Real Characters in Nilpotent Blocks

Abstract

We prove that the number of irreducible real characters in a nilpotent block of a finite group is locally determined. We further conjecture that the Frobenius–Schur indicators of those characters can be computed for $p=2$ in terms of the extended defect group. We derive this from a more general conjecture on the Frobenius–Schur indicator of projective indecomposable characters of 2-blocks with one simple module. This extends results of Murray on 2-blocks with cyclic and dihedral defect groups.

1 Introduction

An important task in representation theory is to determine global invariants of a finite group G by means of local subgroups. Dade’s conjecture, for instance, predicts the number of irreducible characters $\chi \in \text {Irr}(G)$ such that the p-part $\chi (1)_p$ is a given power of a prime p (see [23, Conjecture 9.25]). Since Gow’s work [7], there has been an increasing interest in counting real (i.e. real-valued) characters and more generally characters with a given field of values.

The quaternion group $Q_8$ testifies that a real irreducible character $\chi$ is not always afforded by a representation over the real numbers. The precise behavior is encoded by the Frobenius–Schur indicator (F-S indicator, for short)

$$\begin{aligned} \epsilon (\chi ):=\frac{1}{|G|}\sum _{g\in G}\chi (g^2)= \left\{ \begin{array}{ll} 0&{}\quad \text {if }\overline{\chi }\ne \chi ,\\ 1&{}\quad \text {if } \chi \text { is realized by a real representation},\\ -1&{}\quad \text {if } \chi \text { is real, but not realized by a real representation}. \end{array}\right. \end{aligned}$$

(1)

A new interpretation of the F-S indicator in terms of superalgebras has been given recently in [13]. The case of the dihedral group $D_8$ shows that $\epsilon (\chi )$ is not determined by the character table of G. The computation of F-S indicators can be a surprisingly difficult task, which has not been fully completed for the simple groups of Lie type, for instance (see [25]). Problem 14 on Brauer’s famous list [2] asks for a group-theoretical interpretation of the number of $\chi \in \text {Irr}(G)$ with $\epsilon (\chi )=1$.

To obtain deeper insights, we fix a prime p and assume that $\chi$ lies in a p-block B of G with defect group D. By complex conjugation we obtain another block $\overline{B}$ of G. If $\overline{B}\ne B$, then clearly $\epsilon (\chi )=0$ for all $\chi \in \text {Irr}(B)$. Hence, we assume that B is real, i.e. $\overline{B}=B$. John Murray [18, 19] has computed the F-S indicators when D is a cyclic 2-group or a dihedral 2-group (including the Klein four-group). His results depend on the fusion system of B, on Erdmann’s classification of tame blocks and on the structure of the so-called extended defect group E of B (see Definition 7 below). For $p>2$ and D cyclic, he obtained in [20] partial information on the F-S indicators in terms of the Brauer tree of B.

The starting point of my investigation is the well-known fact that 2-blocks with cyclic defect groups are nilpotent. Assume that B is nilpotent and real. If B is the principal block, then $G=\textrm{O}_{p'}(G)D$ and $\textrm{Irr}(B)=\textrm{Irr}(G/\textrm{O}_{p'}(G))=\textrm{Irr}(D)$. In this case the F-S indicators of B are determined by D alone. Thus, suppose that B is non-principal. By Broué–Puig [4], there exists a height-preserving bijection $\textrm{Irr}(D)\rightarrow \textrm{Irr}(B)$, $\lambda \mapsto \lambda *\chi _0$, where $\chi _0\in \textrm{Irr}(B)$ is a fixed character of height 0 (see also [16, Definition 8.10.2]). However, this bijection does not in general preserve F-S indicators. For instance, the dihedral group $D_{24}$ has a nilpotent 2-block with defect group $C_4$ and a nilpotent 3-block with defect group $C_3$, although every character of $D_{24}$ is real. Our main theorem asserts that the number of real characters in a nilpotent block is nevertheless locally determined. To state it, we introduce the extended inertial group

$$\begin{aligned} \textrm{N}_G(D,b_D)^*:=\left\{ g\in \textrm{N}_G(D):b_D^g\in \{b_D,\overline{b_D}\}\right\} , \end{aligned}$$

where $b_D$ is a Brauer correspondent of B in $D\textrm{C}_G(D)$.

Theorem A

Let B be a real, nilpotent p-block of a finite group G with defect group D. Let $b_D$ be a Brauer correspondent of B in $D\textrm{C}_G(D)$. Then the number of real characters in $\textrm{Irr}(B)$ of height h coincides with the number of characters $\lambda \in \textrm{Irr}(D)$ of degree $p^h$ such that $\lambda ^t=\overline{\lambda },$ where

$$\begin{aligned} \textrm{N}_G(D,b_D)^*/D\textrm{C}_G(D)=\langle tD\textrm{C}_G(D)\rangle . \end{aligned}$$

If $p>2,$ then all real characters in $\textrm{Irr}(B)$ have the same F-S indicator.

In contrast to arbitrary blocks, Theorem A implies that nilpotent real blocks have at least one real character (cf. [20, p. 92] and [8, Theorem 5.3]). If $\overline{b_D}=b_D$, then B and D have the same number of real characters, because $\textrm{N}_G(D,b_D)=D\textrm{C}_G(D)$. This recovers a result of Murray [18, Lemma 2.2]. As another consequence, we will derive in Proposition 5 a real version of Eaton’s conjecture [5] for nilpotent blocks as put forward by Héthelyi–Horváth–Szabó [12].

The F-S indicators of real characters in nilpotent blocks seem to lie somewhat deeper. We still conjecture that they are locally determined by a defect pair (see Definition 6) for $p=2$ as follows.

Conjecture B

Let B be a real, nilpotent, non-principal 2-block of a finite group G with defect pair (D, E). Then there exists a height preserving bijection $\Gamma :\textrm{Irr}(D)\rightarrow \textrm{Irr}(B)$ such that

$$\begin{aligned} \epsilon (\Gamma (\lambda ))=\frac{1}{|D|}\sum \limits _{e\in E\setminus D}\lambda (e^2) \end{aligned}$$

(2)

for all $\lambda \in \textrm{Irr}(D)$.

