# Weights for compact connected Lie groups

Radha Kessar  Department of Mathematics, University of Manchester, United
Kingdom<EMAIL_ADDRESS>,  Gunter Malle  FB Mathematik, TU
Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany.
<EMAIL_ADDRESS>and  Jason Semeraro  Heilbronn Institute for
Mathematical Research, Department of Mathematics, University of Leicester,
United Kingdom<EMAIL_ADDRESS>
###### Abstract.

Let $\ell$ be a prime. If $\mathbf{G}$ is a compact connected Lie group, or a
connected reductive algebraic group in characteristic different from $\ell$,
and $\ell$ is a good prime for $\mathbf{G}$, we show that the number of
weights of the $\ell$-fusion system of $\mathbf{G}$ is equal to the number of
irreducible characters of its Weyl group. The proof relies on the
classification of $\ell$-stubborn subgroups in compact Lie groups.

###### Key words and phrases:

fusion systems, compact Lie groups, Alperin weight conjecture

###### 2010 Mathematics Subject Classification:

20C20, 55R35, 57T10

We thank Bob Oliver for illuminating conversations. The second author
gratefully acknowledges support by the DFG— Project-ID 286237555 – TRR 195.

##  1\. Introduction

In this paper we prove the following identity, first conjectured in [15]:

######  Theorem 1.

Suppose that $\mathbf{G}$ is a compact connected Lie group. Let $\ell$ be a
prime which is good for $\mathbf{G}$, $S$ a Sylow $\ell$-subgroup of
$\mathbf{G}$, $\mathcal{F}=\mathcal{F}_{S}(\mathbf{G})$ the corresponding
saturated fusion system and $W$ the Weyl group of $\mathbf{G}$. Then,

$\mathbf{w}(\mathcal{F})=|{{\operatorname{Irr}}}(W)|.$

Here $\mathbf{w}(\mathcal{F})$ denotes the number of weights of the
$\ell$-fusion system attached to $\mathbf{G}$ (see Section 2 for precise
definitions) and ${{\operatorname{Irr}}}(W)$ denotes the set of ordinary
irreducible characters of $W$. We recall the definition of good primes later
in this introduction. While the stated identity is rather easy to see for
primes not dividing $|W|$, it seems totally mysterious for the other primes.

The origin of our interest in this theorem is Alperin’s Weight Conjecture
(AWC) in finite group representation theory which relates the number of simple
modules of a block of a finite group algebra over a field of characteristic
$\ell$ to the number of simple, projective modules of smaller “local” groups.
Indeed, via some deep representation theory, it can be shown that for the
principal block of the group algebra of a finite group of Lie type whose
defining characteristic is different from $\ell$, and under some extra
conditions, AWC is equivalent to an equation of the above type (see [10, Prop.
4.1]). In [10, Thm 1], we proved this equality for finite groups of Lie type
(under the relevant conditions) and made the surprising discovery that it
continues to hold in a broader setting than that of finite groups, in
particular for certain exotic fusion systems on finite $\ell$-groups arising
from homotopy fixed point spaces of connected $\ell$-compact groups. We view
Theorem 1, where the underlying $\ell$-group is an infinite group, as
additional evidence for AWC and a further indication that it is not restricted
to the world of finite groups. We also note that Theorem 1 yields the
analogous identity for $\ell$-weights of fusion systems of connected reductive
algebraic groups in characteristic different from $\ell$:

######  Theorem 2.

Let $\ell$ and $p$ be distinct primes. Suppose that $\mathbf{G}$ is a
connected reductive algebraic group over $\overline{{\mathbb{F}}}_{p}$ such
that $\ell$ is good for $\mathbf{G}$, let $S$ be a Sylow $\ell$-subgroup of
$\mathbf{G}$, $\mathcal{F}=\mathcal{F}_{S}(\mathbf{G})$ the corresponding
saturated fusion system and $W$ the Weyl group of $\mathbf{G}$. Then,

$\mathbf{w}(\mathcal{F})=|{{\operatorname{Irr}}}(W)|.$

The proof of Theorem 1 proceeds by transferring from the discrete setting to
the continuous one in which the role of centric, radical subgroups in fusion
systems is taken over by $\ell$-stubborn subgroups (in the sense of
Jackowski–McClure–Oliver [8]) of compact Lie groups. This leads to a
straightforward reduction to the case that the underlying compact group is
simple, at which stage we are able to invoke known classifications of
$\ell$-stubborn subgroups [13], [9], [16]. The proof for the classical groups
involves counting arguments similar to those used for [10, Thm 1] which
originated in the work of Alperin–Fong [1].

Recall that a prime $\ell$ is _bad_ for a root system $\Phi$ if
${\mathbb{Z}}\Phi/{\mathbb{Z}}\Psi$ has $\ell$-torsion for some closed
subsystem $\Psi\subset\Phi$ (see e.g. [12, Def. B.24]). Thus, it is bad for
$\Phi$ if it is so for one of its indecomposable summands. The bad primes for
indecomposable root systems are: none for type $A_{n}$, $\ell=2$ for
$B_{n},C_{n}$ and $D_{n}$, $\ell=2,3$ for $G_{2},F_{4},E_{6},E_{7}$, and
$\ell=2,3,5$ for $E_{8}$ (see [12, Tab. 14.1]). In particular, all primes
$\ell\geq 7$ are good (that is, non bad) for all types. In Proposition 4.1 we
show that the conclusion of Theorem 1 does not hold for the compact symplectic
groups at the bad prime $\ell=2$, and in Proposition 4.5 that it does not hold
for $\mathbf{G}=F_{4}$ at the bad prime $\ell=3$. Still in both cases we
obtain $\mathbf{w}(\mathcal{F})\leq|{{\operatorname{Irr}}}(W)|$, as
conjectured in [15]. In Example 4.3 we observe that for finite reductive
groups in bad characteristic we may have
$\mathbf{w}(\mathcal{F})>|{{\operatorname{Irr}}}(W)|$.

Recall that by results of Friedlander (see [4, Thm 3.1] and Section 2.3), the
$\ell$-completed classifying space (and therefore the $\ell$-fusion system) of
a finite group of Lie type in non-describing characteristic can be recovered
by taking homotopy fixed points under suitable unstable Adams operations on
the $\ell$-completed classifying space of the corresponding compact Lie group.
It would be desirable to obtain a direct connection between the weight
equation demonstrated in Theorems 1 and 2 and that shown in [10, Thm 1], but
as yet we do not see such a connection either via unstable Adams operations or
via Frobenius morphisms on algebraic groups. In particular, we do not see how
to deduce the equation in the finite setting from the infinite one or vice-
versa. Note that our results show that the number of weights for compact
groups does not depend on the isogeny type, while for example the two
isogenous groups $\operatorname{SL}_{3}(4)$ and $\operatorname{PGL}_{3}(4)$
have five respectively three 3-weights. In another direction, we believe that
the purview of Theorem 1 could be expanded to also cover fusion systems of
$\ell$-compact groups as defined in [5, Sec. 10].

The paper is organised as follows. In Section 2, we recall the relevant
background material. Section 3 contains the proofs of Theorems 1 and 2 and in
Section 4, we present our calculations for bad primes.

##  2\. Preliminaries

In this section we recall some aspects of the theory of saturated fusion
systems associated to compact Lie groups as set up by Broto, Levi and Oliver
in [5]. For subgroups $Q,R$ of a group $G$, ${{\operatorname{Hom}}}_{G}(Q,R)$
consists of the group homomorphisms from $Q$ to $R$ induced by conjugation by
elements of $G$,
${{\operatorname{Aut}}}_{G}(Q):={{\operatorname{Hom}}}_{G}(Q,R)\cong
N_{G}(Q)/C_{G}(Q)$ is the group of $G$-automorphisms of $Q$, and
${{\operatorname{Out}}}_{G}(Q)={{\operatorname{Aut}}}_{G}(Q)/{{\operatorname{Inn}}}(Q)\cong
N_{G}(Q)/QC_{G}(Q)$ is the corresponding group of outer automorphisms. For
$H\leq G$ denote by $\mathcal{F}_{H}(G)$ the category with objects the
subgroups of $H$, in which the $\mathcal{F}$-morphisms from $Q$ to $R$ are the
elements of ${{\operatorname{Hom}}}_{G}(Q,R)$ for $Q,R\leq H$, and composition
of morphisms is the usual composition of maps. Let $\ell$ be a prime number.

###  2.1. Fusion systems on discrete $\ell$-toral groups.

A _discrete $\ell$-toral group_ is a group $P$ with normal subgroup
$P^{\circ}$ such that $P^{\circ}$ is isomorphic to a finite product of copies
of ${\mathbb{Z}}/\ell^{\infty}$ and $P/P^{\circ}$ is a finite $\ell$-group. In
particular, any finite $\ell$-group is a discrete $\ell$-toral group. Here
$P^{\circ}$ is characterised as the subset of infinitely divisible elements of
$P$ as well as the minimal subgroup of finite index of $P$. In particular,
$P^{\circ}$ is characteristic in $P$. It is called the _identity component of
$P$_ and $P$ is _connected_ if $P=P^{\circ}$.

