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9 The basic theory behind Wedderga
In this chapter we describe the theory that is behind the algorithms used by Wedderga.
All the rings considered in this chapter are associative and have an identity.
We use the following notation: ℚℚ denotes the field of rationals and Fq𝔽q the finite field of order qq. For every positive integer kk, we denote a complex kk-th primitive root of unity by ξkξk and so ℚ(ξk)ℚ(ξk) is the kk-th cyclotomic extension of ℚℚ.
9.1 Group rings and group algebras
Given a group GG and a ring RR, the group ring RGRG over the group GG with coefficients in RR is the ring whose underlying additive group is a right R−R−module with basis GG such that the product is defined by the following rule
(gr)(hs)=(gh)(rs)
(gr)(hs)=(gh)(rs)
for r,s∈Rr,s∈R and g,h∈Gg,h∈G, and extended to RGRG by linearity.
A group algebra is a group ring in which the coefficient ring is a field.
9.2 Semisimple group algebras
We say that a ring RR is semisimple if it is a direct sum of simple left (alternatively right) ideals or equivalently if RR is isomorphic to a direct product of simple algebras each one isomorphic to a matrix ring over a division ring.
By Maschke's Theorem, if GG is a finite group then the group algebra FGFG is semisimple if and only the characteristic of the coefficient field FF does not divide the order of GG.
In fact, an arbitrary group ring RGRG is semisimple if and only if the coefficient ring RR is semisimple, the group GG is finite and the order of GG is invertible in RR.
Some authors use the notion semisimple ring for rings with zero
Jacobson radical. To avoid confusion we usually refer to semisimple
rings as semisimple artinian rings.
9.3 Wedderburn components
If RR is a semisimple ring (9.2) then the Wedderburn decomposition of RR is the decomposition of RR as a direct product of simple algebras. The factors of this Wedderburn decomposition are called Wedderburn components of RR. Each Wedderburn component of RR is of the form ReRe for ee a primitive central idempotent (9.4) of RR.
Let FGFG be a semisimple group algebra (9.2). If FF has positive characteristic, then the Wedderburn components of FGFG are matrix algebras over finite extensions of FF. If FF has zero characteristic then by the Brauer-Witt Theorem [Yam74], the Wedderburn components of FGFG are Brauer equivalent (9.5) to cyclotomic algebras (9.11).
The main functions of Wedderga compute the Wedderburn components of a semisimple group algebra FGFG,
such that the coefficient field is either an abelian number field (i.e.
a subfield of a finite cyclotomic extension of the rationals) or a
finite field. In the finite case, the Wedderburn components are matrix
algebras over finite fields and so can be described by the size of the
matrices and the size of the finite field.
In the zero characteristic case each Wedderburn component AA is Brauer equivalent (9.5) to a cyclotomic algebra (9.11) and therefore AA is a (possibly fractional) matrix algebra over cyclotomic algebra and can be described numerically in one of the following three forms:
[n,K],
[n,K],
[n,K,k,[d,α,β]],
[n,K,k,[d,α,β]],
[n,K,k,[di,αi,βi]mi=1,[γi,j]1≤i<j≤n],
[n,K,k,[di,αi,βi]mi=1,[γi,j]1≤i<j≤n],
where nn is the matrix size, KK is the centre of AA (a finite field extension of FF) and the remaining data are integers whose interpretation is explained in 9.12.
In some cases (for the zero characteristic coefficient field) the size nn of the matrix algebras is not a positive integer but a positive rational number. This is a consequence of the fact that the Brauer-Witt Theorem [Yam74] only ensures that each Wedderburn component (9.3) of a semisimple group algebra is Brauer equivalent (9.5) to a cyclotomic algebra (9.11), but not necessarily isomorphic to a full matrix algebra of a cyclotomic algebra. For example, a Wedderburn component DD of a group algebra can be a division algebra but not a cyclotomic algebra. In this case Mn(D)Mn(D) is a cyclotomic algebra CC for some nn and therefore DD can be described as M1/n(C)M1/n(C) (see last Example in WedderburnDecomposition
(2.1-1)).
