Group rings of nite strongly monomial groups: central units (and ...homepages.vub.ac.be/~ivgelder/Publications/Lecce-jun2013.pdf · Lecce, June 10-14, 2013. Background Group rings

Group rings of finite strongly monomial groups: centralunits (and primitive idempotents)

joint work with E. Jespers, G. Olteanu and A. del Rıo

Inneke Van Gelder

Vrije Universiteit Brussel

Advances in Group Theory and Applications,Lecce, June 10-14, 2013

Background

Group rings

Let G be a group and R a ring.

RG is the set of all linear combinations∑g∈G

rgg ,

with rg ∈ R and rg 6= 0 for only finitely many coefficients.Operations: ∑

g∈Grgg

+

∑g∈G

sgg

=∑g∈G

(rg + sg )g

∑g∈G

rgg

·∑

g∈Gsgg

=∑

g ,h∈Grg shgh

Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 2 / 17

Background

Group rings

Let G be a group and R a ring.RG is the set of all linear combinations∑

g∈Grgg ,

with rg ∈ R and rg 6= 0 for only finitely many coefficients.

Operations: ∑g∈G

rgg

+

∑g∈G

sgg

=∑g∈G

(rg + sg )g

∑g∈G

rgg

·∑

g∈Gsgg

=∑

g ,h∈Grg shgh


Background

Group rings

Let G be a group and R a ring.RG is the set of all linear combinations∑

g∈Grgg ,

with rg ∈ R and rg 6= 0 for only finitely many coefficients.Operations: ∑

g∈Grgg

+

∑g∈G

sgg

=∑g∈G

(rg + sg )g

∑g∈G

rgg

·∑

g∈Gsgg

=∑

g ,h∈Grg shgh


Background

Units

Let R be a ring with unity 1.

U(R) is the set of all units of R:

U(R) = {r ∈ R | ∃s ∈ R : rs = 1 = sr}.

Example

Let G be a finite group.U(ZG ) = ?


Background

Units

Let R be a ring with unity 1.U(R) is the set of all units of R:

U(R) = {r ∈ R | ∃s ∈ R : rs = 1 = sr}.

Example



Background

Units

Let R be a ring with unity 1.U(R) is the set of all units of R:

U(R) = {r ∈ R | ∃s ∈ R : rs = 1 = sr}.

Example



Background

Example: Bass units

Let G be a group, g an element of G of order n and k and m positiveintegers such that km ≡ 1 mod n.

The Bass (cyclic) unit withparameters g , k ,m is the element in ZG

uk,m(g) = (1 + g + · · ·+ gk−1)m +(1− km)

n(1 + g + g2 + · · ·+ gn−1)

with inverse in ZG

(1 + gk + · · ·+ gk(i−1))m +(1− im)

n(1 + g + g2 + · · ·+ gn−1)

where i is any integer such that ki ≡ 1 mod n.

Theorem [Bass-Milnor]

If G is a finite abelian group, then the Bass units generate a subgroup offinite index in U(ZG ).


Background

Example: Bass units

Let G be a group, g an element of G of order n and k and m positiveintegers such that km ≡ 1 mod n.The Bass (cyclic) unit withparameters g , k ,m is the element in ZG

uk,m(g) = (1 + g + · · ·+ gk−1)m +(1− km)

n(1 + g + g2 + · · ·+ gn−1)

with inverse in ZG

(1 + gk + · · ·+ gk(i−1))m +(1− im)

n(1 + g + g2 + · · ·+ gn−1)





Background

Example: Bass units


uk,m(g) = (1 + g + · · ·+ gk−1)m +(1− km)

n(1 + g + g2 + · · ·+ gn−1)

with inverse in ZG

(1 + gk + · · ·+ gk(i−1))m +(1− im)

n(1 + g + g2 + · · ·+ gn−1)





Background

Example: Bass units


uk,m(g) = (1 + g + · · ·+ gk−1)m +(1− km)

n(1 + g + g2 + · · ·+ gn−1)

with inverse in ZG

(1 + gk + · · ·+ gk(i−1))m +(1− im)

n(1 + g + g2 + · · ·+ gn−1)





Background

Wedderburn decomposition

The Wedderburn decomposition of QG is the decomposition into simplealgebras

A1 ⊕ · · · ⊕ Ak .

Each simple component is determined by a primitive central idempotentei , i.e.

Ai = QGei .


Background

Wedderburn decomposition

The Wedderburn decomposition of QG is the decomposition into simplealgebras

A1 ⊕ · · · ⊕ Ak .

Each simple component is determined by a primitive central idempotentei , i.e.

Ai = QGei .


Background

Strongly monomial groups

Let χ be an irreducible (complex) character of G .

χ is strongly monomial if there is a strong Shoda pair (H,K ) of G anda linear character θ of H with kernel K such that χ = θG .The group G is strongly monomial if every irreducible character of G isstrongly monomial.

Example

Abelian-by-supersolvable groups are strongly monomial.