Let $S$ be a discrete $\ell$-toral group and let $\mathcal{F}$ be a saturated
fusion system on $S$ as defined in [5, Sec. 2]. Following [15], the the number
of weights of $\mathcal{F}$ is defined by

$\mathbf{w}(\mathcal{F}):=\sum_{Q}z({{\operatorname{Out}}}_{\mathcal{F}}(Q)),$

where $Q$ runs over a set of representatives of $\mathcal{F}$-conjugacy
classes of $\mathcal{F}$-centric, $\mathcal{F}$-radical subgroups of $S$,
${{\operatorname{Out}}}_{\mathcal{F}}(Q)={{\operatorname{Aut}}}_{\mathcal{F}}(Q)/{{\operatorname{Inn}}}(Q)$
is the group of $\mathcal{F}$-outer automorphisms of $Q$, and
$z({{\operatorname{Out}}}_{\mathcal{F}}(Q))$ is the number of ordinary
irreducible characters of ${{\operatorname{Out}}}_{\mathcal{F}}(Q)$ characters
of zero $\ell$-defect. Note that by [5, Cor. 3.5, Defn. 2.2], $S$ has only
finitely many classes of $\mathcal{F}$-centric, $\mathcal{F}$-radical
subgroups and moreover, the $\mathcal{F}$-outer automorphism group of any
$\mathcal{F}$-centric, $\mathcal{F}$-radical subgroup of $S$ is finite. Thus,
$\mathbf{w}(\mathcal{F})$ is well defined. Also, note that if $G$ is a finite
group, $S$ is a Sylow $\ell$-subgroup of $G$ and
$\mathcal{F}=\mathcal{F}_{S}(G)$ is the corresponding saturated fusion system,
then $\mathbf{w}(\mathcal{F})$ is the number of Alperin weights associated to
the principal block of $kG$, $k$ an algebraically closed field of
characteristic $\ell$ (see [11, Thm 8.14.4]).

Following [5, Sec. 8], we say a group $G$ “has Sylow $\ell$-subgroups” if
there is a discrete $\ell$-toral subgroup $S\leq G$ which contains all
discrete $\ell$-toral subgroups of $G$ up to conjugacy. Such a subgroup, if it
exists, is called a _Sylow $\ell$-subgroup of $G$_. Note that the set of Sylow
$\ell$-subgroups of $G$ is a single $G$-conjugacy class. An _$\ell$ -centric
subgroup_ of $G$ is a discrete $\ell$-toral subgroup $P\leq G$ such that
$C_{G}(P)$ has a Sylow $\ell$-subgroup, which is $Z(P)$ (and unique).
Equivalently, a discrete $\ell$-toral $P\leq G$ is $\ell$-centric in $G$ if
$C_{G}(P)/Z(P)$ contains no elements of order $\ell$.

###  2.2. Fusion systems of compact Lie groups.

Let $\mathbf{G}$ be a compact Lie group. The following definitions and results
are taken from [5, Sec. 9].

  * •

An _$\ell$ -toral group_ is a compact Lie group whose identity component is a
torus and whose group of components is an $\ell$-group. If $P$ is a discrete
$\ell$-toral subgroup of $\mathbf{G}$, then the closure $\bar{P}$ of $P$ in
$\mathbf{G}$ is an $\ell$-toral group.

  * •

For any $\ell$-toral group $\mathbf{P}$,
${{\operatorname{Syl}}}_{\ell}(\mathbf{P})$ denotes the set of all discrete
$\ell$-toral subgroups $P$ of $\mathbf{P}$ such that
$\mathbf{P}^{\circ}.P=\mathbf{P}$, where $\mathbf{P}^{\circ}$ is the identity
component of $\mathbf{P}$ and $P$ contains all $\ell$-power torsion of
$\mathbf{P}$. The elements of ${{\operatorname{Syl}}}_{\ell}(\mathbf{P})$ form
a single $\mathbf{P}$-conjugacy class.

  * •

We denote by $\overline{{\operatorname{Syl}}}_{\ell}(\mathbf{G})$ the set of
all $\ell$-toral subgroups $\mathbf{S}$ of $\mathbf{G}$ such that the identity
component $\mathbf{S}^{\circ}$ is a maximal torus of $\mathbf{G}$ and
$\mathbf{S}/\mathbf{S}^{\circ}$ is a Sylow $\ell$-subgroup of
$N_{\mathbf{G}}(\mathbf{S}^{\circ})/\mathbf{S}^{\circ}$;
${{\operatorname{Syl}}}_{\ell}(\mathbf{G})$ is the set of all discrete
$\ell$-toral subgroups $P$ of $\mathbf{G}$ such that
$\bar{P}\in\overline{{\operatorname{Syl}}}_{\ell}(\mathbf{G})$ and
$P\in{{\operatorname{Syl}}}_{\ell}(\bar{P})$. Any two elements of
$\overline{{\operatorname{Syl}}}_{\ell}(\mathbf{G})$ are
$\mathbf{G}$-conjugate and each $\ell$-toral subgroup of $\mathbf{G}$ is
contained in an element of
$\overline{{\operatorname{Syl}}}_{\ell}(\mathbf{G})$. Any two elements of
${{\operatorname{Syl}}}_{\ell}(\mathbf{G})$ are $\mathbf{G}$-conjugate and
each discrete $\ell$-toral subgroup of $\mathbf{G}$ is contained in an element
of ${{\operatorname{Syl}}}_{\ell}(\mathbf{G})$ (see [5, Prop. 9.3]). In
particular, the elements of ${{\operatorname{Syl}}}_{\ell}(\mathbf{G})$ are
Sylow $\ell$-subgroups of $\mathbf{G}$.

If $S\leq\mathbf{G}$ is a Sylow $\ell$-subgroup then by [5, Lemma 9.5],
$\mathcal{F}:=\mathcal{F}_{S}(\mathbf{G})$ is a saturated fusion system on
$S$.

  * •

An _$\ell$ -stubborn subgroup_ of $\mathbf{G}$ is an $\ell$-toral subgroup
$\mathbf{P}$ of $\mathbf{G}$ such that $N_{\mathbf{G}}(\mathbf{P})/\mathbf{P}$
is a finite group which satisfies
$O_{\ell}(N_{\mathbf{G}}(\mathbf{P})/\mathbf{P})=1$.

  * •

An $\ell$-toral subgroup $\mathbf{P}\leq\mathbf{G}$ is called _$\ell$
-centric_ if
$Z(\mathbf{P})\in\overline{{\operatorname{Syl}}}_{\ell}(C_{\mathbf{G}}(\mathbf{P}))$.

  * •

A discrete $\ell$-toral subgroup $P$ of a compact Lie group $\mathbf{G}$ is
said to be _snugly embedded_ in $\mathbf{G}$ if
$P\in{{\operatorname{Syl}}}_{\ell}(\bar{P})$.

The following well-known result is the first step in the proof of Theorem 1
(see also [3, Sec. 4]).

######  Lemma 2.1.

Let $\mathbf{G}$ be a compact Lie group with Sylow $\ell$-subgroup $S$ and let
$\mathcal{F}=\mathcal{F}_{S}(\mathbf{G})$.

  1. (a)

For any $\ell$-stubborn subgroup $\mathbf{P}$ of $\mathbf{G}$,
$C_{\mathbf{G}^{\circ}}(\mathbf{P})\leq Z(\mathbf{P})$ and if
$\mathbf{G}/\mathbf{G}^{\circ}$ is an $\ell$-group, then
$C_{\mathbf{G}}(\mathbf{P})\leq Z(\mathbf{P})$, and consequently
${{\operatorname{Out}}}_{\mathbf{G}}(\mathbf{P})\cong
N_{\mathbf{G}}(\mathbf{P})/\mathbf{P}$.

  2. (b)

Suppose that $P\leq S$ is $\mathcal{F}$-centric and $\mathcal{F}$-radical.
Then $\mathbf{P}:=\bar{P}$ is an $\ell$-stubborn subgroup of $\mathbf{G}$ and
${{\operatorname{Out}}}_{\mathcal{F}}(P)\cong{{\operatorname{Out}}}_{\mathbf{G}}(\mathbf{P})$.

  3. (c)

The map $P\mapsto\bar{P}$ induces an injective map from the set of
$\mathcal{F}$-classes of $\mathcal{F}$-centric, $\mathcal{F}$-radical
subgroups of $S$ to the set of $\mathbf{G}$-classes of $\ell$-stubborn
subgroups of $\mathbf{G}$.

  4. (d)

If $\mathbf{G}/\mathbf{G}^{\circ}$ is an $\ell$-group or an
$\ell^{\prime}$-group, then the map in (c) is a bijection.

