Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
Euler characteristics of p-subgroup categories
Martin Wedel Jacobsen and Jesper Michael Møl·ler
University of Copenhagen
UAB Topology Seminar, December 17, 2010
MW Jacobsen, JM Møl·ler
Euler characteristics
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
Outline of talk
G is a finite group and p is a prime number
SG∗ the poset of nonidentity p-subgroups of G
TG∗ (H, K ) = NG (H, K ) the transporter category of G
L∗G (H, K ) = O p CG (H)\NG (H, K ) the linking category of G
FG∗ (H, K ) = CG (H)\NG (H, K ) the fusion category of G
∗ (H, K ) = N (H, K )/K the orbit category of G
OG
G
e
FG (H, K ) = CG (H)\NG (H, K )/K the exterior quotient of
the fusion category of G
What are the Euler characteristics of these finite categories?
MW Jacobsen, JM Møl·ler
Euler characteristics
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
Outline of talk
1
Euler characteristics of matrices
Euler characteristic of an invertible matrix
Euler characteristic of the poset SG∗
Euler characteristic of square matrix
2
Euler characteristics of fusion categories
3
Euler characteristics of p-subgroup categories
Categories of nonidentity subgroups
Categories of centric subgroups
Typical result
χ(FG∗ ) =
X −µ(K )
|FG∗ (K )|
[K ]
MW Jacobsen, JM Møl·ler
Euler characteristics
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
Euler characteristic of an invertible matrix
∗
Euler characteristic of the poset SG
Euler characteristic of square matrix
Definition (The Euler characteristic of an invertible matrix)
The Euler characteristic of ζ = ζ(a, b) is the sum
χ(ζ) =
X
µ(a, b)
a,b
of the entries in the inverse µ = ζ −1 =
P∞
k
k =0 (−1) (ζ
− E)k .
Example
ζ= g
ζ=
1
0
0
0
0
0
1
0
0
0
1
1
1
0
0
0
1
0
1
0
1!
1
1
1
1
µ = g −1
 1 0 −1
µ=
0
0
0
0
MW Jacobsen, JM Møl·ler
χ(ζ) = g −1

