Finite group rings of nilpotent groups:
a complete set of orthogonal primitive idempotents
Inneke Van Gelder
Groups, Rings, and Group-Rings
University of Alberta, Edmonton, Canada
July 11-15, 2011
Aim
Rational group ring
Finite group ring
Our aim is to describe a complete set of orthogonal primitive
idempotents in each simple component of a semisimple group
algebra in such a way that it provides a straightforward
implementation.
Problems
Aim
Rational group ring
Finite group ring
Definition (Olivieri, del Rı́o, Simón; 2004)
A strong Shoda pair of G is a pair (H, K ) of subgroups of G
satisfying
I
K ≤ H E NG (K )
I
H/K is cyclic and a maximal abelian subgroup of NG (K )/K
I
for every g ∈ G \ NG (K ), ε(H, K )ε(H, K )g = 0
Problems
Aim
Rational group ring
Finite group ring
Definition (Olivieri, del Rı́o, Simón; 2004)
A strong Shoda pair of G is a pair (H, K ) of subgroups of G
satisfying
I
K ≤ H E NG (K )
I
H/K is cyclic and a maximal abelian subgroup of NG (K )/K
I
for every g ∈ G \ NG (K ), ε(H, K )ε(H, K )g = 0
Here, ε(H, K ) is defined as
(
e
K
Q
ε(H, K ) =
if H = K
e
e
6 K,
M/K ∈M(H/K ) (K − M) if H =
e= 1 P
with N
n∈N n for each group N and M(H/K ) the set of
|N|
all minimal normal non-trivial subgroups of H/K .
Problems
Aim
Rational group ring
Finite group ring
Pci for abelian-by-supersolvable groups
Theorem (Olivieri, del Rı́o, Simón; 2004)
Let G be an abelian-by-supersolvable
group. Then e is a pci of
P
QG iff e = e(G , H, K ) := t∈T ε(H, K )t for a strong Shoda
pair (H, K ) of G .
Problems
Aim
Rational group ring
Finite group ring
Pci for abelian-by-supersolvable groups
Theorem (Olivieri, del Rı́o, Simón; 2004)
Let G be an abelian-by-supersolvable
group. Then e is a pci of
P
QG iff e = e(G , H, K ) := t∈T ε(H, K )t for a strong Shoda
pair (H, K ) of G .
Theorem (Olivieri, del Rı́o, Simón; 2004)
Let G be an abelian-by-supersolvable group and (H, K ) a
strong Shoda pair of G . Then the Wedderburn component
QGe(G , H, K ) ' Mn (Q(ξk ) ∗ NG (K )/H),
for k = [H : K ] and n = [G : NG (K )].
Problems
Aim
Rational group ring
Finite group ring
Problems
Pi for nilpotent groups
Theorem (Jespers, Olteanu, del Rı́o; 2011)
Let G be a finite nilpotent group, e a pci of QG . A complete
set of orthogonal pi of QGe consists of the conjugates of β by
the elements of T , where β and T are defined algorithmically in
4 cases.
Aim
Rational group ring
Finite group ring
Problems
Pi for nilpotent groups
Theorem (Jespers, Olteanu, del Rı́o; 2011)
Let G be a finite nilpotent group, e a pci of QG . A complete
set of orthogonal pi of QGe consists of the conjugates of β by
the elements of T , where β and T are defined algorithmically in
4 cases.
Example odd groups
Let G be a finite nilpotent group of odd order and e = e(G , H, K )
for (H, K ) a strong Shoda pair of G , H/K = hai. Then hai has a
cyclic complement hbi in NG (K )/K . A complete set of
orthogonal pi of QGe consists of all G -conjugates of e
bε(H, K ).
Aim
Rational group ring
Finite group ring
q-cyclotomic classes
Let F be a finite field of order q. Let G be a group of order n,
(n, q) = 1 and let Z∗n be the group of units of Zn . Define
Q = hq n i ⊆ Z∗n and consider the action
Q × G → G : (m, g ) 7→ g m .
Problems
Aim
Rational group ring
Finite group ring
q-cyclotomic classes
Let F be a finite field of order q. Let G be a group of order n,
(n, q) = 1 and let Z∗n be the group of units of Zn . Define
Q = hq n i ⊆ Z∗n and consider the action
Q × G → G : (m, g ) 7→ g m .
Definition
The q-cyclotomic classes of G are the orbits of G under the
action of Q on G .
Problems
Aim
Rational group ring
Finite group ring
q-cyclotomic classes
Let F be a finite field of order q. Let G be a group of order n,
(n, q) = 1 and let Z∗n be the group of units of Zn . Define
Q = hq n i ⊆ Z∗n and consider the action
Q × G → G : (m, g ) 7→ g m .
Definition
The q-cyclotomic classes of G are the orbits of G under the
action of Q on G .