  5. (e)

Suppose that $\mathbf{G}/\mathbf{G}^{\circ}$ is an $\ell$-group. Let
$\bar{\mathcal{F}}$ be the orbit category of $\mathcal{F}$ and
$\bar{\mathcal{F}}^{cr}$ be the full subcategory whose objects are
$\mathcal{F}$-centric, $\mathcal{F}$-radical subgroups of $S$. Let
$\mathcal{R}_{\ell}(\mathbf{G})$ be the full subcategory of the orbit category
of $\mathbf{G}$ whose objects are the $\ell$-toral subgroups of $\mathbf{G}$.
There is an equivalence of categories
$\bar{\mathcal{F}}^{cr}\simeq\mathcal{R}_{\ell}(\mathbf{G})$ which for
$\mathcal{F}$-centric, $\mathcal{F}$-radical subgroups $P$, $Q$ of $S$ sends
$P$ to $\mathbf{G}/\bar{P}$ and which sends the
$\bar{\mathcal{F}}^{cr}$-morphism from $P$ to $Q$ induced by conjugation by
$g\in\mathbf{G}$ to the $\mathcal{R}_{\ell}(\mathbf{G})$-morphism from
$\mathbf{G}/\bar{P}$ to $\mathbf{G}/\bar{Q}$ defined by $x\bar{P}\to
xg^{-1}\bar{Q}$, $x\in\mathbf{G}$.

###### Proof.

Part (a) is Lemma 7 of [13]. Let $P\leq S$ be $\mathcal{F}$-centric and
radical. By [5, Lemma 3.2 and Cor. 3.5], $P=P^{\bullet}$, where $P^{\bullet}$
is as defined in Section 3 of [5] and hence by [5, Lemma 9.9], $P$ is snugly
embedded in $\mathbf{G}$. By Lemma 9.4 of [5] and its proof we have
${{\operatorname{Out}}}_{\mathcal{F}}(P)={{\operatorname{Out}}}_{\mathbf{G}}(P)\cong{{\operatorname{Out}}}_{\mathbf{G}}(\mathbf{P})$.
Since $P$ is $\mathcal{F}$-centric, by [5, Lemma 8.4], $P$ is $\ell$-centric
in $\mathbf{G}$ and hence by [5, Lemma 9.6(a),(b)], $\mathbf{P}$ is
$\ell$-centric in $\mathbf{G}$, $N_{\mathbf{G}}(\mathbf{P})/\mathbf{P}$ is
finite and $\mathbf{P}C_{\mathbf{G}}(\mathbf{P})/\mathbf{P}\cong
C_{\mathbf{G}}(\mathbf{P})/Z(\mathbf{P})$ is finite of order prime to $\ell$.
On the other hand, since
${{\operatorname{Out}}}_{\mathcal{F}}(P)\cong{{\operatorname{Out}}}_{\mathbf{G}}(\mathbf{P})\cong
N_{\mathbf{G}}(\mathbf{P})/\mathbf{P}C_{\mathbf{G}}(\mathbf{P})$ and $P$ is
$\mathcal{F}$-radical,
$O_{\ell}(N_{\mathbf{G}}(\mathbf{P})/\mathbf{P}C_{\mathbf{G}}(\mathbf{P}))=1$,
hence
$O_{\ell}(N_{\mathbf{G}}(\mathbf{P})/\mathbf{P})\leq\mathbf{P}C_{\mathbf{G}}(\mathbf{P})/\mathbf{P}$.
Since the latter is an $\ell^{\prime}$-group,
$O_{\ell}(N_{\mathbf{G}}(\mathbf{P})/\mathbf{P})=1$, showing that $\mathbf{P}$
is $\ell$-stubborn. This proves (b).

The assignment $P\mapsto\mathbf{P}:=\bar{P}$ induces a map from the set of
$\mathbf{G}$-classes of discrete $\ell$-toral subgroups of $\mathbf{G}$ which
are snugly embedded in $\mathbf{G}$ to the set of $\mathbf{G}$-classes of
$\ell$-toral subgroups of $\mathbf{G}$. By [5, Prop. 9.3], this map is a
bijection with the inverse being the map which sends the $\mathbf{G}$-class of
an $\ell$-toral subgroup $\mathbf{P}$ of $\mathbf{G}$ to the
$\mathbf{G}$-class of $P$, where
$P\in{{\operatorname{Syl}}}_{\ell}(\mathbf{P})$. Now (c) follows from (b) and
the fact that by the definition of $\mathcal{F}$, $P,P^{\prime}\leq S$ are
$\mathcal{F}$-conjugate if and only if $P$ and $P^{\prime}$ are
$\mathbf{G}$-conjugate.

Now suppose that $P\leq\mathbf{G}$ is a snugly embedded subgroup such that
$\mathbf{P}:=\bar{P}\leq\mathbf{G}$ is $\ell$-stubborn. From the above, we see
that in order to prove (d) it suffices to show that $P$ is
$\mathcal{F}$-centric and $\mathcal{F}$-radical provided that
$\mathbf{G}/\mathbf{G}^{\circ}$ is an $\ell$-group or an
$\ell^{\prime}$-group. For this, note that as above since $P$ is snugly
embedded in $\mathbf{G}$,
${{\operatorname{Out}}}_{\mathbf{G}}(P)\cong{{\operatorname{Out}}}_{\mathbf{G}}(\mathbf{P})$
by [5, Lemma 9.4]. Suppose first that $\mathbf{G}/\mathbf{G}^{\circ}$ is an
$\ell$-group. By (a), $C_{\mathbf{G}}(\mathbf{P})=Z(\mathbf{P})$, hence
${{\operatorname{Out}}}_{\mathbf{G}}(P)={{\operatorname{Out}}}_{\mathcal{F}}(\mathbf{P})\cong
N_{\mathbf{G}}(\mathbf{P})/\mathbf{P}$ and by hypothesis
$O_{\ell}(N_{\mathbf{G}}(\mathbf{P})/\mathbf{P})=1$. This shows that $P$ is
$\mathcal{F}$-radical. Also, since $C_{\mathbf{G}}(\mathbf{P})\leq
Z(\mathbf{P})$, $\mathbf{P}$ is $\ell$-centric in $\mathbf{G}$. Hence by [5,
Lemma 9.6], $P$ is $\ell$-centric in $\mathbf{G}$ and consequently by [5,
Lemma 8.4], $P$ is $\mathcal{F}$-centric. Now suppose that
$\mathbf{G}/\mathbf{G}^{\circ}$ is an $\ell^{\prime}$-group. We claim that
$\mathbf{P}\leq\mathbf{G}^{\circ}$. Indeed, we have
$\mathbf{P}=\mathbf{P}^{\circ}.P$ and
$\mathbf{P}^{\circ}\leq\mathbf{G}^{\circ}$. On the other hand, every element
of $P$ has finite order a power of $\ell$ and $\mathbf{G}/\mathbf{G}^{\circ}$
is an $\ell^{\prime}$-group, hence $P\leq\mathbf{G}^{\circ}$. This proves the
claim. Since $\mathbf{G}^{\circ}$ is normal in $\mathbf{G}$, $\mathbf{P}$ is
$\ell$-stubborn in $\mathbf{G}^{\circ}$, hence again by [13, Lemma 7],
$C_{\mathbf{G}^{\circ}}(\mathbf{P})=Z(\mathbf{P})$ and hence
$C_{\mathbf{G}}(\mathbf{P})/Z(\mathbf{P})=C_{\mathbf{G}}(\mathbf{P})/C_{\mathbf{G}^{\circ}}(\mathbf{P})\leq\mathbf{G}/\mathbf{G}^{\circ}$
is a finite $\ell^{\prime}$-group. This shows that $Z(\mathbf{P})$ contains
all torsion elements of $\ell$-power order of $C_{\mathbf{G}}(\mathbf{P})$ and
hence that $\mathbf{P}$ is $\ell$-centric in $\mathbf{G}$. Now it follows as
in the previous case that $P$ is $\mathcal{F}$-centric. This completes the
proof of (d).

Now suppose that $P,Q\leq S$ are $\mathcal{F}$-centric and
$\mathcal{F}$-radical. As above, $P$ and $Q$ are snugly embedded in
$\mathbf{G}$. By [5, Lemma 9.4(c)], there is a bijection

${{\operatorname{Inn}}}(Q)\backslash{{\operatorname{Hom}}}_{\mathbf{G}}(P,Q)\longrightarrow{{\operatorname{Inn}}}(\mathbf{Q})\backslash{{\operatorname{Hom}}}_{\mathbf{G}}(\mathbf{P},\mathbf{Q})$

induced by the canonical map from ${{\operatorname{Hom}}}_{\mathbf{G}}(P,Q)$
to ${{\operatorname{Hom}}}_{\mathbf{G}}(\mathbf{P},\mathbf{Q})$. On the one
hand, ${{\operatorname{Hom}}}_{\bar{\mathcal{F}}^{cr}}(P,Q)$ may be identified
with
${{\operatorname{Inn}}}(Q)\backslash{{\operatorname{Hom}}}_{\mathbf{G}}(P,Q)$.
On the other hand, by (a), $C_{\mathbf{G}}(\bar{P})\leq Z(\bar{P})$ from which
it follows that
${{\operatorname{Inn}}}(\mathbf{Q})\backslash{{\operatorname{Hom}}}_{\mathbf{G}}(\mathbf{P},\mathbf{Q})$
may be identified with
$\mathrm{Mor}_{\mathcal{R}_{\ell}(\mathbf{G})}(\mathbf{G}/\mathbf{P},\mathbf{G}/\mathbf{Q})$.
Now (e) follows by (d). ∎

As an immediate consequence of the previous lemma we get:

######  Lemma 2.2.

Suppose that $\mathbf{G}$ is a compact connected Lie group, $\ell$ a prime,
$S\in{{\operatorname{Syl}}}_{\ell}(\mathbf{G})$ a Sylow $\ell$-subgroup of
$\mathbf{G}$, $\mathcal{F}=\mathcal{F}_{S}(\mathbf{G})$. Then

$\mathbf{w}(\mathcal{F})=\sum_{\mathbf{P}}z(N_{\mathbf{G}}(\mathbf{P})/\mathbf{P})$

where $\mathbf{P}$ runs over a set of representatives of $\mathbf{G}$-classes
of $\ell$-stubborn subgroups of $\mathbf{G}$ and
$z(N_{\mathbf{G}}(\mathbf{P})/\mathbf{P})$ is the number of irreducible
characters of $N_{\mathbf{G}}(\mathbf{P})/\mathbf{P}$ of zero $\ell$-defect.

###  2.3. Fusion systems of connected reductive algebraic groups.

Let $p,\ell$ be distinct prime numbers and let $\mathbf{G}$ be a connected
reductive algebraic group over $\overline{{\mathbb{F}}}_{p}$. Then since
$\mathbf{G}$ has a finite dimensional faithful representation over
$\overline{{\mathbb{F}}}_{p}$, $\mathbf{G}$ is a linear torsion group in
characteristic $p$ (in the sense of [5, Sec. 8]). Hence, by [5, Thm 8.10],
$\mathbf{G}$ has a Sylow $\ell$-subgroup $S$ and $\mathcal{F}_{S}(\mathbf{G})$
is a saturated fusion system. Also, $\mathbf{G}$ is the group of
$\overline{{\mathbb{F}}}_{p}$-points of a connected split reductive algebraic
group scheme ${\mathbb{G}}$ over ${\mathbb{Z}}$. By a theorem of
Friedlander–Mislin [6, Thm 1.4], there is a homotopy equivalence of
$\ell$-completed classifying spaces

$B{\mathbb{G}}({\mathbb{C}})^{\wedge}_{\ell}\simeq(B\mathbf{G})^{\wedge}_{\ell}.$

Further, by the Cartan–Chevalley–Iwasawa–Malcev–Mostow theorem (see [7, Chap.
XV, Thm 3.1]), ${\mathbb{G}}({\mathbb{C}})$ has a maximal compact subgroup $K$
and there is a homotopy equivalence $B{\mathbb{G}}({\mathbb{C}})\simeq BK$.
Hence,

$BK^{\wedge}_{\ell}\simeq
B{\mathbb{G}}({\mathbb{C}})^{\wedge}_{\ell}\simeq(B\mathbf{G})^{\wedge}_{\ell}$

and as a consequence of [5, Thms 7.4, 8.10, and 9.10] we obtain the following.

######  Proposition 2.3.

With the above notation there is an isomorphism $S^{\prime}\cong S$ between a
Sylow $\ell$-subgroup $S^{\prime}$ of $K$ and a Sylow $\ell$-subgroup $S$ of
$\mathbf{G}$ inducing an isomorphism
$\mathcal{F}_{S^{\prime}}(K)\cong\mathcal{F}_{S}(\mathbf{G})$ of the
corresponding fusion systems.

##  3\. Proofs of Theorem 1 and Theorem 2.

We recall some notation from Section 7 of [10]. Denote by $\mathcal{C}$ the
set of finite sequences $\mathfrak{c}=(c_{1},\ldots,c_{t})$ of strictly
positive integers including the empty sequence $()$. For
$\mathfrak{c}\in\mathcal{C}$, write $|\mathfrak{c}|:=c_{1}+\cdots+c_{t}$ and
let
$G_{\mathfrak{c}}:=\operatorname{GL}_{c_{1}}(\ell)\times\cdots\times\operatorname{GL}_{c_{t}}(\ell)$.
Denote by $\mathcal{A}(\mathfrak{c})$ the set of irreducible characters of
$G_{\mathfrak{c}}=\operatorname{GL}_{c_{1}}(\ell)\times\cdots\times\operatorname{GL}_{c_{t}}(\ell)$
of the form $\chi_{1}\cdots\chi_{t}$, where each $\chi_{j}$ is an extension to
$\operatorname{GL}_{c_{j}}(\ell)$ of a Steinberg character of
$\operatorname{SL}_{c_{j}}(\ell)$. Note that
$|\mathcal{A}(\mathfrak{c})|=(\ell-1)^{t}$.

We recall the description of $\ell$-stubborn subgroups in classical groups due
to Oliver [13]. As noted already by Oliver this is akin to the Alperin–Fong
type classification of $\ell$-radical subgroups of finite general linear and
symmetric groups in [1].

######  Proposition 3.1.

Suppose that $\mathbf{G}$ is the compact connected Lie group
$\operatorname{U}(n)$, ${{\operatorname{Sp}}}(n)$, $\operatorname{SO}(2n+1)$
or $\operatorname{SO}(2n)$, $n\geq 1$. Let $\ell$ be a prime, assumed odd
unless $\mathbf{G}=\operatorname{U}(n)$. If
$\mathbf{G}\neq\operatorname{U}(n)$, let $\mathcal{X}$ denote the set of
functions

$f:{\mathbb{N}}\times\mathcal{C}\to{\mathbb{N}}$

such that
$\sum_{(\gamma,\mathfrak{c})}\ell^{|\gamma|+|\mathfrak{c}|}f(\gamma,\mathfrak{c})=n$.
If $\mathbf{G}=\operatorname{U}(n)$, let $\mathcal{X}$ denote the subset of
the above set of functions $f$ which additionally have the property
$f(0,())\neq 2,4$ if $\ell=2$ and $f(0,())\neq 3$ if $\ell=3$. There is a
bijection between the set of $\mathbf{G}$-conjugacy classes of $\ell$-stubborn
subgroups of $\mathbf{G}$ and $\mathcal{X}$ satisfying the following: Let
$\mathbf{P}\leq\mathbf{G}$ be an $\ell$-stubborn subgroup whose class
corresponds to $f\in\mathcal{X}$.

  1. (a)

If $\mathbf{G}=\operatorname{U}(n)$, then

$N_{\mathbf{G}}(\mathbf{P})/\mathbf{P}\cong\prod_{(\gamma,\mathfrak{c})}({{\operatorname{Sp}}}_{2\gamma}(\ell)\times
G_{\mathfrak{c}})\wr\mathfrak{S}_{f(\gamma,\mathfrak{c})}.$

  2. (b)

If $\mathbf{G}={{\operatorname{Sp}}}(n)$ or
$\mathbf{G}=\operatorname{SO}(2n+1)$, then

$N_{\mathbf{G}}(\mathbf{P})/\mathbf{P}\cong\prod_{(\gamma,\mathfrak{c})}(C_{2}\times{{\operatorname{Sp}}}_{2\gamma}(\ell)\times
G_{\mathfrak{c}})\wr\mathfrak{S}_{f(\gamma,\mathfrak{c})}.$

  3. (c)

If $\mathbf{G}=\operatorname{SO}(2n)$, then
$N_{\mathbf{G}}(\mathbf{P})/\mathbf{P}$ is isomorphic to the subgroup of

$\prod_{(\gamma,\mathfrak{c})}(C_{2}\times{{\operatorname{Sp}}}_{2\gamma}(\ell)\times
G_{\mathfrak{c}})\wr\mathfrak{S}_{f(\gamma,\mathfrak{c})}$

consisting of those elements for which the number of non-trivial entries from
the $C_{2}$ components is even.

###### Proof.

For $\mathfrak{c}\in\mathcal{C}$ and a non-negative integer $\gamma$ let
$\mathbf{R}_{\gamma,\mathfrak{c}}$ denote the irreducible subgroup
$\Gamma_{\ell^{\gamma}}^{U}\wr E_{\ell^{c_{1}}}\wr\cdots\wr E_{\ell^{c_{t}}}$
of $\operatorname{U}(\ell^{\gamma+|\mathfrak{c}|})$ defined in [13, Def. 2].
We regard $\mathbf{R}_{\gamma,\mathfrak{c}}$ as a subgroup of
${{\operatorname{Sp}}}(\ell^{\gamma+|\mathfrak{c}|})$ (respectively
$\operatorname{SO}(2\ell^{\gamma+|\mathfrak{c}|})$ via a standard embedding
$\operatorname{U}(\ell^{\gamma+|\mathfrak{c}|})\leq{{\operatorname{Sp}}}(\ell^{\gamma+|\mathfrak{c}|})$
(respectively
$\operatorname{U}(\ell^{\gamma+|\mathfrak{c}|})\leq\operatorname{O}(2\ell^{\gamma+|\mathfrak{c}|})$.
By [13, Thm 6],

$N_{\operatorname{U}(\ell^{\gamma+|\mathfrak{c}|})}(\mathbf{R}_{\gamma,\mathfrak{c}})/\mathbf{R}_{\gamma,\mathfrak{c}}\cong{{\operatorname{Sp}}}_{2\gamma}(\ell)\times
G_{\mathfrak{c}},$
$N_{{{\operatorname{Sp}}}(\ell^{\gamma+|\mathfrak{c}|})}(\mathbf{R}_{\gamma,\mathfrak{c}})/\mathbf{R}_{\gamma,\mathfrak{c}}\cong
C_{2}\times{{\operatorname{Sp}}}_{2\gamma}(\ell)\times G_{\mathfrak{c}},$

and

$N_{\operatorname{O}(2\ell^{\gamma+|\mathfrak{c}|})}(\mathbf{R}_{\gamma,\mathfrak{c}})/\mathbf{R}_{\gamma,\mathfrak{c}}\cong
C_{2}\times{{\operatorname{Sp}}}_{2\gamma}(\ell)\times G_{\mathfrak{c}}.$

Moreover, from the proof of [13, Thm 6] it can be checked that the full
inverse image of ${{\operatorname{Sp}}}_{2\gamma}(\ell)\times
G_{\mathfrak{c}}$ in
$N_{\operatorname{O}(2\ell^{\gamma+|\mathfrak{c}|})}(\mathbf{R}_{\gamma,\mathfrak{c}})$
lies in $\operatorname{SO}(2\ell^{\gamma+|\mathfrak{c}|})$, and that the
generator of the $C_{2}$ factor lifts to an element of determinant $-1$ in
$\operatorname{O}(2\ell^{\gamma+|\mathfrak{c}|})$, specifically to an element
of order $2$ with $-1$ as an eigenvalue of multiplicity
$\ell^{\gamma+|\mathfrak{c}|}$.