The right hand side of (2) was introduced and studied by Gow [8, Lemma 2.1] more generally for any groups $D\le E$ with $|E:D|=2$. This invariant was later coined the Gow indicator by Murray [20, (2)]. For 2-blocks of defect 0, Conjecture B confirms the known fact that real characters of 2-defect 0 have F-S indicator 1 (see [8, Theorem 5.1]). There is no such result for odd primes p. As a matter of fact, every real character has p-defect 0 whenever p does not divide |G|. In Theorem 10 we prove Conjecture B for abelian defect groups D. Then it also holds for all quasisimple groups G by work of An–Eaton [1]. Murray’s results mentioned above, imply Conjecture B also for dihedral D.

For $p>2$, the common F-S indicator in the situation of Theorem A is not locally determined. For instance, $G=Q_8\rtimes C_9=$ $\texttt{SmallGroup}$ (72, 3) has a non-principal real 3-block with $D\cong C_9$ and common F-S indicator $-1$, while its Brauer correspondent in $\textrm{N}_G(D)\cong C_{18}$ has common F-S indicator 1. Nevertheless, for cyclic defect groups D we find another way to compute this F-S indicator in Theorem 3 below.

Our second conjecture applies more generally to blocks with only one simple module.

Conjecture C

Let B be a real, non-principal 2-block with defect pair (D, E) and a unique projective indecomposable character $\Phi$. Then

$$\begin{aligned} \epsilon (\Phi )=|\{x\in E\setminus D:x^2=1\}|. \end{aligned}$$

Here $\epsilon (\Phi )$ is defined by extending (1) linearly. If $\epsilon (\Phi )=0$, then E does not split over D and Conjecture C holds (see Proposition 8 below). Conjecture C implies a stronger, but more technical statement on 2-blocks with a Brauer correspondent with one simple module (see Theorem 13 below). This allows us to prove the following.

Theorem D

Conjecture C implies Conjecture B.

We remark that our proof of Theorem D does not work block-by-block. For solvable groups we offer a purely group-theoretical version of Conjecture C at the end of Section 4.

Theorem E

Conjectures B and C hold for all nilpotent 2-blocks of solvable groups.

We have checked Conjectures B and C with GAP [6] in many examples using the libraries of small groups, perfect groups and primitive groups.

2 Theorem A and Its Consequences

Our notation follows closely Navarro’s book [22]. In particular, $G^0$ denotes the set of p-regular elements of a finite group G. Let B be a p-block of G with defect group D. Recall that a B-subsection is a pair (u, b), where $u\in D$ and b is a Brauer correspondent of B in $\textrm{C}_G(u)$. For $\chi \in \textrm{Irr}(B)$ and $\varphi \in \textrm{IBr}(b)$ we denote the corresponding generalized decomposition number by $d^u_{\chi \varphi }$. If $u=1$, we obtain the (ordinary) decomposition number $d_{\chi \varphi }=d^1_{\chi \varphi }$. We put $l(b)=|\textrm{IBr}(b)|$ as usual.

Following [22, p. 114], we define a class function $\chi ^{(u,b)}$ by

$$\begin{aligned} \chi ^{(u,b)}(us):=\sum _{\varphi \in \textrm{IBr}(b)}d^u_{\chi \varphi }\varphi (s) \end{aligned}$$

for $s\in \textrm{C}_G(u)^0$ and $\chi ^{(u,b)}(x)=0$ whenever x is outside the p-section of u. If $\mathcal {R}$ is a set of representatives for the G-conjugacy classes of B-subsections, then $\chi =\sum _{(u,b)\in \mathcal {R}}\chi ^{(u,b)}$ by Brauer’s second main theorem (see [22, Problem 5.3]). Now suppose that B is nilpotent and $\lambda \in \textrm{Irr}(D)$. By [16, Proposition 8.11.4], each Brauer correspondent b of B is nilpotent and in particular $l(b)=1$. Broué–Puig [4] have shown that, if $\chi$ has height 0, then

$$\begin{aligned} \lambda *\chi :=\sum _{(u,b)\in \mathcal {R}}\lambda (u)\chi ^{(u,b)}\in \textrm{Irr}(B) \end{aligned}$$

and $(\lambda *\chi )(1)=\lambda (1)\chi (1)$. Note also that $d_{\lambda *\chi ,\varphi }^u=\lambda (u)d^u_{\chi \varphi }$.

Proof of Theorem A

Let $\mathcal {R}$ be a set of representatives for the G-conjugacy classes of B-subsections $(u,b_u)\le (D,b_B)$ (see [22, p. 219]). Since B is nilpotent, we have $\textrm{IBr}(b_u)=\{\varphi _u\}$ for all $(u,b_u)\in \mathcal {R}$. Since the Brauer correspondence is compatible with complex conjugation, $(u,\overline{b_u})^t\le (D,\overline{b_D})^t=(D,b_D)$, where $\textrm{N}_G(D,b_D)^*/D\textrm{C}_G(D)=\langle tD\textrm{C}_G(D)\rangle$. Thus, $(u,\overline{b_u})^t$ is D-conjugate to some $(u',b_{u'})\in \mathcal {R}$.

If $p>2$, there exists a unique p-rational character $\chi _0\in \textrm{Irr}(B)$ of height 0, which must be real by uniqueness (see [4, Remark after Theorem 1.2]). If $p=2$, there is a 2-rational real character $\chi _0\in \textrm{Irr}(B)$ of height 0 by [8, Theorem 5.1]. Then $d_{\chi _0,\varphi _u}^u=d_{\chi _0,\overline{\varphi _u}}^u\in \mathbb {Z}$ and

$$\begin{aligned} \overline{\chi _0^{(u,b_u)}}=\chi _0^{(u,\overline{b_u})}=\chi _0^{(u,\overline{b_u})^t}=\chi _0^{(u',b_{u'})}. \end{aligned}$$

Now let $\lambda \in \textrm{Irr}(D)$. Then

$$\begin{aligned} \overline{\lambda *\chi _0}=\sum _{(u,b_u)\in \mathcal {R}}\overline{\lambda }(u)\overline{\chi _0^{(u,b_u)}}= \sum _{(u,b_u)\in \mathcal {R}}\overline{\lambda }(u)\chi _0^{(u',b_{u'})}. \end{aligned}$$

Since the class functions $\chi _0^{(u,b)}$ have disjoint support, they are linearly independent. Therefore, $\lambda *\chi _0$ is real if and only if $\lambda (u^t)=\lambda (u')=\overline{\lambda }(u)$ for all $(u,b_u)\in \mathcal {R}$. Since every conjugacy class of D is represented by some u with $(u,b_u)\in \mathcal {R}$, we conclude that $\lambda *\chi _0$ is real if and only $\lambda ^t=\overline{\lambda }$. Moreover, if $\lambda (1)=p^h$, then $\lambda *\chi _0$ has height h. This proves the first claim.

