Groups, Rings, and Group-Rings University of Alberta...

Finite group rings of nilpotent groups:a complete set of orthogonal primitive idempotents

Inneke Van Gelder

Groups, Rings, and Group-RingsUniversity of Alberta, Edmonton, Canada

July 11-15, 2011

Aim Rational group ring Finite group ring Problems

Our aim is to describe a complete set of orthogonal primitiveidempotents in each simple component of a semisimple groupalgebra in such a way that it provides a straightforwardimplementation.

Definition (Olivieri, del Rıo, Simon; 2004)

A strong Shoda pair of G is a pair (H,K ) of subgroups of Gsatisfying

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

ε(H,K ) =

{K if H = K∏

M/K∈M(H/K)(K − M) if H 6= K ,

with N = 1|N|

∑n∈N n for each group N and M(H/K ) the set of

all minimal normal non-trivial subgroups of H/K .

Definition (Olivieri, del Rıo, Simon; 2004)

A strong Shoda pair of G is a pair (H,K ) of subgroups of Gsatisfying

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

ε(H,K ) =

{K if H = K∏

M/K∈M(H/K)(K − M) if H 6= K ,

with N = 1|N|

∑n∈N n for each group N and M(H/K ) the set of

all minimal normal non-trivial subgroups of H/K .

Pci for abelian-by-supersolvable groups

Theorem (Olivieri, del Rıo, Simon; 2004)

Let G be an abelian-by-supersolvable group. Then e is a pci ofQG iff e = e(G ,H,K ) :=

∑t∈T ε(H,K )t for a strong Shoda

pair (H,K ) of G .

Let G be an abelian-by-supersolvable group and (H,K ) astrong 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 )].

Let G be an abelian-by-supersolvable group. Then e is a pci ofQG iff e = e(G ,H,K ) :=

∑t∈T ε(H,K )t for a strong Shoda

pair (H,K ) of G .

Let G be an abelian-by-supersolvable group and (H,K ) astrong 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 )].

Pi for nilpotent groups

Theorem (Jespers, Olteanu, del Rıo; 2011)

Let G be a finite nilpotent group, e a pci of QG . A completeset of orthogonal pi of QGe consists of the conjugates of β bythe elements of T , where β and T are defined algorithmically in4 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 = 〈a〉. Then 〈a〉 has acyclic complement 〈b〉 in NG (K )/K . A complete set oforthogonal pi of QGe consists of all G -conjugates of bε(H,K ).

Theorem (Jespers, Olteanu, del Rıo; 2011)

Let G be a finite nilpotent group, e a pci of QG . A completeset of orthogonal pi of QGe consists of the conjugates of β bythe elements of T , where β and T are defined algorithmically in4 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 = 〈a〉. Then 〈a〉 has acyclic complement 〈b〉 in NG (K )/K . A complete set oforthogonal pi of QGe consists of all G -conjugates of bε(H,K ).

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. DefineQ = 〈qn〉 ⊆ Z∗n and consider the action

Q× G → G : (m, g) 7→ gm.

Definition

The q-cyclotomic classes of G are the orbits of G under theaction of Q on G .

Let Irr(G ) be the group of irreducible characters of G . Wedenote by C(G ) the set of q-cyclotomic classes of Irr(G ) whichconsist of linear faithful characters of G .

Q× G → G : (m, g) 7→ gm.

Definition

Q× G → G : (m, g) 7→ gm.

Definition

Building blocks

Definition (Broche, del Rıo; 2007)

Let K E H ≤ G be such that H/K is cyclic of order k andC ∈ C(H/K ). If χ ∈ C and tr = trF(ξk )/F denotes the field trace ofthe Galois extension F(ξk)/F, then we set

εC (H,K ) = |H|−1∑h∈H

tr(χ(hK ))h−1.

Furhermore, eC (G ,H,K ) denotes the sum of the differentG -conjugates of εC (H,K ).

Connection QG and FG

Theorem (Broche, del Rıo; 2007)

Let G be a finite group and F a finite field such that FG issemisimple. Let X be a set of strong Shoda pairs of G . If everypci of QG is of the form e(G ,H,K ) for (H,K ) ∈ X , then everypci of FG is of the form eC (G ,H,K ) for (H,K ) ∈ X andC ∈ C(H/K ).

Some definitions

Let K E H ≤ G be such that H/K is cyclic. ThenN = NG (H) ∩ NG (K ) acts on H/K by conjugation and thisinduces 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 classof H/K containing generators of H/K .

Theorem (Broche, del Rıo; 2007)

Let G be an abelian-by-supersolvable group and F a finite fieldof order q such that FG is semisimple, then each pci of FG is ofthe form eC (G ,H,K ) for (H,K ) a strong Shoda pair of G andC ∈ 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 ].

Theorem (Olteanu, VG; 2011)

Let F be a finite field and G a finite nilpotent group such thatFG is semisimple and let e be a pci of FG . A complete set oforthogonal pi of FGe consists of the conjugates of β by theelements of T , where β and T are defined algorithmically in 3cases.

Example odd groups

Let G be a finite nilpotent group of odd order ande = eC (G ,H,K ) for (H,K ) a strong Shoda pair of G , H/K = 〈a〉and C ∈ C(H/K ). Then 〈a〉 has a cyclic complement 〈b〉 inEG (H,K )/K . A complete set of orthogonal pi of FGe consistsof all G -conjugates of bεC (H,K ).

Theorem (Olteanu, VG; 2011)

Let F be a finite field and G a finite nilpotent group such thatFG is semisimple and let e be a pci of FG . A complete set oforthogonal pi of FGe consists of the conjugates of β by theelements of T , where β and T are defined algorithmically in 3cases.

Example odd groups

Let G be a finite nilpotent group of odd order ande = eC (G ,H,K ) for (H,K ) a strong Shoda pair of G , H/K = 〈a〉and C ∈ C(H/K ). Then 〈a〉 has a cyclic complement 〈b〉 inEG (H,K )/K . A complete set of orthogonal pi of FGe consistsof all G -conjugates of bεC (H,K ).

Structure of the proof

I Follow proof of QG

I Project pi of QGe on FGeC (Lemma Broche, del Rıo; 2007)

I Use technical arguments

Problems

Next step: finite metacyclic groups

I G = Cp,qm = 〈a, b | aqm = 1 = bp, b−1ab = ar 〉, p, q prime:Making use of

I description of simple components as classical crossed productswith trivial twisting

I explicit isomorphism ψ of classical crossed products with trivialtwisting as matrix algebra

I observe that ψ(b) is permutation matrix and diagonalize

ψ(b) = PE11P−1

Problem: Describe ψ−1(P) in terms of elements of QG

I G = Cp2,q = 〈a, b | aq = 1 = bp2, b−1ab = ar 〉, p, q prime:

Problem: Simple components do not always have trivialtwisting→ Needs new idea

Problems

ψ(b) = PE11P−1

Problems

ψ(b) = PE11P−1

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