If $\mathbf{R}$ is a direct product of $m$ copies of
$\mathbf{R}_{\gamma,\mathfrak{c}}$ diagonally embedded via

$\operatorname{U}(\ell^{\gamma+|\mathfrak{c}|})\times\cdots\times\operatorname{U}(\ell^{\gamma+|\mathfrak{c}|})\leq\operatorname{U}(m\ell^{\gamma+|\mathfrak{c}|}),$

then

$N_{\operatorname{U}(m\ell^{\gamma+|\mathfrak{c}|})}(\mathbf{R})/\mathbf{R}\cong(N_{\operatorname{U}(\ell^{\gamma+|\mathfrak{c}|})}(\mathbf{R}_{\gamma,\mathfrak{c}})/\mathbf{R}_{\gamma,\mathfrak{c}})\wr\mathfrak{S}_{m}\cong({{\operatorname{Sp}}}_{2\gamma}(\ell)\times
G_{\mathfrak{c}})\wr\mathfrak{S}_{m}$

where $\mathfrak{S}_{m}$ acts by usual permutation of factors. Similarly,
regarding $\mathbf{R}$ as a diagonally embedded subgroup via
${{\operatorname{Sp}}}(\ell^{\gamma+|\mathfrak{c}|})^{m}\leq{{\operatorname{Sp}}}(m\ell^{\gamma+|\mathfrak{c}|})$,
then

$N_{{{\operatorname{Sp}}}(m\ell^{\gamma+|\mathfrak{c}|})}(\mathbf{R})/\mathbf{R}\cong
N_{{{\operatorname{Sp}}}(\ell^{\gamma+|\mathfrak{c}|})}(\mathbf{R}_{\gamma,\mathfrak{c}})/\mathbf{R}_{\gamma,\mathfrak{c}}\wr\mathfrak{S}_{m}\cong(C_{2}\times{{\operatorname{Sp}}}_{2\gamma}(\ell)\times
G_{\mathfrak{c}})\wr\mathfrak{S}_{m}$

and regarding $\mathbf{R}$ as a diagonally embedded subgroup via
$\operatorname{O}(2\ell^{\gamma+|\mathfrak{c}|})^{m}\leq\operatorname{O}(2m\ell^{\gamma+|\mathfrak{c}|})$,
then

$N_{\operatorname{O}(2m\ell^{\gamma+|\mathfrak{c}|})}(\mathbf{R})/\mathbf{R}\cong
N_{\operatorname{O}(2\ell^{\gamma+|\mathfrak{c}|})}(\mathbf{R}_{\gamma,\mathfrak{c}})/\mathbf{R}_{\gamma,\mathfrak{c}}\wr\mathfrak{S}_{m}\cong(C_{2}\times{{\operatorname{Sp}}}_{2\gamma}(\ell)\times
G_{\mathfrak{c}})\wr\mathfrak{S}_{m}.$

Note that since the space underlying $\mathbf{R}_{\gamma,\mathfrak{c}}$ in the
orthogonal case is even-dimensional, the elements of $\mathfrak{S}_{m}$ lift
to elements of determinant $1$ in
$\operatorname{O}(2m\ell^{\gamma+|\mathfrak{c}|})$.

Suppose that $\mathbf{G}=\operatorname{U}(n)$. Then (a) follows from [13, Thms
6 and 8] (see also the last part of Definition 2 of [13]) with the bijection
between $\mathcal{X}$ and the $\mathbf{G}$-conjugacy classes of
$\ell$-stubborn subgroups sending $f\in\mathcal{X}$ to the class of

$\mathbf{P}=\prod_{(\gamma,\mathfrak{c})}\mathbf{R}_{\gamma,\mathfrak{c}}^{f(\gamma,\mathfrak{c})}\leq\prod_{(\gamma,\mathfrak{c})}\operatorname{U}(\ell^{\gamma+|\mathfrak{c}|})^{f(\gamma,\mathfrak{c})}\leq\prod_{(\gamma,\mathfrak{c})}\operatorname{U}(f(\gamma,\mathfrak{c})\ell^{\gamma+|\mathfrak{c}|})\leq\operatorname{U}(n),$

where the last inclusion is via a decomposition of the vector space underlying
$\operatorname{U}(n)$, and where

$N_{\mathbf{G}}(\mathbf{P})/\mathbf{P}\cong\prod_{(\gamma,\mathfrak{c})}N_{\operatorname{U}(f(\gamma,\mathfrak{c})\ell^{\gamma+|\mathfrak{c}|})}(\mathbf{R}_{\gamma,\mathfrak{c}}^{f(\gamma,\mathfrak{c})})/\mathbf{R}_{\gamma,\mathfrak{c}}^{f(\gamma,\mathfrak{c})}\cong\prod_{(\gamma,\mathfrak{c})}({{\operatorname{Sp}}}_{2\gamma}(\ell)\times
G_{\mathfrak{c}})\wr\mathfrak{S}_{f(\gamma,\mathfrak{c})},$

proving (a). The proof for the case that $\mathbf{G}={{\operatorname{Sp}}}(n)$
is entirely similar.

Suppose that $\mathbf{G}=\operatorname{SO}(2n)$ and set
$\hat{\mathbf{G}}=\operatorname{O}(2n)$. Since $\mathbf{G}$ is the connected
component of $\hat{\mathbf{G}}$ of index $2$, and $\ell$ is odd, the set of
$\ell$-stubborn subgroups of $\mathbf{G}$ coincides with that of
$\hat{\mathbf{G}}$. By [13, Thms 6 and 8], the set of
$\hat{\mathbf{G}}$-conjugacy classes of $\ell$-stubborn subgroups is in one to
one correspondence with $\mathcal{X}$ in the same way as above where

$\mathbf{P}=\prod_{(\gamma,\mathfrak{c})}\mathbf{R}_{\gamma,\mathfrak{c}}^{f(\gamma,\mathfrak{c})}\leq\prod_{(\gamma,\mathfrak{c})}\operatorname{U}(\ell^{\gamma+|\mathfrak{c}|})^{f(\gamma,\mathfrak{c})}\leq\prod_{(\gamma,\mathfrak{c})}\operatorname{O}(2\ell^{\gamma+|\mathfrak{c}|})^{f(\gamma,\mathfrak{c})}\leq\prod_{(\gamma,\mathfrak{c})}\operatorname{O}(2f(\gamma,\mathfrak{c})\ell^{\gamma+|\mathfrak{c}|})\leq\operatorname{O}(2n)$

and

$N_{\hat{\mathbf{G}}}(\mathbf{P})/\mathbf{P}\cong\prod_{(\gamma,\mathfrak{c})}(C_{2}\times{{\operatorname{Sp}}}_{2\gamma}(\ell)\times
G_{\mathfrak{c}})\wr\mathfrak{S}_{f(\gamma,\mathfrak{c})}\leq\prod_{(\gamma,\mathfrak{c})}\operatorname{O}(2f(\gamma,\mathfrak{c})\ell^{\gamma+|\mathfrak{c}|})\leq\operatorname{O}(2n).$

From the description above, it follows that $N_{\mathbf{G}}(\mathbf{P})$ is
the index $2$-subgroup of $N_{\hat{\mathbf{G}}}(\mathbf{P})$ described in the
statement of (c), and in particular, the $\hat{\mathbf{G}}$-conjugacy class of
$\mathbf{P}$ is the same as the $\mathbf{G}$-conjugacy class of $\mathbf{P}$.
This proves (c).

Finally suppose that $\mathbf{G}=\operatorname{SO}(2n+1)$, and set
$\hat{\mathbf{G}}=\operatorname{O}(2n+1)$. Then since
$\hat{\mathbf{G}}=\mathbf{G}\times\\{\pm I\\}$ and $\ell$ is odd, the
$\mathbf{G}$-conjugacy classes of $\ell$-stubborn subgroups of $\mathbf{G}$
are the same as the $\hat{\mathbf{G}}$-conjugacy classes of $\ell$-stubborn
subgroups of $\hat{\mathbf{G}}$. By [13, Thms 6 and 8], the set of
$\hat{\mathbf{G}}$-conjugacy classes of $\ell$-stubborn subgroups of
$\hat{\mathbf{G}}$ are in one to one correspondence with $\mathcal{X}$ in the
same way as above where now

$\mathbf{P}=\prod_{(\gamma,\mathfrak{c})}\mathbf{R}_{\gamma,\mathfrak{c}}^{f(\gamma,\mathfrak{c})}\leq\prod_{(\gamma,\mathfrak{c})}\operatorname{U}(\ell^{\gamma+|\mathfrak{c}|})^{f(\gamma,\mathfrak{c})}\leq\prod_{(\gamma,\mathfrak{c})}\operatorname{O}(2\ell^{\gamma+|\mathfrak{c}|})^{f(\gamma,\mathfrak{c})}\leq\operatorname{O}(2n)\times\operatorname{O}(1)\leq\operatorname{O}(2n+1)$

and

$N_{\hat{\mathbf{G}}}(\mathbf{P})/\mathbf{P}\cong\big{(}\prod_{(\gamma,\mathfrak{c})}(C_{2}\times{{\operatorname{Sp}}}_{2\gamma}(\ell)\times
G_{\mathfrak{c}})\wr\mathfrak{S}_{f(\gamma,\mathfrak{c})}\big{)}\times\operatorname{O}(1)\leq\operatorname{O}(2n)\times\operatorname{O}(1)\leq\operatorname{O}(2n+1).$

Here, note that $\operatorname{O}(1)\cong C_{2}$. Now the restriction to
$N_{\mathbf{G}}(\mathbf{P})/\mathbf{P}$ of the canonical surjection
$N_{\hat{\mathbf{G}}}(\mathbf{P})/\mathbf{P}\to
N_{\hat{\mathbf{G}}}(\mathbf{P})/(\mathbf{P}\times\operatorname{O}(1))$ is an
isomorphism and this proves (b). ∎

######  Theorem 3.2.

Suppose that $\mathbf{G}$ is the compact connected Lie group
$\operatorname{U}(n)$, ${{\operatorname{Sp}}}(n)$, or $\operatorname{SO}(n)$,
$n\geq 1$. Let $\ell$ be a prime, assumed odd unless
$\mathbf{G}=\operatorname{U}(n)$,
$S\in{{\operatorname{Syl}}}_{\ell}(\mathbf{G})$ a Sylow $\ell$-subgroup of
$\mathbf{G}$, $\mathcal{F}=\mathcal{F}_{S}(\mathbf{G})$ the associated fusion
system and $W$ the Weyl group of $\mathbf{G}$. Then,

$\mathbf{w}(\mathcal{F})=|{{\operatorname{Irr}}}(W)|.$

###### Proof.

By Lemma 2.2 we have

$\mathbf{w}(\mathcal{F})=\sum_{\mathbf{P}}z(N_{\mathbf{G}}(\mathbf{P})/\mathbf{P})$

where $\mathbf{P}$ runs over representatives of $\mathbf{G}$-conjugacy classes
of $\ell$-stubborn subgroups of $\mathbf{G}$ and
$z(N_{\mathbf{G}}(\mathbf{P})/\mathbf{P})$ is the number of irreducible
characters of $N_{\mathbf{G}}(\mathbf{P})/\mathbf{P}$ of zero $\ell$-defect.

Suppose first that $\mathbf{G}=\operatorname{U}(n)$. Let $\mathcal{X}$ be as
in Proposition 3.1, let $f\in\mathcal{X}$ and let $\mathbf{P}$ be an
$\ell$-stubborn of $\mathbf{G}$ corresponding to $f$. The only characters of
zero $\ell$-defect of $\operatorname{GL}_{c}(\ell)$ or
${{\operatorname{Sp}}}_{2\gamma}(\ell)$ are Steinberg characters and
${{\operatorname{Sp}}}_{2\gamma}(\ell)$ has a unique Steinberg character which
we denote by $\chi_{\gamma}$. Thus by Proposition 3.1(i) and [10, Prop. 9.3],
the weights contributed by $\mathbf{P}$ are in bijection with the set

${\mathcal{W}}_{f}:=\Big{\\{}w:\bigcup_{(\gamma,\mathfrak{c})}\\{\chi_{\gamma}\\}\times\mathcal{A}(\mathfrak{c})\to\\{\ell\text{-cores}\\}\quad\text{with}\sum_{\varphi\in\\{\chi_{\gamma}\\}\times\mathcal{A}(\mathfrak{c})}|w(\varphi)|=f(\gamma,\mathfrak{c})\Big{\\}}.$

Letting $f$ range over $\mathcal{X}$ and identifying
$\\{\chi_{\gamma}\\}\times\mathcal{A}(\mathfrak{c})$ with
$\mathcal{A}(\mathfrak{c})$, we see that the set of weights of $\mathcal{F}$
is in bijection with the set $\mathcal{W}$ of assignments

$w:{\mathbb{N}}\times\bigcup_{\mathfrak{c}\in\mathcal{C}}\mathcal{A}(\mathfrak{c})\to\\{\ell\text{-cores}\\}$

such that

$\sum_{\mathfrak{c}}\sum_{(\gamma,\varphi)\in{\mathbb{N}}\times\mathcal{A}(\mathfrak{c})}\ell^{\gamma+|c|}|w(\gamma,\varphi)|=n.$

Here note that the restrictions on $\mathcal{X}$ in Proposition 3.1(a) do not
play a role. By [10, Lemma 7.15], applied with $e=r=1$ and by the argument of
the proof of Theorem 4.2 at the end of Section 7 of [10] it follows that the
number of $\mathcal{F}$-weights equals the number of ordinary irreducible
characters of $G(1,1,n)\cong\mathfrak{S}_{n}$, the Weyl group of
$\operatorname{U}(n)$.

Suppose next that $\mathbf{G}={{\operatorname{Sp}}}(n)$ or
$\operatorname{SO}(2n+1)$. Let $\mathcal{X}$ be as in Proposition 3.1, let
$f\in\mathcal{X}$ and let $\mathbf{P}$ be an $\ell$-stubborn subgroup of
$\mathbf{G}$ corresponding to $f$. Since $\ell$ is odd, all irreducible
characters of $C_{2}$ are of $\ell$-defect $0$, we obtain from Proposition
3.1(b) and by [10, Prop. 9.3] that the weights contributed by $\mathbf{P}$ are
in bijection with the set

$\tilde{\mathcal{W}}_{f}:=\Big{\\{}w:\bigcup_{(\gamma,\mathfrak{c})}\\{\chi_{\gamma}\\}\times
C_{2}\times\mathcal{A}(\mathfrak{c})\to\\{\ell\text{-cores}\\}\quad\text{with}\sum_{\varphi\in\\{\chi_{\gamma}\\}\times
C_{2}\times\mathcal{A}(\mathfrak{c})}|w(\varphi)|=f(\gamma,\mathfrak{c})\Big{\\}}.$

Letting $f$ range over $\mathcal{X}$ and identifying
$\\{\chi_{\gamma}\\}\times C_{2}\times\mathcal{A}(\mathfrak{c})$ with
$C_{2}\times\mathcal{A}(\mathfrak{c})$, it follows that the set of
$\mathcal{F}$-weights is in bijection with the set $\tilde{\mathcal{W}}$ of
assignments

$w:{\mathbb{N}}\times
C_{2}\times\bigcup_{\mathfrak{c}\in\mathcal{C}}\mathcal{A}(\mathfrak{c})\to\\{\ell\text{-cores}\\}$

such that

$\sum_{\mathfrak{c}}\sum_{(\gamma,x,\varphi)\in{\mathbb{N}}\times
C_{2}\times\mathcal{A}(\mathfrak{c})}\ell^{\gamma+|c|}|w(\gamma,x,\varphi)|=n.$

As in the previous case by [10, Lemma 7.15] now applied with $e=2$, $r=1$ and
with $\tilde{\mathcal{W}}$ in place of $\mathcal{W}$ and by the argument of
the proof of Theorem 4.2 of [10] we obtain that the number of
$\mathcal{F}$-weights equals the number of ordinary irreducible characters of
$G(2,1,n)\cong W(C_{n})=W(B_{n})$.