To prove the second claim, let $p>2$ and $\text {IBr}(B)=\{\varphi \}$. Then the decomposition numbers $d_{\lambda *\chi _0,\varphi }=\lambda (1)$ are powers of p; in particular they are odd. A theorem of Thompson and Willems (see [26, Theorem 2.8]) states that all real characters $\chi$ with $d_{\chi ,\varphi }$ odd have the same F-S indicator. So in our situation all real characters in $\text {Irr}(B)$ have the same F-S indicator.$\square$

Since the automorphism group of a p-group is “almost always” a p-group (see [11]), the following consequence is of interest.

Corollary 1

Let B be a real, nilpotent p-block with defect group D such that p and $|\text {Aut}(D)|$ are odd. Then B has a unique real character.

Proof

The hypothesis on $\text {Aut}(D)$ implies that $\text {N}_G(D,b_D)^*=D\text {C}_G(D)$. Hence by Theorem A, the number of real characters in $\text {Irr}(B)$ is the number of real characters in D. Since $p>2$, the trivial character is the only real character of D.$\square$

The next lemma is a consequence of Brauer’s second main theorem and the fact that $|\{g\in G:g^2=x\}|=|\{g\in \textrm{C}_G(x):g^2=x\}|$ is locally determined for $g,x\in G$.

Lemma 2

(Brauer) For every p-block B of G and every B-subsection (u, b) with $\varphi \in \textrm{IBr}(b)$ we have

$$\begin{aligned} \sum _{\chi \in \textrm{Irr}(B)}\epsilon (\chi )d^u_{\chi \varphi }=\sum _{\psi \in \textrm{Irr}(b)}\epsilon (\psi )d^u_{\psi \varphi }=\sum _{\psi \in \textrm{Irr}(b)}\epsilon (\psi )\frac{\psi (u)}{\psi (1)}d_{\psi \varphi }. \end{aligned}$$

If $l(b)=1$, then

$$\begin{aligned} \sum _{\chi \in \textrm{Irr}(B)}\epsilon (\chi )d^u_{\chi \varphi }=\frac{1}{\varphi (1)}\sum _{\psi \in \textrm{Irr}(b)}\epsilon (\psi )\psi (u). \end{aligned}$$

Proof

The first equality is [3, Theorem 4A]. The second follows from $u\in \textrm{Z}(\textrm{C}_G(u))$. If $l(b)=1$, then $\psi (1)=d_{\psi \varphi }\varphi (1)$ for $\psi \in \textrm{Irr}(b)$ and the last claim follows.$\square$

Recall that a canonical character of B is a character $\theta \in \textrm{Irr}(D\textrm{C}_G(D))$ lying in a Brauer correspondent of B such that $D\le \textrm{Ker}(\theta )$ (see [22, Theorem 9.12]). We define the extended stabilizer

$$\begin{aligned} \textrm{N}_G(D)_\theta ^*:=\left\{ g\in \textrm{N}_G(D):\theta ^g\in \{\theta ,\overline{\theta }\}\right\} . \end{aligned}$$

The following results adds some detail to the nilpotent case of [20, Theorem 1].

Theorem 3

Let B be a real, nilpotent p-block with cyclic defect group $D=\langle u\rangle$ and $p>2$. Let $\theta \in \textrm{Irr}(\textrm{C}_G(D))$ be a canonical character of B and set $T:=\textrm{N}_G(D)_\theta ^*$. Then one of the following holds:

• 1) $\overline{\theta }\ne \theta$. All characters in $\textrm{Irr}(B)$ are real with F-S indicator $\epsilon (\theta ^T)$.

• 2) $\overline{\theta }=\theta$. The unique non-exceptional character $\chi _0\in \textrm{Irr}(B)$ is the only real character in $\textrm{Irr}(B)$ and $\epsilon (\chi _0)=\textrm{sgn}(\chi _0(u))\epsilon (\theta )$, where $\textrm{sgn}(\chi _0(u))$ is the sign of $\chi _0(u)$.

Proof

Let $b_D$ be a Brauer correspondent of B in $\textrm{C}_G(D)$ containing $\theta$. Then $T=\textrm{N}_G(D,b_D)^*$. If $\overline{\theta }\ne \theta$, then T inverts the elements of D since $p>2$. Thus, Theorem A implies that all characters in $\textrm{Irr}(B)$ are real. By [20, Theorem 1(v)], the common F-S indicator is the Gow indicator of $\theta$ with respect to T. This is easily seen to be $\epsilon (\theta ^T)$ (see [20, after (2)]).

Now assume that $\overline{\theta }=\theta$. Here Theorem A implies that the unique p-rational character $\chi _0\in \textrm{Irr}(B)$ is the only real character. In particular, $\chi _0$ must be the unique non-exceptional character. Note that $(u,b_D)$ is a B-subsection and $\textrm{IBr}(b_D)=\{\varphi \}$. Since $\chi _0$ is p-rational, $d^u_{\chi _0\varphi }=\pm 1$. Since all Brauer correspondents of B in $\textrm{C}_G(u)$ are conjugate under $\textrm{N}_G(D)$, the generalized decomposition numbers are Galois conjugate, in particular $d^u_{\chi _0\varphi }$ does not depend on the choice of $b_D$. Hence,

$$\begin{aligned} \chi _0(u)=|\textrm{N}_G(D):\textrm{N}_G(D)_\theta |d^u_{\chi _0\varphi }\varphi (1) \end{aligned}$$

and $d^u_{\chi _0\varphi }=\textrm{sgn}(\chi _0(u))$. Moreover, $\theta$ is the unique non-exceptional character of $b_D$ and $\theta (u)=\theta (1)$. By Lemma 2, we obtain

$$\begin{aligned} \epsilon (\chi _0)= & {} \textrm{sgn}(\chi _0(u))\sum _{\chi \in \textrm{Irr}(B)}\epsilon (\chi )d^u_{\chi \varphi }\\= & {} \frac{\textrm{sgn}(\chi _0(u))}{\varphi (1)}\sum _{\psi \in \textrm{Irr}(b_D)}\epsilon (\psi )\psi (u)=\textrm{sgn}(\chi _0(u))\epsilon (\theta ). \end{aligned}$$

$\square$

If B is a nilpotent block with canonical character $\theta \ne \overline{\theta }$, the common F-S indicator of the real characters in $\textrm{Irr}(B)$ is not always $\epsilon (\theta ^T)$ as in Theorem 3 . A counterexample is given by a certain 3-block of $G=\texttt{SmallGroup}(288,924)$ with defect group $D\cong C_3\times C_3$.

