1}$ and $\rho\in N_{i}$. According to [D, Theorem 1, Part 3] there are signs $\varepsilon_{0}^{\prime},\varepsilon_{0},\varepsilon_{1}\dots,\varepsilon_{e}$ and $\gamma_{i}$ such that

[TABLE]

Here $EC_{i}/C_{i}$ is a set of representatives for the cosets of $C_{i}$ in $EC_{i}$. Note that Feit uses the notation $\delta_{0}=-\varepsilon_{0}$ and $\delta_{j}=\varepsilon_{j}$, for $j=1,\dots,e$. Now let $i=0,\dots,a-1$ and $x\in D_{i}-D_{i+1}$. Then it follows from [D, Corollary 1.9] that $X_{j}(x)=|N_{i}:EC_{i}|\varphi_{i}(1)\delta_{j}\gamma_{i}$. So $\delta_{j}\gamma_{i}$ is the sign of the integer $X_{j}(x)$.

There is a nice relationship between the signs $\varepsilon_{0},\varepsilon_{1}\dots,\varepsilon_{e}$ and the Brauer tree of $B$. Suppose that $j$ and $k$ are adjacent vertices in the Brauer tree. Then $X_{j}+X_{k}$ is a principal indecomposable character of $G$. So it vanishes on $D^{\times}$, and hence $\delta_{j}+\delta_{k}=0$ (see [F, V11, Section 9]). So suppose that there are $d_{j}$ edges between the vertex $j$ and the exceptional vertex [math] in the Brauer tree. Then $\delta_{j}=(-1)^{d_{j}}\delta_{0}$. So $\varepsilon_{j}=(-1)^{d_{j}-1}\varepsilon_{0}$, for $j=1,\dots,e$.

We now prove part (i) of our main theorem. But note that this proof does not depend on Propositions 10 and 11:

Proof of part (i) of Theorem 1.

Applying (1), with $\rho\in N_{i}$ and $x\in D_{i}-D_{i+1}$, we get

[TABLE]

Now set $\sigma:=\varepsilon_{0}\sum_{j=1}^{e}\epsilon(X_{j})\varepsilon_{j}$. So $\sigma$ is independent of $i$, $\rho$ and $x$. Then the above equality transforms to

[TABLE]

where the right hand side is independent of $\rho$ and $x$. Let $\rho$ range over a set of representatives for the $\frac{|N_{i}:C_{i}|}{e}$ cosets of $EC_{i}$ in $N_{i}$ and let $x$ range over a set of representatives for the $\frac{p^{a-i}-p^{a-i-1}}{|N_{i}:C_{i}|}$ orbits of $N_{i}$ on the generators of $D_{i}$. Then ${{}^{\rho\tau}}x$ will range over all generators of $D_{i}$. Summing the resulting equalities gives

[TABLE]

We use $|\Lambda_{u}|=\frac{p^{u}-p^{u-1}}{e}$ and Lemma 5 to transform this equality to

[TABLE]

After cancelling the factor $\frac{p^{a-i-1}(p-1)}{e}$, we get

[TABLE]

Here $\sum_{u=1}^{0}(p^{u}-p^{u-1})\epsilon_{u}$ is taken to be [math], when $i=0$. We write down the equalities (5) for $i=0,1,2,\dots$ in turn:

[TABLE]

Subtract the first equality from the second to get

[TABLE]

The left hand side equals $-p,0$ or $p$ and the right hand equals $-2,0$ or $2$. As $p$ is odd, the common value is [math]. So $\epsilon_{2}=\epsilon_{1}$ and $\gamma_{1}\nu_{1}=\gamma_{0}\nu_{0}$. Substitute these values back into all equations in (6). Now subtract the first from the third equality to get

[TABLE]

Once again both sides are [math]. So $\gamma_{2}\nu_{2}=\gamma_{0}\nu_{0}$ and $\epsilon_{3}=\epsilon_{1}$. Proceeding in this way, we get

[TABLE]

∎

Following the above proof, and the discussion before the proof, we obtain:

Corollary 13**.**

Suppose that $b_{0}$ is real and let $D=\langle x\rangle$. Then for each $i=0,\dots,a-1$ and $j=0,\dots,e$, the integer $X_{j}(x^{p^{i}})X_{j}(x)$ has sign $\epsilon(X_{i,1}^{\prime})\epsilon(X_{0,1}^{\prime})$.