The main algorithm of Wedderga is based on a computational oriented proof of the Brauer-Witt Theorem due to Olteanu [Olt07] which uses previous work by Olivieri, del Río and Simón [ORS04] for rational group algebras of strongly monomial groups (9.16).
9.4 Characters and primitive central idempotents
A primitive central idempotent of a ring RR is a non-zero central idempotent ee which cannot be written as the sum of two non-zero central idempotents of ReRe, or equivalently, such that ReRe is indecomposable as a direct product of two non-trivial two-sided ideals.
The Wedderburn components (9.3) of a semisimple ring RR are the rings of the form ReRe for ee running over the set of primitive central idempotents of RR.
Let FGFG be a semisimple group algebra (9.2) and χχ an irreducible character of GG (in an algebraic closure of FF). Then there is a unique Wedderburn component A=AF(χ)A=AF(χ) of FGFG such that χ(A)≠0χ(A)≠0. Let eF(χ)eF(χ) denote the unique primitive central idempotent of FGFG in AF(χ)AF(χ), that is the identity of AF(χ)AF(χ), i.e.
AF(χ)=FGeF(χ).
AF(χ)=FGeF(χ).
The centre of AF(χ)AF(χ) is F(χ)=F(χ(g):g∈G)F(χ)=F(χ(g):g∈G), the field of character values of χχ over FF.
The map χ↦AF(χ)χ↦AF(χ) defines a surjective map from the set of irreducible characters of GG (in an algebraic closure of FF) onto the set of Wedderburn components of FGFG.
Equivalently, the map χ↦eF(χ)χ↦eF(χ) defines a surjective map from the set of irreducible characters of GG (in an algebraic closure of FF) onto the set of primitive central idempontents of FGFG.
If the irreducible character χχ of GG takes values in FF then
eF(χ)=e(χ)=χ(1)|G|∑g∈Gχ(g−1)g.
eF(χ)=e(χ)=χ(1)|G|∑g∈Gχ(g−1)g.
In general one has
eF(χ)=∑σ∈Gal(F(χ)/F)e(σ∘χ).
eF(χ)=∑σ∈Gal(F(χ)/F)e(σ∘χ).
9.5 Central simple algebras and Brauer equivalence
Let KK be a field. A central simple KK-algebra is a finite dimensional KK-algebra with center KK which has no non-trivial proper ideals. Every central simple KK-algebra is isomorphic to a matrix algebra Mn(D)Mn(D) where DD is a division algebra (which is finite-dimensional over KK and has centre KK). The division algebra DD is unique up to KK-isomorphisms.
Two central simple KK-algebras AA and BB are said to be Brauer equivalent, or simply equivalent, if there is a division algebra DD and two positive integers mm and nn such that AA is isomorphic to Mm(D)Mm(D) and BB is isomorphic to Mn(D)Mn(D).
9.6 Crossed Products
Let RR be a ring and GG a group.
Intrinsic definition. A crossed product [Pas89] of GG over RR (or with coefficients in RR) is a ring R∗GR∗G with a decomposition into a direct sum of additive subgroups
R∗G=⨁g∈GAg
R∗G=⨁g∈GAg
such that for each g,hg,h in GG one has:
* A1=RA1=R (here 11 denotes the identity of GG),
* AgAh=AghAgAh=Agh and
* AgAg has a unit of R∗GR∗G.
Extrinsic definition. Let Aut(R)Aut(R) denote the group of automorphisms of RR and let R∗R∗ denote the group of units of RR.
Let a:G→Aut(R)a:G→Aut(R) and t:G×G→R∗t:G×G→R∗ be mappings satisfying the following conditions for every gg, hh and kk in GG:
(1) a(gh)−1a(g)a(h)a(gh)−1a(g)a(h) is the inner automorphism of RR induced by t(g,h)t(g,h) (i.e. the automorphism x↦t(g,h)−1xt(g,h)x↦t(g,h)−1xt(g,h)) and
(2) t(gh,k)t(g,h)k=t(g,hk)t(h,k)t(gh,k)t(g,h)k=t(g,hk)t(h,k), where for g∈Gg∈G and x∈Rx∈R we denote a(g)(x)a(g)(x) by xgxg.