Now suppose that $\mathbf{G}=\operatorname{SO}(2n)$ and let $f$,
$\tilde{\mathcal{W}}_{f}$ be as just defined above. Let $C_{2}$ act on
$\tilde{\mathcal{W}}_{f}$ via $y.w(\gamma,x,\varphi):=w(\gamma,y+x,\varphi)$
for $y\in C_{2}$, $w\in\tilde{\mathcal{W}}_{f}$ and
$(\gamma,x,\varphi)\in{\mathbb{N}}\times C_{2}\times\mathcal{A}$. By
Proposition 3.1(c) and [10, Prop. 9.3], applied with
$N_{\operatorname{O}(2n)}(\mathbf{P})/\mathbf{P}$ in place of $G$ and
$N_{\operatorname{SO}(2n)}(\mathbf{P})/\mathbf{P}$ in place of $M$, the
weights contributed by $\mathbf{P}$ are indexed by
${\mathbb{Z}}/2{\mathbb{Z}}$-orbits of $\tilde{\mathcal{W}}_{f}$ with each
orbit contributing as many weights as the order of the stabiliser of a point
of the orbit. Again, letting $f$ range over all elements of $\mathcal{X}$, we
obtain that the $\mathcal{F}$-weights are indexed by the
${\mathbb{Z}}/2{\mathbb{Z}}$-orbits of $\mathcal{W}$ with each orbit
contributing as many weights as the order of the stabiliser of a point of the
orbit, where $C_{2}$ acts on $\tilde{\mathcal{W}}_{f}$ as above, that is via
$y.w(\gamma,x,\varphi):=w(\gamma,y+x,\varphi)$ for $y\in C_{2}$,
$w\in\tilde{\mathcal{W}}_{f}$ and $(\gamma,x,\varphi)\in{\mathbb{N}}\times
C_{2}\times\mathcal{A}$. As before, by [10, Lemma 7.15] now applied with
$e=2$, $r=2$ and with $\tilde{\mathcal{W}}$ in place of $\mathcal{W}$ and by
the argument of the proof of Theorem 4.2 of [10] we obtain that the number of
$\mathcal{F}$-weights equals the number of ordinary irreducible characters of
$G(2,2,n)\cong W(D_{n})$. ∎

###### Proof of Theorem 1.

We use again Lemma 2.2. By [8, Lemma 1.5(ii)], any $\ell$-stubborn subgroup of
$\mathbf{G}$ contains $Z(\mathbf{G})$ and a subgroup
$\mathbf{P}\leq\mathbf{G}$ which contains $Z(\mathbf{G})$ is $\ell$-stubborn
if and only if $\bar{\mathbf{P}}:=\mathbf{P}/Z(\mathbf{G})$ is an
$\ell$-stubborn subgroup of $\bar{\mathbf{G}}:=\mathbf{G}/Z(\mathbf{G})$. For
any $\mathbf{P}\leq\mathbf{G}$ containing $Z(\mathbf{G})$,
$N_{\mathbf{G}}(\mathbf{P})/\mathbf{P}\cong
N_{\bar{\mathbf{G}}}(\bar{\mathbf{P}})/\bar{\mathbf{P}}$ and the Weyl groups
of $\mathbf{G}$ and $\bar{\mathbf{G}}$ are isomorphic. Thus, we may assume
that $Z(\mathbf{G})=1$ and hence that $\mathbf{G}$ is a direct product of
simple compact Lie groups. By [8, Lemma 1.5(i)], the $\ell$-stubborn subgroups
of a direct product of compact connected Lie groups are the direct products of
$\ell$-stubborn subgroups of the factors and the decomposition into direct
factors respects conjugacy, normalisers, and Weyl groups hence we may assume
that $\mathbf{G}$ is simple.

Assume first that $\mathbf{G}$ is of classical type, so $\ell\geq 3$. Applying
the above arguments again, we may assume that $\mathbf{G}$ is one of the
groups $\operatorname{U}(n)$, ${{\operatorname{Sp}}}(n)$,
$\operatorname{SO}(2n+1)$ or $\operatorname{SO}(2n)$ and we are done by
Theorem 3.2. Now assume that $\mathbf{G}$ is an exceptional group. Suppose
first that the Weyl group of $\mathbf{G}$ is an $\ell^{\prime}$-group. Then
every $\ell$-toral subgroup of $\mathbf{G}$ is contained in a maximal torus of
$\mathbf{G}$, and by Lemma 2.1(a) the only $\ell$-stubborn subgroups of
$\mathbf{G}$ are the maximal tori and the result is immediate. Thus the only
cases left are $\mathbf{G}=E_{6}$ with $\ell=5$, $\mathbf{G}=E_{7}$ with
$\ell=5,7$ and $\mathbf{G}=E_{8}$ with $\ell=7$. These are handled by the next
proposition. Note that in all of these cases Sylow $\ell$-subgroups of $W$ are
cyclic. ∎

######  Proposition 3.3.

Suppose $\mathbf{G}=E_{n}$ for $n\in\\{6,7,8\\}$ and let $\ell$ be a good
prime dividing $|W(\mathbf{G})|$. Then
$\mathbf{w}(\mathcal{F}_{\ell}(\mathbf{G}))=|{{\operatorname{Irr}}}(W)|$ where
$\mathcal{F}_{\ell}(\mathbf{G})$ denotes the $\ell$-fusion system of
$\mathbf{G}$.

###### Proof.

The fusion system $\mathcal{F}=\mathcal{F}_{\ell}(\mathbf{G})$ is calculated
in [14, Thm B(c)] and its structure is described in [14, Tab. 5.1]. Apart from
$\\{S\\}$ and $\\{T\\}$ there is one further class
$\mathcal{B}=E^{\mathcal{F}}$ of $\mathcal{F}$-centric radical subgroups with
$E\cong\ell_{+}^{1+2}$. Now
${{\operatorname{Out}}}_{\mathcal{F}}(T)=W(\mathbf{G})$, and [15, Thm 4.5]
shows that this determines ${{\operatorname{Out}}}_{\mathcal{F}}(P)$ for the
remaining $P\in\mathcal{F}^{cr}$. We deduce that
${{\operatorname{Out}}}_{\mathcal{F}}(P)$ and
$z=z({{\operatorname{Out}}}_{\mathcal{F}}(P))$ are given by the following
table:

$\begin{array}[]{c|cccccccc|c}(n,\ell)&S&z&&T&z&&E&z&|{{\operatorname{Irr}}}(W(\mathbf{G}))|\\\
\hline\cr(6,5)&C_{4}\times
C_{2}&\bf{8}&&W&\bf{15}&&\operatorname{SL}_{2}(5).2&\bf{2}&25\\\
(7,5)&C_{4}\times
C_{2}\times\mathfrak{S}_{3}&\bf{24}&&W&\bf{30}&&\operatorname{SL}_{2}(5).2\times\mathfrak{S}_{3}&\bf{6}&60\\\
(7,7)&C_{6}\times
C_{2}&\bf{12}&&W&\bf{46}&&\operatorname{SL}_{2}(7).2&\bf{2}&60\\\
(8,7)&C_{6}\times(C_{2})^{2}&\bf{24}&&W&\bf{84}&&\operatorname{SL}_{2}(7).2\times
C_{2}&\bf{4}&112\\\ \end{array}$

The equality ensues. ∎

###### Proof of Theorem 2.

This is immediate from Proposition 2.3 and Theorem 1. ∎

##  4\. Bad primes

######  Proposition 4.1.

Suppose that $\mathbf{G}={{\operatorname{Sp}}}(n)$ with $n\geq 2$ and
$\ell=2$. Then $\mathbf{w}(\mathcal{F})<|{{\operatorname{Irr}}}(W)|$.

###### Proof.

For $\gamma\in{\mathbb{N}}$ and $\mathfrak{c}\in\mathcal{F}$, let
$\mathbf{S}_{\gamma,\mathfrak{c}}$ (respectively
$\bar{\mathbf{S}}_{\gamma,\mathfrak{c}}$) denote the irreducible subgroup
$\Gamma_{2^{\gamma}}^{{\operatorname{Sp}}}\wr E_{2^{c_{1}}}\wr\cdots\wr
E_{2^{c_{t}}}$ (respectively
$\bar{\Gamma}_{2^{\gamma}}^{{\operatorname{Sp}}}\wr E_{2^{c_{1}}}\wr\cdots\wr
E_{2^{c_{t}}}$) of ${{\operatorname{Sp}}}(2^{\gamma+|\mathfrak{c}|})$ defined
in [13, Def. 2]. By [13, Thms 6 and 8], the $\mathbf{G}$-classes of
$2$-stubborn subgroups of $\mathbf{G}$ are in bijection with the set of
ordered pairs of functions $(f,f^{\prime})$ where

$f:{\mathbb{N}}\times\mathcal{C}\to{\mathbb{N}}\quad\text{and}\quad
f^{\prime}:{\mathbb{N}}\times\mathcal{C}\to{\mathbb{N}}$

are such that

$\sum_{(\gamma,\mathfrak{c})}2^{\gamma+|\mathfrak{c}|}f({\gamma,\mathfrak{c}})+\sum_{(\gamma^{\prime},\mathfrak{c}^{\prime})}2^{\gamma^{\prime}+|\mathfrak{c}^{\prime}|}f^{\prime}({\gamma^{\prime},\mathfrak{c}^{\prime}})=n$

and $f^{\prime}((0,()))\neq 2,4$. If $\mathbf{P}\leq\mathbf{G}$ is a
$2$-stubborn subgroup corresponding to $(f,f^{\prime})$ then