There is no apparent relationship between the F-S indicators $\nu_{0},\dots,\nu_{a-1}$:

Example: The $2$-nilpotent group $G=\langle a,b,c\mid a^{4},a^{2}=b^{2},a^{b}=a^{-1},c^{9},a^{c}=b,b^{c}=ab\rangle$ has isomorphism type $3.\operatorname{SL}(2,3)$. Set $D=\langle c\rangle$. Then $D$ is cyclic of order $9$, with $C_{0}=D\times\langle a^{2}\rangle$ and $C_{1}=G$. Let $\theta$ be the non-trivial irreducible character of $C_{0}/D$, and let $b_{0}$ be the $3$-block of $C_{0}$ which contains $\theta$. Then $\theta=X_{0,1}^{\prime}$ is the unique non-exceptional character in $b_{0}$. So $\nu_{0}=\epsilon(X_{0,1}^{\prime})=+1$. Set $b_{1}=b_{0}^{G}$. Then $b_{1}$ also has a unique non-exceptional character $X_{1,1}^{\prime}$. But now $\nu_{1}=\epsilon(X_{1,1}^{\prime})=-1$, as $X_{1,1}^{\prime}$ restricts to the non-linear irreducible character of $\langle a,b\rangle\cong Q_{8}$.

This example arises from the fact that the Glauberman correspondence [NT, 5.12] does not preserve the F-S indicators of characters.

proof of part (iii) of Theorem 1.

This is an immediate consequence of Lemma 8 and part (i) of Theorem 1. ∎

Consider the real-stem of $B$ as a horizontal line segment with $s$ vertices and $s-1$ edges, where $s\geq 1$. We label the vertices using an interval $[-\ell,\dots,-2,-1,0,1,2,\dots,r]$ so that [math] labels the exceptional vertex. Thus $s=r+\ell+1$, and there are $\ell$ real non-exceptional characters on the left of the exceptional vertex, and $r$ on the right (the choice of left and right is unimportant).

As above, $X_{0}$ is the sum of the exceptional characters in $B$. Now we relabel the non-exceptional characters in $B$ so that $X_{i}$ is the real non-exceptional character corresponding to vertex $i$, for $i=-\ell,\dots,r$ and $i\neq 0$. In view of parts (i) and (ii) of Theorem 1 there are signs $\epsilon_{\pm}$ such that

[TABLE]

Next let $\sigma$ be a generator of $D$. It follows from [D, Corollary 1.9] that $X_{0}(\sigma)=-\varepsilon_{0}\gamma_{0}|N_{0}:E|\varphi_{0}(1)$. So $X_{i}(\sigma)=(-1)^{i}X_{0}(\sigma)$, as $X_{i}+X_{i+1}$ is a projective character of $G$, for $i=-\ell,\dots,r-1$ (see [F, VII,2.19(ii)]).

Recall from Section (5) that there are $|N_{0}:E|$ blocks of $C_{0}$ which induce to $B$; these are the blocks $b_{0}^{\tau}$, where $\tau$ ranges over $N_{0}/E$. We note also that $X_{0,1}^{\prime}({{}^{\tau}}\sigma)=\varphi_{0}(1)$. Now [B, Theorem(4B)] is an immediate consequence of [B, Theorem(4A)]. In our context, this states that

[TABLE]

In view of the previous paragraph this simplifies to

[TABLE]

We consider a number of cases.

Suppose first that $\epsilon_{0}\neq 0$. Then $\epsilon_{-}=\epsilon_{0}=\epsilon_{+}$, by part (iii) of Theorem 1. So (7) becomes

[TABLE]

In particular $b_{0}$ is not real if $s$ is even. As $e$ is odd when $s$ is even, this already follows from Corollary 12.

Suppose then that $\epsilon_{0}=0$. Now (7) evaluates as

[TABLE]

proof of part (iv) of Theorem 1.

The hypothesis is that $\epsilon_{0}=0$, at least one of $\epsilon_{-},\epsilon_{+}$ is not zero and $\ell\equiv r\equiv 1\ (\mathrm{mod}\ 2)$. Now $B$ has $e$ non-exceptional characters, of which $\ell+r$ are real-valued. So $e\equiv\ell+r$ is even. Then $b_{0}$ is not real, according to Corollary 12. This in turn implies that $\nu_{0}=0$. So $\epsilon_{-}+\epsilon+=0$, according to (9). We conclude that $\epsilon_{-}\epsilon_{+}=-1$, which gives the conclusion of (iv). ∎