Let Irr(G ) be the group of irreducible characters of G . We
denote by C(G ) the set of q-cyclotomic classes of Irr(G ) which
consist of linear faithful characters of G .
Problems
Aim
Rational group ring
Finite group ring
Problems
Building blocks
Definition (Broche, del Rı́o; 2007)
Let K E H ≤ G be such that H/K is cyclic of order k and
C ∈ C(H/K ). If χ ∈ C and tr = trF(ξk )/F denotes the field trace of
the Galois extension F(ξk )/F, then we set
X
εC (H, K ) = |H|−1
tr(χ(hK ))h−1 .
h∈H
Furhermore, eC (G , H, K ) denotes the sum of the different
G -conjugates of εC (H, K ).
Aim
Rational group ring
Finite group ring
Connection QG and FG
Theorem (Broche, del Rı́o; 2007)
Let G be a finite group and F a finite field such that FG is
semisimple. Let X be a set of strong Shoda pairs of G . If every
pci of QG is of the form e(G , H, K ) for (H, K ) ∈ X , then every
pci of FG is of the form eC (G , H, K ) for (H, K ) ∈ X and
C ∈ C(H/K ).
Problems
Aim
Rational group ring
Finite group ring
Problems
Some definitions
Let K E H ≤ G be such that H/K is cyclic. Then
N = NG (H) ∩ NG (K ) acts on H/K by conjugation and this
induces an action of N on the set of q-cyclotomic classes of H/K .
Now define EG (H, K ) as the stabilizer of any q-cyclotomic class
of H/K containing generators of H/K .
Aim
Rational group ring
Finite group ring
Pci for abelian-by-supersolvable groups
Theorem (Broche, del Rı́o; 2007)
Let G be an abelian-by-supersolvable group and F a finite field
of order q such that FG is semisimple, then each pci of FG is of
the form eC (G , H, K ) for (H, K ) a strong Shoda pair of G and
C ∈ C(H/K ).
Furthermore, the Wedderburn component
FGeC (G , H, K ) ' M[G :H] (Fqo/[E :K ] ),
where E = EG (H, K ) and o the multiplicative order of q modulo
[H : K ].
Problems
Aim
Rational group ring
Finite group ring
Pi for nilpotent groups
Theorem (Olteanu, VG; 2011)
Let F be a finite field and G a finite nilpotent group such that
FG is semisimple and let e be a pci of FG . A complete set of
orthogonal pi of FGe consists of the conjugates of β by the
elements of T , where β and T are defined algorithmically in 3
cases.
Problems
Aim
Rational group ring
Finite group ring
Problems
Pi for nilpotent groups
Theorem (Olteanu, VG; 2011)
Let F be a finite field and G a finite nilpotent group such that
FG is semisimple and let e be a pci of FG . A complete set of
orthogonal pi of FGe consists of the conjugates of β by the
elements of T , where β and T are defined algorithmically in 3
cases.
Example odd groups
Let G be a finite nilpotent group of odd order and
e = eC (G , H, K ) for (H, K ) a strong Shoda pair of G , H/K = hai
and C ∈ C(H/K ). Then hai has a cyclic complement hbi in
EG (H, K )/K . A complete set of orthogonal pi of FGe consists
of all G -conjugates of e
bεC (H, K ).
Aim
Rational group ring
Finite group ring
Structure of the proof
I
Follow proof of QG
I
Project pi of QGe on FGeC (Lemma Broche, del Rı́o; 2007)
I
Use technical arguments
Problems
Aim
Rational group ring
Problems
Next step: finite metacyclic groups
Finite group ring
Problems
Aim
Rational group ring
Finite group ring
Problems
Problems
Next step: finite metacyclic groups
I
m
G = Cp,qm = ha, b | aq = 1 = b p , b −1 ab = ar i, p, q prime:
Making use of
I
I
I
description of simple components as classical crossed products
with trivial twisting
explicit isomorphism ψ of classical crossed products with trivial
twisting as matrix algebra
observe that ψ(b) is permutation matrix and diagonalize
ψ(e
b) = PE11 P −1
Problem: Describe ψ −1 (P) in terms of elements of QG
Aim
Rational group ring
Finite group ring
Problems
Problems
Next step: finite metacyclic groups
I
m
G = Cp,qm = ha, b | aq = 1 = b p , b −1 ab = ar i, p, q prime:
Making use of
I
I
I
description of simple components as classical crossed products
with trivial twisting
explicit isomorphism ψ of classical crossed products with trivial
twisting as matrix algebra
observe that ψ(b) is permutation matrix and diagonalize
ψ(e
b) = PE11 P −1
Problem: Describe ψ −1 (P) in terms of elements of QG
I
2
G = Cp2 ,q = ha, b | aq = 1 = b p , b −1 ab = ar i, p, q prime:
Problem: Simple components do not always have trivial
twisting
→ Needs new idea