0
0
1 −1 −1 1
0 1 0 −1 
0 0 1 −1
0 0 0 1
Euler characteristics
χ(ζ) = 1
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
Euler characteristic of an invertible matrix
∗
Euler characteristic of the poset SG
Euler characteristic of square matrix
Definition (The incidence matrix of a poset S)
(
1 a≤b
ζ(a, b) =
ζ(S) = ζ(a, b) a,b ,
0 otherwise
Definition (The Euler characteristic of a finite poset S)
χ(S) = χ(ζ(S))
Example (A poset with a terminal element)
e44
44
4
c44
d
444 a
b
MW Jacobsen, JM Møl·ler
ζ(S) =
1
0
0
0
0
0
1
0
0
0
1
1
1
0
0
0
1
0
1
0
1!
1
1
1
1
χ(S) = χ(ζ(S)) = 1
Euler characteristics
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
Euler characteristic of an invertible matrix
∗
Euler characteristic of the poset SG
Euler characteristic of square matrix
Definition (Simplices in a poset)
A k -simplex, k ≥ 0, (from a to b) is a totally ordered subset of
k + 1 points (with a as smallest and b as greatest element).
Example ((ζ − E)k counts k -simplices)
0-simplices
1-simplices
2-simplices
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
MW Jacobsen, JM Møl·ler
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
1
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0!
0
0
0
1
1!
1
1
1
1
1!
2
0
0
0
=E
e4
444
4
c4
4
d
4
44 a
b
= ζ(S) − E
= (ζ(S) − E)2
Euler characteristics
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
Euler characteristic of an invertible matrix
∗
Euler characteristic of the poset SG
Euler characteristic of square matrix
Lemma (Counting simplices in poset S)
(ζ − E)k (a, b) = #{k -simplices from a to b} (k ≥ 0)
P
k
a,b (ζ − E) (a, b) = #{k -simplices in S} (k ≥ 0)
Topological Euler characteristic of the realization |S|
∞
X
χ(|S|) =
(−1)k #{k -simplices in S}
=
k =0
∞
X
(−1)k
k =0
=
X
a,b∈S
∞
XX
(−1)k (ζ − E)k (a, b)
a,b k =0
=
X
(ζ − E)k (a, b)
x −1 = (1 + (x − 1))−1 =
ζ −1 (a, b) =
a,b
MW Jacobsen, JM Møl·ler
X
P∞
k =0 (−1)
µ(a, b) = ζ(S)
a,b
Euler characteristics
k
(x − 1)k
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
Euler characteristic of an invertible matrix
∗
Euler characteristic of the poset SG
Euler characteristic of square matrix
Assumptions
G is a finite group and p is a prime number
Definition (The posets SG and SG∗ )
SG is the poset of all p-subgroups of G
SG∗ is the poset of nonidentity p-subgroups of G
Proposition (Euler characteristic of the poset SG∗ I)
P
P
χ(SG∗ ) = K ∈Ob(S ∗ ) −µ(K ) = [K ]6=1 −µ(K )|G : NG (K )|
G
Proof
1 = χ(SG ) =
X
µ(H, K ) =
H,K
= µ(1, 1) +
X
X
X
µ(1, K ) +
1≤K
1H≤K ≤G
µ(1, K ) + χ(SG∗ )
1K
MW Jacobsen, JM Møl·ler
Euler characteristics
µ(H, K )
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
Euler characteristic of an invertible matrix
∗
Euler characteristic of the poset SG
Euler characteristic of square matrix
Lemma (The Möbius function on SG [1, Lemme 4.1])
Let H and K be p-subgroups of G. Then µ(H, K ) = 0 unless
H C K with elementary abelian factor group where
n
µ(H, K ) = (−1)n p(2) ,
|K : H| = pn
In particular, µ(K ) = µ(1, K ) = 0 unless K is elementary
abelian where
n
µ(K ) = (−1)n p(2) ,
pn = |K |
Example (Alternating groups at p = 2)
n
χ(SA∗n )
4
1
5
5
6
−15
7
−175
MW Jacobsen, JM Møl·ler
8
65
9
5121
Euler characteristics
10
15105
11
55935
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
Euler characteristic of an invertible matrix
∗
Euler characteristic of the poset SG
Euler characteristic of square matrix
Proposition (Euler characteristic of the poset SG∗ II)
X
χ(SG∗ ) =
1 − χ(SN∗ G (H)/H )
H
Recursive calculations?
Proof
χ(SG∗ ) =
X
=
X
=
X
µ(H, K ) =
H,K
XX
H
X
µ(H, K ) =
X
X
X
H K ∈[H,NG (H)]
K
µ(K /H) =
H K ∈[H,NG (H)]
1 − χ(SN∗ G (H)/H )
X
H K̄ ∈NG (H)/H
H
MW Jacobsen, JM Møl·ler
Euler characteristics
µ(K̄ )
µ(H, K )
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
Euler characteristic of an invertible matrix
∗
Euler characteristic of the poset SG
Euler characteristic of square matrix
What is known about χ(SG∗ )?
Theorem (Product formula)
1−
∗
χ(SQ
n
i=1
Gi )
=
n
Y
1 − χ(SG∗ i )
i=1
Proposition
If G has a nonidentity normal p-subgroup then χ(SG∗ ) = 1
Theorem (Brown 1975, Quillen 1978)
1 − χ(SG∗ ) is divisible by |G|p
Example
∗
χ(SSL
)
n (Fq )
n
= 1 + (−1)n q (2) when q is a power of p
MW Jacobsen, JM Møl·ler
Euler characteristics
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
Euler characteristic of an invertible matrix
∗
Euler characteristic of the poset SG
Euler characteristic of square matrix
What is unknown about χ(SG∗ )!
(Euler characteristics of Chevalley groups) Is
1 − χ(SG∗ n (q) ) = (−1)n q #{positive roots} for G = A, B, C, D, E?
(Euler characteristics of alternating groups) Describe the
sequence n → χ(SA∗n )
(Quillen conjecture 1978) Op (G) > 1 ⇐⇒ SG∗ ' ∗
MW Jacobsen, JM Møl·ler
Euler characteristics
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
Euler characteristic of an invertible matrix
∗
Euler characteristic of the poset SG
Euler characteristic of square matrix
The Euler characteristic of a matrix ζ = ζ(a, b)
How do we define χ(ζ) when ζ is not invertible?
Definition (Weightings and coweightings)
A weighting for ζ is a column vector (k • ) such that
 