$\mathbf{P}\cong\prod_{(\gamma,\mathfrak{c})}\mathbf{S}_{\gamma,\mathfrak{c}}^{f(\gamma,\mathfrak{c})}\times\prod_{(\gamma^{\prime},\mathfrak{c}^{\prime})}\bar{\mathbf{S}}_{\gamma^{\prime},\mathfrak{c}^{\prime}}^{f(\gamma^{\prime},\mathfrak{c}^{\prime})}$

and

$N_{\mathbf{G}}(\mathbf{P})/\mathbf{P}\cong\prod_{(\gamma,\mathfrak{c})}(\operatorname{GO}_{2\gamma+2}^{-}(2)\times
G_{\mathfrak{c}})\wr\mathfrak{S}_{f({\gamma,\mathfrak{c})}}\times\prod_{(\gamma^{\prime},\mathfrak{c}^{\prime})}({{\operatorname{Sp}}}_{2\gamma^{\prime}}(2)\times
G_{\mathfrak{c}^{\prime}})\wr\mathfrak{S}_{f^{\prime}(\gamma^{\prime},\mathfrak{c}^{\prime})}.$

Now $\operatorname{GO}_{2}^{-}(2)\cong\mathfrak{S}_{3}$ has a unique
irreducible character of $2$-defect $0$ while
$\operatorname{GO}_{2\gamma+2}^{-}(2)$ has no irreducible characters of
$2$-defect $0$ for $\gamma\geq 1$ so the base subgroup
$\operatorname{GO}_{2\gamma+2}^{-}(2)\times G_{\mathfrak{c}}$ has a unique
irreducible character of $2$-defect $0$ if $\gamma=0$ and none if $\gamma\geq
1$. The base subgroup ${{\operatorname{Sp}}}_{2\gamma^{\prime}}(2)\times
G_{\mathfrak{c}^{\prime}}$ has a unique irreducible character of $2$-defect
$0$ for all $\gamma^{\prime}\geq 0$. Further, the symmetric group
$\mathfrak{S}_{m}$ has one block of 2-defect $0$ if $m$ is a triangular number
and none otherwise. From this and [5, Lemma 8.4], it follows that the weight
contribution of $\mathbf{P}$ is $1$ if $\gamma=0$ and the values of $f$ and
$f^{\prime}$ are triangular numbers (including 0), and is $0$ otherwise. So,
the number of $\mathcal{F}$-weights equals the number of pairs
$(f,f^{\prime})$ of functions

$f:\mathcal{C}\to{\mathbb{N}}\quad\text{and}\quad
f^{\prime}:{\mathbb{N}}\times\mathcal{C}\to{\mathbb{N}}$

such that

$\sum_{\mathfrak{c}}2^{|\mathfrak{c}|}f(\mathfrak{c})+\sum_{(\gamma^{\prime},\mathfrak{c}^{\prime})}2^{\gamma^{\prime}+|\mathfrak{c}^{\prime}|}f^{\prime}({\gamma^{\prime},\mathfrak{c}^{\prime}})=n$

and such that all values of $f$ and $f^{\prime}$ are triangular numbers. On
the other, hand using the analysis for the odd $\ell$ case and noting that
when $\ell=2$, $\mathcal{A}(\mathfrak{c})$ is a singleton and that the set of
$2$-cores is in bijection with the set of triangular numbers, we have that the
number of bipartitions of $n$, i.e., $|{{\operatorname{Irr}}}(W)|$, equals the
number of pairs $(f,f^{\prime})$ of functions

$f:{\mathbb{N}}\times\mathcal{C}\to{\mathbb{N}}\quad\text{and}\quad
f^{\prime}:{\mathbb{N}}\times\mathcal{C}\to{\mathbb{N}}$

such that

$\sum_{(\gamma,\mathfrak{c})}2^{|\gamma|+|\mathfrak{c}|}f(\gamma,\mathfrak{c})+\sum_{(\gamma^{\prime},\mathfrak{c}^{\prime})}2^{\gamma^{\prime}+|\mathfrak{c}^{\prime}|}f^{\prime}({\gamma^{\prime},\mathfrak{c}^{\prime}})=n$

and such that all values of $f$ and $f^{\prime}$ are triangular numbers. Thus,
the number of weights is strictly less than $|{{\operatorname{Irr}}}(W)|$ for
all $n\geq 2$. ∎

######  Example 4.2.

For $n=2$, we get 4 weights for ${{\operatorname{Sp}}}(2)$, namely for
$((1);(0,()))$ with $f=1,f^{\prime}=0$, $(();(0,()))$ with $f=f^{\prime}=1$,
$(();(1,()))$ and $(();(0,(1)))$ with $f=0,f^{\prime}=1$, and so by the proof
of Theorem 2,
$4=\mathbf{w}(\mathcal{F})<|{{\operatorname{Irr}}}(W(\mathbf{G}))|=5$, where
$\mathbf{G}$ is a simple algebraic group over $\overline{{\mathbb{F}}}_{p}$ of
type $C_{2}$ and $\mathcal{F}$ is its $2$-fusion system. Note, however, that
if $\mathcal{F}$ is the $2$-fusion system of ${{\operatorname{CSp}}}_{4}(q)$
with $\nu_{2}(q-1)>2$ there are five $\mathcal{F}$-centric radical subgroups
each contributing a single weight (so $\mathbf{w}(\mathcal{F})=5$), but two of
these subgroups are fused in ${{\operatorname{CSp}}}_{4}(q^{2})$.

######  Example 4.3.

Let $\mathbf{G}$ be a connected reductive group of symplectic or odd
dimensional orthogonal type over a field of odd characteristic and
$F:\mathbf{G}\to\mathbf{G}$ a Frobenius map. Then the number
$\mathbf{w}(\mathcal{F})$ of 2-weights for the principal 2-block of
$\mathbf{G}^{F}$ equals the number of unipotent conjugacy classes of
$\mathbf{G}^{F}$ (see [2, Prop. (2A)] in general). Now the Springer
correspondence defines an injective map from ${{\operatorname{Irr}}}(W)$ to
the set of unipotent classes of $\mathbf{G}^{F}$, so we conclude
$\mathbf{w}(\mathcal{F})\geq|{{\operatorname{Irr}}}(W)|$. In fact, since the
Springer correspondence is not surjective in general we have
$\mathbf{w}(\mathcal{F})>|{{\operatorname{Irr}}}(W)|$ for large enough rank.
E.g., for $\mathbf{G}$ of type $B_{4}$ or type $C_{6}$ we have
$\mathbf{w}(\mathcal{F})=|{{\operatorname{Irr}}}(W)|+1$.

######  Proposition 4.4.

Suppose $\mathbf{G}=G_{2}$. Then
$\mathbf{w}(\mathcal{F})=|{{\operatorname{Irr}}}(W)|=6$ for all primes $\ell$.

###### Proof.

If $\ell>3$, then we are done by Theorem 3.2. The $2$-stubborn and
$3$-stubborn subgroups of $\mathbf{G}$ are determined in [9, Lemma 3.2]. There
are six classes of $2$-stubborn subgroups $\mathbf{P}$, with
$N(\mathbf{P})/\mathbf{P}$ isomorphic to $1$, $\mathfrak{S}_{3}$ (for three of
the classes), $\mathfrak{S}_{3}\times\mathfrak{S}_{3}$, and
$\operatorname{GL}_{3}(2)$, respectively. Each of these contributes one
weight. There are two classes of $3$-stubborn subgroups $\mathbf{P}$ with
$N(\mathbf{P})/\mathbf{P}$ isomorphic to $C_{2}\times C_{2}$ and
$\operatorname{GL}_{2}(3)$, respectively. The first contributes $4$ weights
and the second contributes $2$ weights. ∎

######  Proposition 4.5.

Suppose $\mathbf{G}=F_{4}$. Then
$\mathbf{w}(\mathcal{F})=22<|{{\operatorname{Irr}}}(W)|=25$ for $\ell=3$.

###### Proof.

The $3$-stubborn subgroups of $\mathbf{G}$ are determined in [16, Prop. 3.6]:
there are seven conjugacy classes, with respective automisers $D_{8}$,
$(C_{2}\times{{\operatorname{Sp}}}_{2}(3)).2$ (twice),
${{\operatorname{Sp}}}_{2}(3)\wr 2$, $\operatorname{GL}_{2}(3)$,
$\operatorname{SL}_{3}(3)$ and $W(F_{4})$. The respective number of weights
for those is 5, 4 (twice), 2, 2, 1, 4, adding up to 22. ∎

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|  Dr Rohitash Chandra is a Senior Lecturer in Data Science at the UNSW School
of Mathematics and Statistics. He leads a program of research encircling
methodologies and applications of artificial intelligence. The methodologies
include Bayesian deep learning, neuroevolution, ensemble learning, and data
augmentation. The applications include climate extremes, geoscientific models,
mineral exploration, biomedicine, and language modelling. Dr Chandra is on the
Editorial Board for Geoscientific Model Development. He also served as an
Associate Editor for IEEE Transactions on Neural Networks and Learning
Systems, and Neurocomputing (2021-2022). Since 2021, Dr Chandra has been
featured in the Stanford’s list of top 2% scientists in the world.
---|---
|  Dr Joshua Simmons received his Ph.D from UNSW Sydney and is currently
working as a Research Engineer at the ARC Training Centre in Data Analytics
for Resources and Environments (DARE). His research has included numerical and
statistical modelling of coastal processes, with a particular focus on storm
erosion of sandy beaches. More recently, his work has focused on interpretable
machine learning/data science approaches and Bayesian methods for uncertainty
quantification in the hydrology domain.
---|---