6. Passing from $B$ to its canonical character

Let $i=0,\dots,a-1$. Then $N_{i}$ contains the normalizer $N_{0}$ of $D$ in $G$. So by Brauer’s first main theorem there is a unique $p$-block $B_{i}$ of $N_{i}$ such that $B_{i}^{G}=B$. As $(B_{i}^{o})^{G}=B^{o}=B$, the uniqueness forces $B_{i}^{o}=B_{i}$. Now $B_{i}$ has defect group $D$ and inertial index $e=|EC_{i}:C_{i}|$. So $\ell(B_{a-1})=e$ and $k(B_{a-1})=e+\frac{p^{a}-1}{e}$. We first consider the block $B_{a-1}$ of the largest subgroup $N_{a-1}$. Following [D, Section 7], write

[TABLE]

and set $\tilde{X}_{0}=\sum\tilde{X}_{\lambda}$.

Proposition 14**.**

The exceptional characters in $B$ and $B_{a-1}$ have the same F-S indicators.

Proof.

Suppose first that $|\Lambda|\geq 2$. According [D, (7.2)] there is a sign $d$ such that

[TABLE]

It follows that $\langle\tilde{X}_{\lambda},X_{\lambda}\rangle$ or $\langle\tilde{X}_{\mu},X_{\lambda}\rangle$ is odd. So in view of part (i) of Theorem 1, the conclusion holds in this case.

From now on we suppose that $|\Lambda|=1$. Then $E$ has a single orbit on $\operatorname{Irr}(D)^{\times}$, which forces $|D|=p$ and $e=p-1$. As $\tilde{X}_{0}$ is the unique exceptional character in $B_{a-1}$, it is real valued. Then it follows from part (iii) of Theorem 1 that all real irreducible characters in $B_{a-1}$ have the same F-S indicators.

Now by [D, (7.3), (7.8), first two paragraphs of p40], there is a sign $\varepsilon_{0}^{\prime}$ such that

[TABLE]

Here $\varepsilon_{0},\dots,\varepsilon_{p-1}$ are as introduced earlier and $X_{0}$ can be chosen to be real, as $p$ is odd. Taking inner-products of characters, and reading modulo $2$, we see that $\langle\tilde{X}_{i}^{G},X_{0}\rangle$ is odd, for some real $\tilde{X}_{i}$. So $\epsilon(\tilde{X}_{i})=\epsilon(X_{0})$. Then by the previous paragraph $\epsilon(\tilde{X}_{0})=\epsilon(X_{0})$. ∎

Proposition 15**.**

All exceptional characters in $B_{0},\dots,B_{a-1}$ and $B$ have the same F-S indicators.

Proof.

We prove this by induction on $|D|$. The base case $|D|=p$ holds, by Proposition 14. Suppose that $|D|>p$. We assume that the conclusion holds for all $p$-blocks with a cyclic defect group of order strictly less than $|D|$.

We use the bar notation for subgroups and objects associated with the quotient group $N_{a-1}/D_{a-1}$. Let $i=0,\dots,a-1$. Then $\overline{N}_{i}$ is the normalizer of $\overline{D}_{i}$ in $\overline{N}_{a-1}$. As $C_{i}$ centralizes $D_{a-1}$, Theorem 5.8.11 of [NT] shows that $b_{i}$ dominates a unique block $\overline{b_{i}}$ of $\overline{C}_{i}$. Moreover $\overline{b}_{i}$ has cyclic defect group $\overline{D}$. Now $b_{i}$ has the unique irreducible Brauer character $\varphi_{i}$, and we can and do identify $\varphi_{i}$ with the unique irreducible Brauer character in $\overline{b}_{i}$. Then the inertia group of $\overline{b}_{i}$ in $\overline{N}_{i}$ is the inertia group of $\varphi_{i}$ in $\overline{N}_{i}$, which is $\overline{EC_{i}}$.

According to [D, Section 4], there is a unique $p$-block of $N_{i}$, denoted here by $\overline{B_{i}}$, which lies over $\overline{b_{i}}$. Moreover $\overline{B_{i}}$ has cyclic defect group $\overline{D}$. As inflation and induction of characters commute, this block is dominated by $B_{i}$. Now $B_{i}$ and $\overline{B_{i}}$ have the same inertial index as $|EC_{i}:C_{i}|=|\overline{EC_{i}}:\overline{C_{i}}|$. So by inflation $\operatorname{IBr}(\overline{B_{i}})=\operatorname{IBr}(B_{i})$. In particular $\overline{B_{i}}$ is the unique block of $\overline{N_{i}}$ that is dominated by $B_{i}$. Also by inflation $\operatorname{Irr}(\overline{B_{i}})\subseteq\operatorname{Irr}(B_{i})$.