The crossed product [Pas89] of GG over RR (or with coefficients in RR), action aa and twisting tt is the ring
R∗taG=⨁g∈GugR
R∗taG=⨁g∈GugR
where {ug:g∈G}{ug:g∈G} is a set of symbols in one-to-one correspondence with GG, with addition and multiplication defined by
(ugr)+(ugs)=ug(r+s),(ugr)(uhs)=ught(g,h)rhs
(ugr)+(ugs)=ug(r+s),(ugr)(uhs)=ught(g,h)rhs
for g,h∈Gg,h∈G and r,s∈Rr,s∈R, and extended to R∗taGR∗taG by linearity.
The associativity of the product defined is a consequence of conditions (1) and (2) [Pas89].
Equivalence of the two definitions. Obviously the crossed product of GG over RR defined using the extrinsic definition is a crossed product of GG over u1Ru1R in the sense of the first definition. Moreover, there is r0r0 in R∗R∗ such that u1r0u1r0 is the identity of R∗taGR∗taG and the map r↦u1r0rr↦u1r0r is a ring isomorphism R→u1RR→u1R.
Conversely, let R∗G=⨁g∈GAgR∗G=⨁g∈GAg be an (intrinsic) crossed product and select for each g∈Gg∈G a unit ug∈Agug∈Ag of R∗GR∗G. This is called a basis of units for the crossed product R∗GR∗G. Then the maps a:G→Aut(R)a:G→Aut(R) and t:G×G→R∗t:G×G→R∗ given by
rg=u−1grug,t(g,h)=u−1ghuguh(g,h∈G,r∈R)
rg=u−1grug,t(g,h)=u−1ghuguh(g,h∈G,r∈R)
satisfy conditions (1) and (2) and R∗G=R∗taGR∗G=R∗taG.
The choice of a basis of units ug∈Agug∈Ag determines the action aa and twisting tt. If {ug∈Ag:g∈G}{ug∈Ag:g∈G} and {vg∈Ag:g∈G}{vg∈Ag:g∈G} are two sets of units of R∗GR∗G then vg=ugrgvg=ugrg for some units rgrg of RR.
Changing the basis of units results in a change of the action and the
twisting and so changes the extrinsic definition of the crossed product
but it does not change the intrinsic crossed product.
It is customary to select u1=1u1=1. In that case a(1)a(1) is the identity map of RR and t(1,g)=t(g,1)=1t(1,g)=t(g,1)=1 for each gg in GG.
9.7 Cyclic Crossed Products
Let R∗G=⨁g∈GAgR∗G=⨁g∈GAg be a crossed product (9.6) and assume that G=⟨g⟩G=⟨g⟩ is cyclic. Then the crossed product can be given using a particularly nice description.
Select a unit uu in AgAg, and let aa be the automorphism of RR given by ra=u−1rura=u−1ru.
If GG is infinite then set ugk=ukugk=uk for every integer kk. Then
R∗G=R[u|ru=ura],
R∗G=R[u|ru=ura],
a skew polynomial ring. Therefore in this case R∗GR∗G is determined by
[R,a].
[R,a].
If GG is finite of order dd then set ugk=ukugk=uk for 0≤k<d0≤k<d. Then b=ud∈Rb=ud∈R and
R∗G=R[u|ru=ura,ud=b]
R∗G=R[u|ru=ura,ud=b]
Therefore, R∗GR∗G is completely determined by the following data:
[R,[d,a,b]]
[R,[d,a,b]]
9.8 Abelian Crossed Products
Let R∗G=⨁g∈GAgR∗G=⨁g∈GAg be a crossed product (9.6) and assume that GG is abelian. Then the crossed product can be given using a simple description.
Express GG as a direct sum of cyclic groups:
G=⟨g1⟩×⋯×⟨gn⟩
G=⟨g1⟩×⋯×⟨gn⟩
and for each i=1,…,ni=1,…,n select a unit uiui in AgiAgi.
Each element gg of GG has a unique expression
g=gk11⋯gknn,
g=gk11⋯gknn,
where kiki is an arbitrary integer, if gigi has infinite order, and 0≤ki<di0≤ki<di, if gigi has finite order didi. Then one selects a basis for the crossed product by taking
ug=ugk11⋯gknn=uk11⋯uknn.
ug=ugk11⋯gknn=uk11⋯uknn.
* For each i=1,…,ni=1,…,n, let aiai be the automorphism of RR given by rai=u−1iruirai=u−1irui.
* For each 1≤i<j≤n1≤i<j≤n, let ti,j=u−1ju−1iujui∈Rti,j=u−1ju−1iujui∈R.
* If gigi has finite order didi, let bi=udii∈Rbi=udii∈R.
Then
R∗G=R[u1,…,un|rui=uirai,ujui=tijuiuj,udii=bi(1≤i<j≤n)],
R∗G=R[u1,…,un|rui=uirai,ujui=tijuiuj,udii=bi(1≤i<j≤n)],
where the last relation vanishes if gigi has infinite order.
Therefore R∗GR∗G is completely determined by the following data:
[R,[di,ai,bi]ni=1,[ti,j]1≤i<j≤n].
[R,[di,ai,bi]ni=1,[ti,j]1≤i<j≤n].
9.9 Classical crossed products
A classical crossed product is a crossed product L∗taGL∗taG, where L/KL/K is a finite Galois extension, G=Gal(L/K)G=Gal(L/K) is the Galois group of L/KL/K and aa is the natural action of GG on LL. Then tt is a 22-cocycle and the crossed product (9.6) L∗taGL∗taG is denoted by (L/K,t)(L/K,t). The crossed product (L/K,t)(L/K,t) is known to be a central simple KK-algebra [Rei03].
9.10 Cyclic Algebras
A cyclic algebra is a classical crossed product (9.9) (L/K,t)(L/K,t) where L/KL/K is a finite cyclic field extension. The cyclic algebras have a very simple form.
Assume that Gal(L/K)Gal(L/K) is generated by gg and has order dd. Let u=ugu=ug be the basis unit (9.6) of the crossed product corresponding to gg and take the remaining basis units for the crossed product by setting ugi=uiugi=ui, (i=0,1,…,d−1i=0,1,…,d−1). Then a=un∈Ka=un∈K. The cyclic algebra is usually denoted by (L/K,a)(L/K,a) and one has the following description of (L/K,t)(L/K,t)
(L/K,t)=(L/K,a)=L[u|ru=urg,ud=a].
(L/K,t)=(L/K,a)=L[u|ru=urg,ud=a].
9.11 Cyclotomic algebras
A cyclotomic algebra over FF is a classical crossed product (9.9) (F(ξ)/F,t)(F(ξ)/F,t), where FF is a field, ξξ is a root of unity in an extension of FF and t(g,h)t(g,h) is a root of unity for every gg and hh in Gal(F(ξ)/F)Gal(F(ξ)/F).
The Brauer-Witt Theorem [Yam74] asserts that every Wedderburn component (9.3) of a group algebra is Brauer equivalent (9.5) (over its centre) to a cyclotomic algebra.
9.12 Numerical description of cyclotomic algebras
Let A=(F(ξ)/F,t)A=(F(ξ)/F,t) be a cyclotomic algebra (9.11), where ξ=ξkξ=ξk is a kk-th root of unity. Then the Galois group G=Gal(F(ξ)/F)G=Gal(F(ξ)/F) is abelian and therefore one can obtain a simplified form for the description of cyclotomic algebras as for any abelian crossed product (9.8).
Then the n×nn×n matrix algebra Mn(A)Mn(A) can be described numerically in one of the following forms:
* If F(ξ)=FF(ξ)=F, (i.e. G=1G=1) then A=Mn(F)A=Mn(F) and thus the only data needed to describe AA are the matrix size nn and the field FF:
[n,F]
[n,F]
* If GG is cyclic (but not trivial) of order dd then AA is a cyclic cyclotomic algebra
A=F(ξ)[u|ξu=uξα,ud=ξβ]
A=F(ξ)[u|ξu=uξα,ud=ξβ]
and so Mn(A)Mn(A) can be described with the following data
[n,F,k,[d,α,β]],
[n,F,k,[d,α,β]],
where the integers kk, dd, αα and ββ satisfy the following conditions:
αd≡1modk,β(α−1)≡0modk.
αd≡1modk,β(α−1)≡0modk.
* If GG is abelian but not cyclic then Mn(A)Mn(A) can be described with the following data (see 9.8):
[n,F,k,[di,αi,βi]mi=1,[γi,j]1≤i<j≤m]
[n,F,k,[di,αi,βi]mi=1,[γi,j]1≤i<j≤m]
representing the n×nn×n matrix ring over the following algebra:
A=F(ξ)[u1,…,um∣ξui=uiξαi,udii=ξβi,usur=ξγrsurus,i=1,…,m,0≤r<s≤m]
A=F(ξ)[u1,…,um∣ξui=uiξαi,udii=ξβi,usur=ξγrsurus,i=1,…,m,0≤r<s≤m]
where
* {g1,…,gm}{g1,…,gm} is an independent set of generators of GG,
* didi is the order of gigi,
* αiαi, βiβi and γrsγrs are integers, and
ξgi=ξαi.
ξgi=ξαi.
9.13 Idempotents given by subgroups
Let GG be a finite group and FF a field whose characteristic does not divide the order of GG. If HH is a subgroup of GG then set
ˆH=|H|−1∑x∈Hx.
Hˆ=|H|−1∑x∈Hx.
The element ˆHHˆ is an idempotent of FGFG which is central in FGFG if and only if HH is normal in GG.
If HH is a proper normal subgroup of a subgroup KK of GG then set
ε(K,H)=∏L(ˆN−ˆL)
ε(K,H)=∏L(Nˆ−Lˆ)
where LL runs on the normal subgroups of KK which are minimal among the normal subgroups of KK containing NN properly. By convention, ε(K,K)=ˆKε(K,K)=Kˆ. The element ε(K,H)ε(K,H) is an idempotent of FGFG.
If HH and KK are subgroups of GG such that HH is normal in KK then e(G,K,H)e(G,K,H) denotes the sum of all different GG-conjugates of ε(K,H)ε(K,H). The element e(G,K,H)e(G,K,H) is central in FGFG. In general it is not an idempotent but if the different conjugates of ε(K,H)ε(K,H) are orthogonal then e(G,K,H)e(G,K,H) is a central idempotent of FGFG.
If (K,H)(K,H) is a Shoda Pair (9.14) of GG then there is a non-zero rational number aa such that ae(G,K,H))ae(G,K,H)) is a primitive central idempotent (9.4) of the rational group algebra ℚGℚG. If (K,H)(K,H) is a strong Shoda pair (9.15) of GG then e(G,K,H)e(G,K,H) is a primitive central idempotent of ℚGℚG.
Assume now that FF is a finite field of order qq, (K,H)(K,H) is a strong Shoda pair of GG and CC is a cyclotomic class of K/HK/H containing a generator of K/HK/H. Then eC(G,K,H)eC(G,K,H) is a primitive central idempotent of FGFG (see 9.17).
9.14 Shoda pairs of a group
Let GG be a finite group. A Shoda pair of GG is a pair (K,H)(K,H) of subgroups of GG for which there is a linear character χχ of KK with kernel HH such that the induced character χGχG in GG is irreducible. By [Sho33] or [ORS04], (K,H)(K,H) is a Shoda pair if and only if the following conditions hold:
* HH is normal in KK,
* K/HK/H is cyclic and
* if Kg∩K⊆HKg∩K⊆H for some g∈Gg∈G then g∈Kg∈K.
If (K,H)(K,H) is a Shoda pair and χχ is a linear character of K≤GK≤G with kernel HH then the primitive central idempotent (9.4) of ℚGℚG associated to the irreducible character χGχG is of the form e=eℚ(χG)=ae(G,K,H)e=eℚ(χG)=ae(G,K,H) for some a∈ℚa∈ℚ [ORS04] (see 9.13 for the definition of e(G,K,H)e(G,K,H)). In that case we say that ee is the primitive central idempotent realized by the Shoda pair (K,H)(K,H) of GG.
A group GG is monomial, that is every irreducible character of GG is monomial, if and only if every primitive central idempotent of ℚGