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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
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∈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
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 ) = ?
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 3 / 17
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 ) = ?
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 3 / 17
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 ) = ?
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 3 / 17
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 ).
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 4 / 17
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 ).
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 4 / 17
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 ).
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 4 / 17
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 ).
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 4 / 17
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 .
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 5 / 17
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 .
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 5 / 17
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.
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 6 / 17
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.
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 6 / 17
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.
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 6 / 17
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.
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 6 / 17
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 .
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 7 / 17
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 )).
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 8 / 17
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 )).
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 8 / 17
Virtual basis of the center
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 .
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 9 / 17
Virtual basis of the center
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 .
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 9 / 17
Virtual basis of the center
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 .
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 9 / 17
Virtual basis of the center
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.
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 10 / 17
Virtual basis of the center
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.
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 10 / 17
Virtual basis of the center
Step 1: Cyclotomic units
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
).
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 11 / 17
Virtual basis of the center
Step 1: Cyclotomic units
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
).
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 11 / 17
Virtual basis of the center
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.
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 12 / 17
Virtual basis of the center
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 ZGfor some integer nG ,M . We call this unit a generalized Bass unit basedon g and M with parameters k and m.
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 12 / 17
Virtual basis of the center
Step 2: Generalized Bass units
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
).
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 13 / 17
Virtual basis of the center
Step 2: Generalized Bass units
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
).
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 13 / 17
Virtual basis of the center
Step 2: Generalized Bass units
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
).
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 13 / 17
Virtual basis of the center
Step 2: Generalized Bass units
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
).
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 13 / 17
Virtual basis of the center
Step 2: Generalized Bass units
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.
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 14 / 17
Virtual basis of the center
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 )).
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 15 / 17
Virtual basis of the center
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
. . .
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 16 / 17
Virtual basis of the center
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
. . .
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 16 / 17
Virtual basis of the center
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
. . .
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 16 / 17
Virtual basis of the center
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
. . .
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 16 / 17
Virtual basis of the center
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
Inneke Van Gelder (Brussels) Central units Lecce, June 10-14, 2013 17 / 17