Background


Let χ be an irreducible (complex) character of G .χ is strongly monomial if there is a strong Shoda pair (H,K ) of G anda linear character θ of H with kernel K such that χ = θG .

The group G is strongly monomial if every irreducible character of G isstrongly monomial.

Example



Background


Let χ be an irreducible (complex) character of G .χ is strongly monomial if there is a strong Shoda pair (H,K ) of G anda linear character θ of H with kernel K such that χ = θG .The group G is strongly monomial if every irreducible character of G isstrongly monomial.

Example



Background


Let χ be an irreducible (complex) character of G .χ is strongly monomial if there is a strong Shoda pair (H,K ) of G anda linear character θ of H with kernel K such that χ = θG .The group G is strongly monomial if every irreducible character of G isstrongly monomial.

Example



Background

The Wedderburn decomposition for strongly monomialgroups

Theorem [Olivieri-del Rıo-Simon]

Let G be a finite strongly monomial group, then the Wedderburndecomposition is as follows:

QG =⊕(H,K)

QGe(G ,H,K ) =⊕(H,K)

Mn(Q(ξk) ∗ NG (K )/H)

with (H,K ) running on strong Shoda pairs of G .


Virtual basis of the center

Virtual basis

It is well known that

Z(U(ZG )) = ±Z(G )× T ,

where T is a finitely generated free abelian group.

A virtual basis of Z(U(ZG )) is a set of multiplicatively independentelements of Z(U(ZG )) which generate a subgroup of finite index inZ(U(ZG )).



Virtual basis

It is well known that

Z(U(ZG )) = ±Z(G )× T ,

where T is a finitely generated free abelian group.A virtual basis of Z(U(ZG )) is a set of multiplicatively independentelements of Z(U(ZG )) which generate a subgroup of finite index inZ(U(ZG )).



Main strategy

Using the Wedderburn decomposition of QG , we see that

ZG ⊂⊕(H,K)

ZGe(G ,H,K ) =∏

(H,K)

(1− e(G ,H,K ) + ZGe(G ,H,K )).

Because of the structure of QGe(G ,H,K ), we know that

Z(ZG ) /∏

(H,K)

(1− e(G ,H,K ) + Z[ζ[H:K ]]NG (K)/H).

Strategy

1 Compute a virtual basis of U(Z[ζ[H:K ]]NG (K)/H)

2 Lift the virtual bases of all components to ZG .



Main strategy


ZG ⊂⊕(H,K)

ZGe(G ,H,K ) =∏

(H,K)

(1− e(G ,H,K ) + ZGe(G ,H,K )).


Z(ZG ) /∏

(H,K)

(1− e(G ,H,K ) + Z[ζ[H:K ]]NG (K)/H).

Strategy





Main strategy


ZG ⊂⊕(H,K)

ZGe(G ,H,K ) =∏

(H,K)

(1− e(G ,H,K ) + ZGe(G ,H,K )).


Z(ZG ) /∏

(H,K)

(1− e(G ,H,K ) + Z[ζ[H:K ]]NG (K)/H).

Strategy





Step 1: Cyclotomic units

If n > 1 and k is an integer coprime with n then

ηk(ζn) =1− ζkn1− ζn

= 1 + ζn + ζ2n + · · ·+ ζk−1n

is a unit of Z[ζn]. These units are called the cyclotomic units of Q(ζn).

Theorem [Washington]

{ηk(ζpn) | 1 < k < pn

2 , p - k} generates a free abelian subgroup of finiteindex in U(Z[ζpn ]) when p is prime.




If n > 1 and k is an integer coprime with n then

ηk(ζn) =1− ζkn1− ζn

= 1 + ζn + ζ2n + · · ·+ ζk−1n

is a unit of Z[ζn]. These units are called the cyclotomic units of Q(ζn).

Theorem [Washington]

{ηk(ζpn) | 1 < k < pn

2 , p - k} generates a free abelian subgroup of finiteindex in U(Z[ζpn ]) when p is prime.




For a subgroup A of Gal(Q(ζpn)/Q) and u ∈ Q(ζpn), we define πA(u) tobe∏σ∈A σ(u).

Lemma [Jespers-del Rıo-Olteanu-VG]

Let A be a subgroup of Gal(Q(ζpn)/Q). Let I be a set of cosetrepresentatives of U(Z/pnZ) modulo 〈A, φ−1〉 containing 1. Then the set

{πA (ηk(ζpn)) | k ∈ I \ {1}}

is a virtual basis of U(Z[ζpn ]A

).




For a subgroup A of Gal(Q(ζpn)/Q) and u ∈ Q(ζpn), we define πA(u) tobe∏σ∈A σ(u).

Lemma [Jespers-del Rıo-Olteanu-VG]

Let A be a subgroup of Gal(Q(ζpn)/Q). Let I be a set of cosetrepresentatives of U(Z/pnZ) modulo 〈A, φ−1〉 containing 1. Then the set

{πA (ηk(ζpn)) | k ∈ I \ {1}}

is a virtual basis of U(Z[ζpn ]A

).