As $|\overline{D}|<|D|$, all exceptional characters in $\overline{B}_{0},\dots,\overline{B}_{a-1}$ have the same F-S indicators, by our inductive hypothesis. But the inclusion $\operatorname{Irr}(\overline{B_{i}})\subseteq\operatorname{Irr}(B_{i})$ identifies the exceptional characters in $\overline{B}_{i}$ with exceptional characters in $B_{i}$. It now follows from part (i) of Theorem 1 that all exceptional characters in $B_{0},\dots,B_{a-1}$ have the same F-S indicators. ∎

Recall that $b_{0}$ has a unique irreducible Brauer character $\varphi_{0}$. This is the canonical character of $B$, in the sense of [NT, 5.8.3]. For the next theorem, we simplify the notation of (4) for the irreducible characters in $b_{0}$ by writing $\chi_{\lambda}$ in place of $X_{0,\lambda}^{\prime}$, for all $\lambda\in\operatorname{Irr}(D)$. Then according to W. Reynolds [NT, 5.8.14], for $c\in C_{0}$ we have

[TABLE]

Then $\operatorname{Irr}(b_{0})=\{\chi_{\lambda}\mid\lambda\in\operatorname{Irr}(D)\}$. Notice that $\chi_{1}$ is the unique irreducible character in $b_{0}$ whose kernel contains $D$.

Theorem 16**.**

Suppose that $B$ has a real exceptional character. Then $N_{0}/C_{0}$ has a unique subgroup $T/C_{0}$ of order $2$, and all exceptional characters in $B$ have F-S indicator equal to the Gow indicator $\epsilon_{T/C_{0}}(\chi_{1})$.

Proof.

Recall that $B$ has a real exceptional character if $b_{0}$ is real and $e$ is even, or if $b_{0}$ is not real and $e$ is odd. In both these cases $|N_{0}:C_{0}|$ is even. As $N_{0}/C_{0}$ is also cyclic, it has a unique subgroup $T/C_{0}$ of order $2$.

In view of Proposition 15, we may assume that $G=N_{0}$. So $B=B_{0}$, $D$ and $C_{0}$ are normal subgroups of $G$ and $E$ is the stabilizer of $b_{0}$ in $G$. Then $\Lambda$ is a set of representatives for the orbits of $N_{0}$ on $\operatorname{Irr}(D)^{\times}$. Set $E^{*}$ as the stabilizer of $\{b_{0},b_{0}^{o}\}$ in $G$. Clifford correspondence defines a bijection between the irreducible characters of $E^{*}$ which lie over $b_{0}$ and the irreducible characters in $B$. This bijection preserves reality, and hence F-S indicators. So from now on we assume that $G=E^{*}$.

As $\chi_{1}$ is invariant in $E$ and $E/C_{0}$ is cyclic, $\chi_{1}$ has $e$ extensions to $E$, which we denote by $\eta_{1},\dots,\eta_{e}$. Then $X_{i}:=\eta_{i}^{G}$, for $i=1,\dots,e$, give the $e$ non-exceptional characters in $B$. Moreover $X_{\lambda}:=\chi_{\lambda}^{G}$, for all $\lambda\in\Lambda$, give the exceptional characters in $B$.

Following Corollary 12, there are three cases we must consider:

Case 1: $b_{0}$ is real, $e$ is even and $B$ has real non-exceptional characters. Then according to part (iii) of Theorem 1 all real irreducible characters in $B$ have the same F-S indicators. We choose notation so that $X_{1}$ is real. As $X_{1}{\downarrow_{T}}$ is a real extension of $\chi_{1}$ to $T$, it follows that $\epsilon(X_{1})=\epsilon(X_{1}{\downarrow_{T}})=\epsilon_{T/C_{0}}(\chi_{1})$. This concludes Case 1.

Case 2: $b_{0}$ is real, $e$ is even but $B$ has no real non-exceptional characters. As $\chi_{1}$ does not extend to a real character of $E$, it does not extend to a real character of $T$, according to Lemma 4. So $\epsilon_{T/C_{0}}(\chi_{1})=-\epsilon(\chi_{1})$, by the definition of the Gow indicator.

Now consider the notation used in the proof of part (i) of Theorem 1. Here $C_{i}=C_{0}$ and $\varphi_{i}=\varphi_{0}$ and $X_{i,1}^{\prime}=\chi_{1}$, for $i=0,\dots,a-1$. If $\lambda\in\Lambda$ then $(X_{\lambda}){\downarrow_{C_{0}}}=\sum_{\tau\in G/C_{0}}\chi_{\lambda^{\tau}}$. So $d_{X_{\lambda},\varphi_{i}}^{(x)}=\sum_{\tau\in G/C_{0}}\lambda({{}^{\tau}}x)$, for all $x\in D^{\times}$. This means that $\varepsilon_{0}\gamma_{i}=1$, for $i=0,\dots,a-1$. Now in (6), the term $\sigma$ is [math], as none of $X_{1},\dots,X_{e}$ are real. So the first equation in (6) simplifies here to $-\epsilon(X_{\lambda})=\epsilon(\chi_{1})$, for all $\lambda\in\Lambda_{1}$. So $\epsilon(X_{\lambda})=\epsilon_{T/C_{0}}(\chi_{1})$, for all $\lambda\in\Lambda$, by the previous paragraph and Proposition 15.

Case 3: The final case is that $b_{0}$ is not real and $e$ is odd. As $B$ has an odd number $e$ of non-exceptional characters, at least one of them must be real valued. So we assume that $X_{1}$ is real. Then, just as in Case 1, all real irreducible characters in $B$ have the same F-S indicators.

As $|E:C_{0}|$ is odd and $|G:E|=2$, we have $G/C_{0}=E/C_{0}\times T/C_{0}$. Now $T/C_{0}$ conjugates $\operatorname{Irr}(b_{0})$ into $\operatorname{Irr}(b_{0}^{o})$. So $\chi_{1}$ is $T$-conjugate to $\overline{\chi}_{1}$. In particular $\chi_{1}{\uparrow^{T}}$ is irreducible and real valued. Now $X_{1}=(\eta_{1}){\uparrow^{G}}$ and $(\eta_{1}){\downarrow_{C_{0}}}=\chi_{1}$. So $(X_{1}){\downarrow_{T}}=(\chi_{1}){\uparrow^{T}}$, by Mackey’s theorem.

Now from above $\epsilon(X_{\lambda})=\epsilon(X_{1})$, for all $\lambda\in\Lambda$. Also $\epsilon(X_{1})=\epsilon((X_{1}){\downarrow_{T}})$, as both are real valued. Finally $\epsilon((X_{1}){\downarrow_{T}})=\epsilon_{T/C_{0}}(\chi_{1})$, by the definition. This completes Case 3. ∎

Finally, we prove the application to ordinary characters as stated in the Introduction:

Proof of Theorem 2.

Let $x$ be a weakly real $p$-element of $G$ of maximal order and set $Q:=\langle x\rangle$ and $N:=\operatorname{N}_{G}(Q)$. Let $\lambda$ be a faithful linear character of $Q$. Then $N_{\lambda}=\operatorname{C}_{N}(x)$ and $N_{\lambda}^{*}=\operatorname{C}_{N}^{*}(x)$. So $N_{\lambda}^{*}$ does not split over $N_{\lambda}$. By Lemma 3 there exists $\chi\in\operatorname{Irr}(N\mid\lambda)$ such that $\epsilon(\chi)=-1$.

Let $\tilde{B}$ be the $p$-block of $N$ which contains $\chi$ and let $D$ be a defect group of $\tilde{B}$. Then $Q\subseteq D$ and $\operatorname{N}_{G}(D)\subseteq N$. In particular $B:=\tilde{B}^{G}$ is defined and $B$ has defect group $D$. So $Q=D_{i}$, $N=N_{i}$ and $\tilde{B}=B_{i}$ for some $i\geq 0$, in cyclic defect group notation.

Notice that $\lambda$ is non-trivial. So $D\not\subseteq\operatorname{ker}(\chi)$. This means that $\chi$ is an exceptional character in $B_{i}$. So all exceptional characters in $B_{i}$, and hence also in $B$, are symplectic. The number of exceptional characters in $B$ is $\frac{|D|-1}{e}$, where $e$ is the inertial index of $B$. The number of weakly real $p$-conjugacy classes of $G$ is equal to the number of $N$-orbits on $Q^{\times}$, which equals $\frac{|D_{i}|-1}{|N_{i}:C_{i}|}$. As $|D_{i}|\leq|D|$ and $e\leq|N_{i}:C_{i}|$, we conclude that the number of symplectic irreducible characters of $G$ is not less than the number of weakly real $p$-conjugacy classes of $G$. ∎

7. Acknowledgement

H. Blau and D. Craven alerted us to examples of real $p$-blocks which do not have a real irreducible character. We thank G. Navarro for permission to include his example. We had a number of interesting discussions with R. Gow on F-S indicators.