We now restrict ourselves to 2-blocks. Héthelyi–Horváth–Szabó [12] introduced four conjectures, which are real versions of Brauer’s conjecture, Olsson’s conjecture and Eaton’s conjecture. We only state the strongest of them, which implies the remaining three. Let $D^{(0)}:=D$ and $D^{(k+1)}:=[D^{(k)},D^{(k)}]$ for $k\ge 0$ be the members of the derived series of D.

Conjecture 4

(Héthelyi–Horváth–Szabó) Let B be a 2-block with defect group D. For every $h\ge 0$, the number of real characters in $\textrm{Irr}(B)$ of height $\le h$ is bounded by the number of elements of $D/D^{(h+1)}$ which are real in $\textrm{N}_G(D)/D^{(h+1)}$.

A conjugacy class K of G is called real if $K=K^{-1}:=\{x^{-1}:x\in K\}$. A conjugacy class K of a normal subgroup $N\unlhd G$ is called real under G if there exists $g\in G$ such that $K^g=K^{-1}$.

Proposition 5

Let B be a nilpotent 2-block with defect group D and Brauer correspondent $b_D$ in $D\textrm{C}_G(D)$. Then the number of real characters in $\textrm{Irr}(B)$ of height $\le h$ is bounded by the number of conjugacy classes of $D/D^{(h+1)}$ which are real under $\textrm{N}_G(D,b_D)^*/D^{(h+1)}$. In particular, Conjecture 4 holds for B.

Proof

We may assume that B is real. As in the proof of Theorem A, we fix some 2-rational real character $\chi _0\in \textrm{Irr}(B)$ of height 0. Now $\lambda *\chi _0$ has height $\le h$ if and only if $\lambda (1)\le p^h$ for $\lambda \in \textrm{Irr}(B)$. By [14, Theorem 5.12], the characters of degree $\le p^h$ in $\textrm{Irr}(D)$ lie in $\textrm{Irr}(D/D^{(h+1)})$. By Theorem A, $\lambda *\chi _0$ is real if and only if $\lambda ^t=\overline{\lambda }$. By Brauer’s permutation lemma (see [23, Theorem 2.3]), the number of those characters $\lambda$ coincides with the number of conjugacy classes K of $D/D^{(h+1)}$ such that $K^t=K^{-1}$. Now Conjecture 4 follows from $\textrm{N}_G(D,b_D)^*\le \textrm{N}_G(D)$.$\square$

3 Extended Defect Groups

We continue to assume that $p=2$. As usual we choose a complete discrete valuation ring $\mathcal {O}$ such that $F:=\mathcal {O}/J(\mathcal {O})$ is an algebraically closed field of characteristic 2. Let $\textrm{Cl}(G)$ be the set of conjugacy classes of G. For $K\in \textrm{Cl}(G)$ let $K^+:=\sum _{x\in K}x\in \textrm{Z}(FG)$ be the class sum of K. We fix a 2-block B of FG with block idempotent $1_B=\sum _{K\in \textrm{Cl}(G)}a_KK^+$, where $a_K\in F$. The central character of B is defined by

$$\begin{aligned} \lambda _B~:~\textrm{Z}(FG)\rightarrow F,\quad K^+\mapsto \left( \frac{|K|\chi (g)}{\chi (1)}\right) ^*, \end{aligned}$$

where $g\in K$, $\chi \in \textrm{Irr}(B)$ and $^*$ denotes the canonical reduction $\mathcal {O}\rightarrow F$ (see [22, Chapter 2]).

Since $\lambda _B(1_B)=1$, there exists $K\in \textrm{Cl}(G)$ such that $a_K\ne 0\ne \lambda _B(K^+)$. We call K a defect class of B. By [22, Corollary 3.8], K consists of elements of odd order. According to [22, Corollary 4.5], a Sylow 2-subgroup D of $\textrm{C}_G(x)$, where $x\in K$, is a defect group of B. For $x\in K$ let

$$\begin{aligned} \textrm{C}_G(x)^*:=\{g\in G:gxg^{-1}=x^{\pm 1}\}\le G \end{aligned}$$

be the extended centralizer of x.

Proposition 6

(Gow, Murray) Every real 2-block B has a real defect class K. Let $x\in K$. Choose a Sylow 2-subgroup E of $\textrm{C}_G(x)^*$ and put $D:=E\cap \textrm{C}_G(x)$. Then the G-conjugacy class of the pair (D, E) does not depend on the choice of K or x.

Proof

For the principal block (which is always real since it contains the trivial character), $K=\{1\}$ is a real defect class and $E=D$ is a Sylow 2-subgroup of G. Hence, the uniqueness follows from Sylow’s theorem. Now suppose that B is non-principal. The existence of K was first shown in [8, Theorem 5.5]. Let L be another real defect class of B and choose $y\in L$. By [9, Corollary 2.2], we may assume after conjugation that E is also a Sylow 2-subgroup of $\textrm{C}_G(y)^*$. Let $D_x:=E\cap \textrm{C}_G(x)$ and $D_y:=E\cap \textrm{C}_G(y)$. We may assume that $|E:D_x|=2=|E:D_y|$ (cf. the remark after the proof).