1
b
 .. 
ζ(a, b) (k ) =  . 
1
A coweighting for ζ is a row vector (k• ) such that
(ka ) ζ(a, b) = 1 · · · 1
MW Jacobsen, JM Møl·ler
Euler characteristics
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
Euler characteristic of an invertible matrix
∗
Euler characteristic of the poset SG
Euler characteristic of square matrix
A matrix may have none or many (co)weightings
If µ(a, b) is an inverse to ζ(a, b) then
 
1
X
.
a
k = µ(a, b)  ..  =
µ(a, b) (column sums)
b∈S
1
X
kb = 1 · · · 1 µ(a, b) =
µ(a, b) (row sums)
a∈S
are the unique weighting and coweighting for ζ
χ(ζ) = 1 for ζ = 11 11 (and ζ is not invertible)
ζ = 11 22 has no Euler characteristic
MW Jacobsen, JM Møl·ler
Euler characteristics
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
Euler characteristic of an invertible matrix
∗
Euler characteristic of the poset SG
Euler characteristic of square matrix
If ζ admits both a weighting k • and a coweighting k• then the
sum of the values of the weighting
X
X X
X
X
X
kb =
ka ζ(a, b)) k b =
ka
ζ(a, b)k b =
ka
b
b
a
a
b
a
Definition (The Euler characteristic of a matrix (Leinster 2008))
X
X
χ(ζ) =
kb =
ka
a
b
If ζ is invertible then
χ(ζ) =
X
ka =
a
X
µ(a, b)
a,b
as before.
MW Jacobsen, JM Møl·ler
Euler characteristics
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
Euler characteristic of an invertible matrix
∗
Euler characteristic of the poset SG
Euler characteristic of square matrix
Definition (The incidence matrix of a finite category C)
ζ(a, b) = |C(a, b)|
ζ(C) = ζ(a, b) a,b
Definition (Euler characteristic of a category (Leinster 2008))
χ(C) = χ(ζ(C))
Proposition (Invarians under equivalence (Leinster 2008))
If there is an adjunction C o
/
D then χ(C) = χ(D)
If C has an initial or terminal element then χ(C) = 1
If C and D are equivalent then χ(C) = χ(D)
MW Jacobsen, JM Møl·ler
Euler characteristics
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
Example (Skeletal subcategory of FA∗6 , p = 2)
4
6
? eO