Step 2: Generalized Bass units

If G is a finite group, M a normal subgroup of G , g ∈ G and k and mpositive integers such that gcd(k, |g |) = 1 and km ≡ 1 mod |g |.

One can show that the element uk,mnG ,M(1− M + gM) is an unit in ZG

for some integer nG ,M . We call this unit a generalized Bass unit basedon g and M with parameters k and m.




If G is a finite group, M a normal subgroup of G , g ∈ G and k and mpositive integers such that gcd(k, |g |) = 1 and km ≡ 1 mod |g |.One can show that the element uk,mnG ,M

(1− M + gM) is an unit in ZGfor some integer nG ,M . We call this unit a generalized Bass unit basedon g and M with parameters k and m.




Let k be a positive integer coprime with p and let r be an arbitrary integer.

For every 0 ≤ j ≤ s ≤ n we construct recursively the following products ofgeneralized Bass units of ZH:

css (H,K , k , r) = 1,

and, for 0 ≤ j ≤ s − 1,

csj (H,K , k , r) =

∏h∈Hj

uk,Opn (k)nH,K(g rpn−s

hK + 1− K )

ps−j−1

s−1∏l=j+1

csl (H,K , k , r)−1

(j−1∏l=0

cs+l−jl (H,K , k , r)−1

).




Let k be a positive integer coprime with p and let r be an arbitrary integer.For every 0 ≤ j ≤ s ≤ n we construct recursively the following products ofgeneralized Bass units of ZH:

css (H,K , k , r) = 1,

and, for 0 ≤ j ≤ s − 1,

csj (H,K , k , r) =

∏h∈Hj


hK + 1− K )

ps−j−1

s−1∏l=j+1

csl (H,K , k , r)−1

(j−1∏l=0

cs+l−jl (H,K , k , r)−1

).





css (H,K , k , r) = 1,

and, for 0 ≤ j ≤ s − 1,

csj (H,K , k , r) =

∏h∈Hj


hK + 1− K )

ps−j−1

s−1∏l=j+1

csl (H,K , k , r)−1

(j−1∏l=0

cs+l−jl (H,K , k , r)−1

).





css (H,K , k , r) = 1,

and, for 0 ≤ j ≤ s − 1,

csj (H,K , k , r) =

∏h∈Hj


hK + 1− K )

ps−j−1

s−1∏l=j+1

csl (H,K , k , r)−1

(j−1∏l=0

cs+l−jl (H,K , k , r)−1

).




Proposition [Jespers-del Rıo-Olteanu-VG]

Let H be a finite group and K a subgroup of H such that H/K = 〈gK 〉 iscyclic of order pn.Then cn0 (H,K , k , r) 7→ 1− e(G ,H,K ) + ηk(ζrpn)m for some integer m.



Main Theorem

Theorem [Jespers-del Rıo-Olteanu-VG]

Let G be a strongly monomial group such that there is a complete andnon-redundant set S of strong Shoda pairs (H,K ) of G with the propertythat each [H : K ] is a prime power. For every (H,K ) ∈ S, let TK be aright transversal of NG (K ) in G , let I(H,K) be a set of representatives ofU(Z/[H : K ]Z) modulo 〈NG (K )/H,−1〉 containing 1 and let[H : K ] = p

n(H,K)

(H,K), with p(H,K) prime. Then∏t∈TK

∏x∈NG (K)/H

cn(H,K)

0 (H,K , k, x)t : (H,K ) ∈ S, k ∈ I(H,K) \ {1}

is a virtual basis of Z(U(ZG )).



Examples of groups satisfying the conditions

Cqm o Cpn with Cpn acting faithfully on Cqm

A4

D2n =⟨a, b | an = b2 = 1, ab = a−1

⟩⇔ n is a power of a prime

Q4n =⟨x , y | x2n = y4 = 1, xn = y2, xy = x−1

⟩⇔ n is a power of 2

. . .





A4

D2n =⟨a, b | an = b2 = 1, ab = a−1


Q4n =⟨x , y | x2n = y4 = 1, xn = y2, xy = x−1


. . .





A4

D2n =⟨a, b | an = b2 = 1, ab = a−1


Q4n =⟨x , y | x2n = y4 = 1, xn = y2, xy = x−1


. . .





A4

D2n =⟨a, b | an = b2 = 1, ab = a−1


Q4n =⟨x , y | x2n = y4 = 1, xn = y2, xy = x−1


. . .



Reference

E. Jespers, G. Olteanu, A. del Rıo, I. Van Gelder, Group rings of finitestrongly monomial groups: Central units and primitive idempotents,Journal of Algebra, Volume 387, 1 August 2013, Pages 99-116


Documents

Group rings of nite strongly monomial groups: central units (and ...homepages.vub.ac.be/~ivgelder/Publications/Lecce-jun2013.pdf · Lecce, June 10-14, 2013. Background Group rings