We now introduce some notation in order to apply [17, Proposition 14]. Let $\Sigma =\langle \sigma \rangle \cong C_2$. We consider FG as an $F[G\times \Sigma ]$-module, where G acts by conjugation and $g^\sigma =g^{-1}$ for $g\in G$ (observe that these actions indeed commute). For $H\le G\times \Sigma$ let

$$\begin{aligned} \textrm{Tr}^{G\times \Sigma }_H~:~(FG)^H\rightarrow (FG)^{G\times \Sigma },\quad \alpha \mapsto \sum _{x\in \mathcal {R}}\alpha ^x \end{aligned}$$

be the relative trace with respect to H, where $\mathcal {R}$ denotes a set of representatives of the right cosets of H in $G\times \Sigma$. By [17, Proposition 14], we have $1_B\in \textrm{Tr}^{G\times \Sigma }_{E_x}(FG)$, where $E_x:=D_x\langle e_x\sigma \rangle$ for some $e_x\in E\setminus D_x$. By the same result we also obtain that $D_y\langle e_y\sigma \rangle$ with $e_y\in E\setminus D_y$ is G-conjugate to $E_x$. This implies that $D_y$ is conjugate to $D_x$ inside $\textrm{N}_G(E)$. In particular, $(D_x,E)$ and $(D_y,E)$ are G-conjugate as desired.$\square$

Definition 7

In the situation of Proposition 6 we call E an extended defect group and (D, E) a defect pair of B.

We stress that real 2-blocks can have non-real defect classes and non-real blocks can have real defect classes (see [10, Theorem 3.5]).

It is easy to show that non-principal real 2-blocks cannot have maximal defect (see [22, Problem 3.8]). In particular, the trivial class cannot be a defect class and consequently, $|E:D|=2$ in those cases. For non-real blocks we define the extended defect group by $E:=D$ for convenience. Every given pair of 2-groups $D\le E$ with $|E:D|=2$ occurs as a defect pair of a real (nilpotent) block. To see this, let $Q\cong C_3$ and $G=Q\rtimes E$ with $\textrm{C}_E(Q)=D$. Then G has a unique non-principal block with defect pair (D, E).

We recall from [14, p. 49] that

$$\begin{aligned} \sum _{\chi \in \textrm{Irr}(G)}\epsilon (\chi )\chi (g)=|\{x\in G:x^2=g\}| \end{aligned}$$

(3)

for all $g\in G$. The following proposition provides some interesting properties of defect pairs.

Proposition 8

(Gow, Murray) Let B be a real 2-block with defect pair (D, E). Let $b_D$ be a Brauer correspondent of B in $D\textrm{C}_G(D)$. Then the following holds:

• (i) $\textrm{N}_G(D,b_D)^*=\textrm{N}_G(D,b_D)E$. In particular, $b_D$ is real if and only if $E=D\textrm{C}_E(D)$.

• (ii) For $u\in D$, we have $\sum _{\chi \in \textrm{Irr}(B)}\epsilon (\chi )\chi (u)\ge 0$ with strict inequality if and only if u is G-conjugate to $e^2$ for some $e\in E\setminus D$. In particular, E splits over D if and only if $\sum _{\chi \in \textrm{Irr}(B)}\epsilon (\chi )\chi (1)>0$.

• (iii) $E/D'$ splits over $D/D'$ if and only if all height zero characters in $\textrm{Irr}(B)$ have non-negative F-S indicator.

Proof

• (i) See [19, Lemma 1.8] and [18, Theorem 1.4].

• (ii) See [19, Lemma 1.3].

• (iii) See [8, Theorem 5.6].$\square$

The next proposition extends [18, Lemma 1.3].

Corollary 9

Suppose that B is a 2-block with defect pair (D, E) where D is abelian. Then E splits over D if and only if all characters in $\textrm{Irr}(B)$ have non-negative F-S indicator.

Proof

If B is non-real, then $E=D$ splits over D and all characters in $\textrm{Irr}(B)$ have F-S indicator 0. Hence, let $\overline{B}=B$. By Kessar–Malle [15], all characters in $\textrm{Irr}(B)$ have height 0. Hence, the claim follows from Proposition 8 (iii).$\square$

Theorem 10

Let B be a real, nilpotent 2-block with defect pair (D, E), where D is abelian. If E splits over D, then all real characters in $\textrm{Irr}(B)$ have F-S indicator 1. Otherwise exactly half of the real characters have F-S indicator 1. In either case, Conjecture B holds for B.

Proof

If E splits over D, then all real characters in $\textrm{Irr}(B)$ have F-S indicator 1 by Corollary 9. Otherwise we have $\sum _{\chi \in \textrm{Irr}(B)}\epsilon (\chi )=0$ by Proposition 8(ii), because all characters in $\textrm{Irr}(B)$ have the same degree. Hence, exactly half of the real characters have F-S indicator 1. Using Theorem A we can determine the number of characters for each F-S indicator. For the last claim, we may therefore replace B by the unique non-principal block of $G=Q\rtimes E$, where $Q\cong C_3$ and $\textrm{C}_E(Q)=D$ (mentioned above). In this case Conjecture B follows from Gow [8, Lemma 2.2] or Theorem E.$\square$

Example 11

Let B be a real block with defect group $D\cong C_4\times C_2$. Then B is nilpotent since $\textrm{Aut}(D)$ is a 2-group and D is abelian. Moreover $|\textrm{Irr}(B)|=8$. The F-S indicators depend not only on E, but also on the way D embeds into E. The following cases can occur (here $M_{16}$ denotes the modular group and [16, 3] refers to the small group library):

$$\begin{aligned} \begin{array}{ll} \text {F-S indicators}&{}~E\\ \hline ++++++++&{}~D_8\times C_2\\ ++++----&{}~Q_8\times C_2,~C_4\rtimes C_4\text { with }\Phi (D)=E'\\ ++++\,0~~0~\,0~0&{}~D,~D\times C_2,~D_8*C_4,~ [16,3]\\ ++--\,0~~0~\,0~0&{}~C_4^2,~C_8\times C_2,~M_{16},~C_4\rtimes C_4\text { with }\Phi (D)\ne E' \end{array} \end{aligned}$$

The F-S indicator $\epsilon (\Phi )$ appearing in Conjecture C has an interesting interpretation as follows. Let $\Omega :=\{g\in G:g^2=1\}$. The conjugation action of G on $\Omega$ turns $F\Omega$ into an FG-module, called the involution module.

Lemma 12

(Murray) Let B be a real 2-block and $\varphi \in \textrm{IBr}(B)$. Then $\epsilon (\Phi _\varphi )$ is the multiplicity of $\varphi$ as a constituent of the Brauer character of $F\Omega$.

Proof

See [18, Lemma 2.6].$\square$

Next we develop a local version of Conjecture C. Let B be a real 2-block with defect pair (D, E) and B-subsection (u, b). If $E=D\textrm{C}_E(u)$, then b is real and $(\textrm{C}_D(u),\textrm{C}_E(u))$ is a defect pair of b by [19, Lemma 2.6] applied to the subpair $(\langle u\rangle ,b)$. Conversely, if b is real, we may assume that $(\textrm{C}_D(u),\textrm{C}_E(u))$ is a defect pair of b by [19, Theorem 2.7]. If b is non-real, we may assume that $(\textrm{C}_D(u),\textrm{C}_D(u))=(\textrm{C}_D(u),\textrm{C}_E(u))$ is a defect pair of b.

Theorem 13

Let B be 2-block of a finite group G with defect pair (D, E). Suppose that Conjecture C holds for all Brauer correspondents of B in sections ofG. Let (u, b) be a B-subsection with defect pair $({\text {C}}_{D}(u), {\text {C}}_{E}(u))$ such that $\text {IBr}(b)=\{\varphi \}$. Then

$$\begin{aligned} \sum _{\chi \in \textrm{Irr}(B)}\epsilon (\chi )d^u_{\chi \varphi }= \left\{ \begin{array}{ll} |\{x\in D:x^2=u\}|&{}\quad if~ {B}~ is~ the ~principal ~block ,\\ |\{x\in E\setminus D:x^2=u\}|&{}\quad otherwise . \end{array}\right. \end{aligned}$$

Proof

If B is not real, then B is non-principal and $E=D$. It follows that $\epsilon (\chi )=0$ for all $\chi \in \textrm{Irr}(B)$ and

$$\begin{aligned} |\{x\in E\setminus D:x^2=u\}|=0. \end{aligned}$$

Hence, we may assume that B is real. By Lemma 2, we have

$$\begin{aligned} \sum _{\chi \in \textrm{Irr}(B)}\epsilon (\chi )d^u_{\chi \varphi }=\sum _{\psi \in \textrm{Irr}(b)}\epsilon (\psi )d^u_{\psi \varphi }=\frac{1}{\varphi (1)}\sum _{\psi \in \textrm{Irr}(b)}\epsilon (\psi )\psi (u). \end{aligned}$$

(4)

Suppose that B is the principal block. Then b is the principal block of $\textrm{C}_G(u)$ by Brauer’s third main theorem (see [22, Theorem 6.7]). The hypothesis $l(b)=1$ implies that $\varphi =1_{\textrm{C}_G(u)}$ and $\textrm{C}_G(u)$ has a normal 2-complement N (see [22, Corollary 6.13]). It follows that $\textrm{Irr}(b)=\textrm{Irr}(\textrm{C}_G(u)/N)=\textrm{Irr}(\textrm{C}_D(u))$ and

$$\begin{aligned} \sum _{\psi \in \textrm{Irr}(b)}\epsilon (\psi )d^u_{\psi \varphi }=\sum _{\lambda \in \textrm{Irr}(\textrm{C}_D(u))}\epsilon (\lambda )\lambda (u)=|\{x\in \textrm{C}_D(u):x^2=u\}| \end{aligned}$$

by (3). Since every $x\in D$ with $x^2=u$ lies in $\textrm{C}_D(u)$, we are done in this case.

Now let B be a non-principal real 2-block. If b is not real, then (4) shows that $\sum _{\chi \in \textrm{Irr}(B)}\epsilon (\chi )d^u_{\chi \varphi }=0$. On the other hand, we have $\textrm{C}_E(u)=\textrm{C}_D(u)\le D$ and $|\{x\in E\setminus D:x^2=u\}|=0$. Hence, we may assume that b is real. Since every $x\in E$ with $x^2=u$ lies in $\textrm{C}_E(u)$, we may assume that $u\in \textrm{Z}(G)$ by (4).

Then $\chi (u)=d^u_{\chi \varphi }\varphi (1)$ for all $\chi \in \textrm{Irr}(B)$. If $u^2\notin \textrm{Ker}(\chi )$, then $\chi (u)\notin \mathbb {R}$ and $\epsilon (\chi )=0$. Thus, it suffices to sum over $\chi$ with $d^u_{\chi \varphi }=\pm d_{\chi \varphi }$. Let $Z:=\langle u\rangle \le \textrm{Z}(G)$ and $\overline{G}:=G/Z$. Let $\hat{B}$ be the unique (real) block of $\overline{G}$ dominated by B. By [19, Lemma 1.7], $(\overline{D},\overline{E})$ is a defect pair for $\hat{B}$. Then, using [14, Lemma 4.7] and Conjecture C for B and $\hat{B}$, we obtain

$$\begin{aligned} \sum _{\chi \in \textrm{Irr}(B)}\epsilon (\chi )d^u_{\chi \varphi }= & {} \sum _{\chi \in \textrm{Irr}(B)}\epsilon (\chi )(d_{\chi \varphi }+d^u_{\chi \varphi })-\sum _{\chi \in \textrm{Irr}(B)}\epsilon (\chi )d_{\chi \varphi }\\= & {} 2\sum _{\chi \in \textrm{Irr}(\hat{B})}\epsilon (\chi )d_{\chi \varphi }-\sum _{\chi \in \textrm{Irr}(B)}\epsilon (\chi )d_{\chi \varphi }\\= & {} 2|\{\overline{x}\in \overline{E}\setminus \overline{D}:\overline{x}^2=1\}|-|\{x\in E\setminus D:x^2=1\}|\\= & {} \sum _{\lambda \in \textrm{Irr}(E)}\epsilon (\lambda )(\lambda (1)+\lambda (u))-\sum _{\lambda \in \textrm{Irr}(D)}\epsilon (\lambda )(\lambda (1)+\lambda (u))\\{} & {} -\sum _{\lambda \in \textrm{Irr}(E)}\epsilon (\lambda )\lambda (1)+\sum _{\lambda \in \textrm{Irr}(D)}\epsilon (\lambda )\lambda (1)\\= & {} \sum _{\lambda \in \textrm{Irr}(E)}\epsilon (\lambda )\lambda (u)-\sum _{\lambda \in \textrm{Irr}(D)}\epsilon (\lambda )\lambda (u)=|\{x\in E\setminus D:x^2=u\}|. \end{aligned}$$

$\square$

4 Theorems D and E

The following result implies Theorem D.

Theorem 14

Suppose thatB is a real, nilpotent, non-principal 2-block fulfilling the statement of Theorem 13. Then Conjecture B holds for B.

Proof

Let (D, E) be defect pair of B. By Gow [8, Theorem 5.1], there exists a 2-rational character $\chi _0\in \textrm{Irr}(B)$ of height 0 and $\epsilon (\chi _0)=1$. Let

$$\begin{aligned} \Gamma :\textrm{Irr}(D)\rightarrow \textrm{Irr}(B),\qquad \lambda \mapsto \lambda *\chi _0 \end{aligned}$$

be the Broué–Puig bijection. Let $(u_1,b_1),\ldots ,(u_k,b_k)$ be representatives for the conjugacy classes of B-subsections. Since B is nilpotent, we may assume that $u_1,\ldots ,u_k\in D$ represent the conjugacy classes of D. Let $\textrm{IBr}(b_i)=\{\varphi _i\}$ for $i=1,\ldots ,k$. Since $\chi _0$ is 2-rational, we have $\sigma _i:=d^u_{\chi _0,\varphi _i}\in \{\pm 1\}$ for $i=1,\ldots ,k$. Hence, the generalized decomposition matrix of B has the form

$$\begin{aligned} Q=(\lambda (u_i)\sigma _i:\lambda \in \textrm{Irr}(D),i=1,\ldots ,k) \end{aligned}$$

(see [16, Section 8.10]). Let $v:=(\epsilon (\Gamma (\lambda )):\lambda \in \textrm{Irr}(D))$ and $w:=(w_1,\ldots ,w_k)$ where $w_i:=|\{x\in E\setminus D:x^2=u_i\}|$. Then Theorem 13 reads as $vQ=w$.

Let $d_i:=|\textrm{C}_D(u_i)|$ and $d=(d_1,\ldots ,d_k)$. Then the second orthogonality relation yields $Q^\textrm{t}\overline{Q}=\textrm{diag}(d)$, where $Q^\textrm{t}$ denotes the transpose of Q. It follows that $Q^{-1}=\textrm{diag}(d)^{-1}\overline{Q}^\textrm{t}$ and

$$\begin{aligned} v=w\textrm{diag}(d)^{-1}\overline{Q}^\textrm{t}=w\textrm{diag}(d)^{-1}Q^\textrm{t}, \end{aligned}$$

because $\overline{v}=v$. Since $w_i=|\{x\in E\setminus D:x^2=u_i^y\}|$ for every $y\in D$, we obtain $\sum _{i=1}^kw_i|D:\textrm{C}_D(u_i)|=|E\setminus D|=|D|$. In particular,

$$\begin{aligned} 1=\epsilon (\chi _0)=\sum _{i=1}^k\frac{w_i\sigma _i}{|\textrm{C}_D(u_i)|}\le \sum _{i=1}^k\frac{w_i|\sigma _i|}{|\textrm{C}_D(u_i)|}=1. \end{aligned}$$

Therefore, $\sigma _i=1$ or $w_i=0$ for each i. This means that the signs $\sigma _i$ have no impact on the solution of the linear system $xQ=w$. Hence, we may assume that $Q=(\lambda (u_i))$ is just the character table of D. Since Q has full rank, v is the only solution of $xQ=w$. Setting $\mu (\lambda ):=\frac{1}{|D|}\sum _{e\in E\setminus D}\lambda (e^2)$, it suffices to show that $(\mu (\lambda ):\lambda \in \textrm{Irr}(D))$ is another solution of $xQ=w$. Indeed,

$$\begin{aligned} \sum _{\lambda \in \textrm{Irr}(D)}\frac{\lambda (u_i)}{|D|}\sum _{e\in E\setminus D}\lambda (e^2)= & {} \frac{1}{|D|}\sum _{e\in E\setminus D}\sum _{\lambda \in \textrm{Irr}(D)}\lambda (u_i)\lambda (e^2)\\= & {} \frac{1}{|D|}\sum _{\begin{array}{c} e\in E\setminus D\\ e^2=u_i^{-1} \end{array}}|D:\textrm{C}_D(u_i)||\textrm{C}_D(u_i)|=w_i \end{aligned}$$

for $i=1,\ldots ,k$. $\square$

Theorem E

Conjectures B and C hold for all nilpotent 2-blocks of solvable groups.

Proof

Let B be a real, nilpotent, non-principal 2-block of a solvable group G with defect pair (D, E). We first prove Conjecture C for B. Since all sections of G are solvable and all blocks dominated by B-subsections are nilpotent, Conjecture C holds for those blocks as well. Hence, the hypothesis of Theorem 13 is fulfilled for B. Now by Theorem 14, Conjecture B holds for B.

Let $N:=\textrm{O}_{2'}(G)$ and let $\theta \in \textrm{Irr}(N)$ such that the block $\{\theta \}$ is covered by B. Since B is non-principal, $\theta \ne 1_N$ and therefore $\overline{\theta }\ne \theta$ as N has odd order. Since B also lies over $\overline{\theta }$, it follow that $G_\theta <G$. Let b be the Fong–Reynolds correspondent of B in the extended stabilizer $G_\theta ^*$. By [22, Theorem 9.14] and [20, p. 94], the Clifford correspondence $\textrm{Irr}(b)\rightarrow \textrm{Irr}(B)$, $\psi \mapsto \psi ^G$ preserves decomposition numbers and F-S indicators. Thus, we need to show that b has defect pair (D, E). Let $\beta$ be the Fong–Reynolds correspondent of B in $G_\theta$. By [22, Theorem 10.20], $\beta$ is the unique block over $\theta$. In particular, the block idempotents $1_\beta =1_\theta$ are the same (we identify $\theta$ with the block $\{\theta \}$). Since b is also the unique block of $G_\theta ^*$ over $\theta$, we have $1_b=1_\theta +1_{\overline{\theta }}=\sum _{x\in N}\alpha _xx$ for some $\alpha _x\in F$. Let S be a set of representatives for the cosets $G/G_\theta ^*$. Then

$$\begin{aligned} 1_B=\sum _{s\in S}(1_\theta +1_{\overline{\theta }})^s=\sum _{s\in S}1_b^s=\sum _{g\in N}\left( \sum _{s\in S}\alpha _{g^{s^{-1}}}\right) g. \end{aligned}$$

Hence, there exists a real defect class K of B such that $\alpha _{g^{s^{-1}}}\ne 0$ for some $g\in K$ and $s\in S$. Of course we can assume that $g=g^{s^{-1}}$. Then $1_b$ does not vanish on g. By [22, Theorem 9.1], the central characters $\lambda _B$, $\lambda _b$ and $\lambda _\theta$ agree on N. It follows that K is also a real defect class of b. Hence, we may assume that (D, E) is a defect pair of b.

It remains to consider $G=G_\theta ^*$ and $B=b$. Then D is a Sylow 2-subgroup of $G_\theta$ by [22, Theorem 10.20] and E is a Sylow 2-subgroup of G. Since $|G:G_\theta |=2$, it follows that $G_\theta \unlhd G$ and $N=\textrm{O}_{2'}(G_\theta )$. By [21, Lemmas 1 and 2], $\beta$ is nilpotent and $G_\theta$ is 2-nilpotent, i.e. $G_\theta =N\rtimes D$ and $G=N\rtimes E$. Let $\widetilde{\Phi }:=\sum _{\chi \in \textrm{Irr}(B)}\chi (1)\chi =\varphi (1)\Phi$, where $\textrm{IBr}(B)=\{\varphi \}$. We need to show that

$$\begin{aligned} \epsilon (\widetilde{\Phi })=\varphi (1)|\{x\in E\setminus D:x^2=1\}|. \end{aligned}$$

Note that $\chi _N=\frac{\chi (1)}{2\theta (1)}(\theta +\overline{\theta })$. By Frobenius reciprocity, it follows that $\widetilde{\Phi }=2\theta (1)\theta ^G$ and

$$\begin{aligned} \widetilde{\Phi }_N=|G:N|\theta (1)(\theta +\overline{\theta }). \end{aligned}$$

Since $\Phi$ vanishes on elements of even order, $\widetilde{\Phi }$ vanishes outside N. Since $\widetilde{\Phi }_{G_\theta }$ is a sum of non-real characters in $\beta$, we have

$$\begin{aligned} \epsilon (\widetilde{\Phi })=\frac{1}{|G|}\sum _{g\in G_\theta }\widetilde{\Phi }(g^2)+\frac{1}{|G|}\sum _{g\in G\setminus G_\theta }\widetilde{\Phi }(g^2)=\frac{1}{|G|}\sum _{g\in G\setminus G_\theta }\widetilde{\Phi }(g^2). \end{aligned}$$

Every $g\in G\setminus G_\theta =NE\setminus ND$ with $g^2\in N$ is N-conjugate to a unique element of the form xy, where $x\in E\setminus D$ is an involution and $y\in \textrm{C}_N(x)$ (Sylow’s theorem). Setting $\Delta :=\{x\in E\setminus D:x^2=1\}$, we obtain

$$\begin{aligned} \epsilon (\widetilde{\Phi })=\frac{\theta (1)}{|N|}\sum _{x\in \Delta }|N:\textrm{C}_N(x)|\sum _{y\in \textrm{C}_N(x)}(\theta (y)+\overline{\theta (y)})=2\theta (1)\sum _{x\in \Delta }\frac{1}{|\textrm{C}_N(x)|}\sum _{y\in \textrm{C}_N(x)}\theta (y). \end{aligned}$$

(5)

For $x\in \Delta$ let $H_x:=N\langle x\rangle$. Again by Sylow’s theorem, the N-orbit of x is the set of involutions in $H_x$. From $\theta ^x=\overline{\theta }$ we see that $\theta ^{H_x}$ is an irreducible character of 2-defect 0. By [8, Theorem 5.1], we have $\epsilon (\theta ^{H_x})=1$. Now applying the same argument as before, it follows that

$$\begin{aligned} 1=\epsilon (\theta ^{H_x})=\frac{1}{|N|}\sum _{g\in H_x\setminus N}\theta ^{H_x}(g^2)=\frac{2}{|\textrm{C}_N(x)|}\sum _{y\in \textrm{C}_N(x)}\theta (y). \end{aligned}$$

Combined with (5), this yields $\epsilon (\widetilde{\Phi })=2\theta (1)|\Delta |$. By Green’s theorem (see [22, Theorem 8.11]), $\varphi _N=\theta +\overline{\theta }$ and $\epsilon (\widetilde{\Phi })=\varphi (1)|\Delta |$ as desired.$\square$

For non-principal blocks B of solvable groups with $l(B)=1$ it is not true in general that $G_\theta$ is 2-nilpotent in the situation of Theorem E. For example, a (non-real) 2-block of a triple cover of $A_4\times A_4$ has a unique simple module. Extending this group by an automorphism of order 2, we obtain the group $G=\texttt{SmallGroup}(864,3988)$, which fulfills the assumptions with $D\cong C_2^4$, $N\cong C_3$ and $|G:NE|=9$.

In order to prove Conjecture C for arbitrary 2-blocks of solvable groups, we may follow the steps in the proof above until E is a Sylow 2-subgroup of G and $|G:G_\theta |=2$. By [24, Theorem 2.1], one gets

$$\begin{aligned} \varphi (1)/\theta (1)=2\sqrt{|G_\theta /N|_{2'}}=\sqrt{|G:EN|}. \end{aligned}$$

With some more effort, the claim then boils down to a purely group-theoretical statement:

Let B be a real, non-principal 2-block of a solvable group G with defect pair (D, E) and $l(B)=1$. Let $N:=\textrm{O}_{2'}(G)$ and $\overline{G}:=G/N$. Let $\theta \in \textrm{Irr}(N)$ such $\{\theta \}$ is covered by B. Then

$$\begin{aligned} |\{\overline{x}\in \overline{G}\setminus \overline{G_\theta }:\overline{x}^2=1\}|=|\{x\in E\setminus D:x^2=1\}|\sqrt{|G:EN|}. \end{aligned}$$

Unfortunately, I am unable to prove this.