6

5
9 b _??
?
3??
?
aY
_?gO?OOO
? OOO
6?? 2OOO
OOO
?
O
6 ?9 c
7d e
o
o
 ooooo

3 ooo1
oooo

1
0

ζ(C) = 
0
0
0
2
3
6
0
0
0
3
0
6
0
0
1
0
0
2
0

5
6

6

2
4
1
 1 −1/2 −1/2 −1/2
0
µ(C) =  0
1/6
0
0 0
0 0
0
1/6
0
0

1/2
0 −1/4

0 −1/4 
1/2 −1/4
0
1/4
MW Jacobsen, JM Møl·ler
χ(C) =
Euler characteristics
X
µ(i, j) = 1/3
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
Theorem (A weighting, a coweighting, and the Euler
characteristic of FG∗ )
−µ(K )
|G : CG (K )|
X µ(H, K )
1
=
=
|G : CG (K )|
|G|
kKF =
kFH
K
χ(FG∗ ) =
X
K ∈Ob(FG∗ )
X
1 − χ(SC∗ N
x∈CG (H)
−µ(K )
=
|G : CG (K )|
X
[K ]∈π0 (FG∗ )
(x)/H
G (H)
−µ(K )
|FG∗ (K )|
In all cases
kKF 6= 0 ⇐⇒ K is elementary abelian
If the Sylow p-subgroup P is normal in G then
kFH 6= 0 ⇐⇒ H = CP (g) for some g ∈ G
MW Jacobsen, JM Møl·ler
Euler characteristics
)
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
What is known about χ(FG∗ )?
Theorem (Product formula)
∗
1 − χ(FQ
n
i=1
G)=
n
Y
i
1 − χ(FG∗ i )
i=1
Proposition
If G has a nonidentity central p-subgroup then χ(FG∗ ) = 1
|G|p0 · χ(FG∗ ) ∈ Z
|{ϕ ∈ FG∗ (P) | CP (ϕ) > 1}
χ(FG∗ ) =
when P, the Sylow
|FG∗ (P)|
p-subgroup, is abelian.
χ(F ∗ ) = χ(F a ) and χ(F ∗ ) = χ(Fe∗ )
G
G
G
MW Jacobsen, JM Møl·ler
G
Euler characteristics
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
Example (Alternating groups An at p = 2)
n
4
5
6
7
8
9
χ(SA∗n )
1
5
−15
−175
68
5121
χ(FA∗n )
1/3
1/3
1/3
1/3
41/63
41/63
n
10
11
12
13
14
15
χ(SA∗n )
55105
55935
−288255
1626625
23664641
150554625
χ(FA∗n )
18/35
18/35
389/567
389/567
233/405
233/405
Example (The smallest group with χ(FG∗ ) > 1)
There is a group G = C24 o H, where H = (C3 × C3 ) o C2 , of
order |G| = 288 with Euler characteristic χ(FG∗ ) = 10/9 at
p = 2.
MW Jacobsen, JM Møl·ler
Euler characteristics
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
What is unknown about χ(FG∗ )
Are FG∗ and FGa homotopy equivalent?
Are FG∗ and FeG∗ homotopy equivalent?
Is χ(FG∗ ) always positive when p divides the order of G?
Can χ(FG∗ ) get arbitrarily large?
What is χ(FA∗n )? Does it converge for n → ∞?
∗
What is χ(FSL
n (Fq )
)?
Is there a |G|p0 -fold covering map E → BFG∗ where E is
(homotopy) finite and χ(E) = |G|p0 χ(FG∗ )?
MW Jacobsen, JM Møl·ler
Euler characteristics
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
Categories of nonidentity subgroups
Categories of centric subgroups
Theorem (Euler characteristics of nonidentity p-subgroup
categories)
C
TG∗
L∗G
χ(C)
X −µ(H)
|TG∗ (H)|
[H]
X −µ(H)
[H]
FG∗
∗
OG
|L∗G (H)|
X −µ(H)
|FG∗ (H)|
[H]
p−1X
1
χ(TG∗ ) +
∗
p
|OG (C)
[C]
MW Jacobsen, JM Møl·ler
Euler characteristics
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
Categories of nonidentity subgroups
Categories of centric subgroups
Combinatorial identities
Corollary
For any finite group G and any prime p,
X
1 − χ(SN∗ G (H)/H ) + µ(H) = 0
H
X
X
1 − χ(SC∗ N
H x∈CG (H)
X
H
(x)/H
G (H)
) + µ(H) = 0
p−1X
|C|
|H| − χ(SN∗ G (H)/H )|H| + µ(H) =
p
C
where H runs over the set of nonidentity p-subgroups of G and
C over the set of nonidentity cyclic p-subgroups of G.
MW Jacobsen, JM Møl·ler
Euler characteristics
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
Categories of nonidentity subgroups
Categories of centric subgroups
Definition
The p-subgroup H ≤ G is p-centric if p - |CG (H) : CH (H)|
Example (Euler characteristics of centric subgroup categories
for alternating groups at p = 2)
n
|An |χ(LcAn )
χ(SAc n )
χ(LcAn )
χ(FAcn )
χ(FeAcn )
4 5
1 5
1 5
1/12
1/3
1/3
6
7
−15 −105
−15 −175
−1/24
1/3
1/3
8
9
65 585
65 585
13/4032
13/63
13/63
Conjecture
χ(FGc ) = χ(FeGc )
MW Jacobsen, JM Møl·ler
Euler characteristics
10
11
11745 129195
11745 107745
29/4480
19/105
19/105
Euler characteristics of matrices
Euler characteristics of fusion categories
Euler characteristics of p-subgroup categories
Categories of nonidentity subgroups
Categories of centric subgroups
References
Charles Kratzer and Jacques Thévenaz, Type d’homotopie
des treillis et treillis des sous-groupes d’un groupe fini,
Comment. Math. Helv. 60 (1985), no. 1, 85–106. MR
787663 (87b:06017)
MW Jacobsen, JM Møl·ler
Euler characteristics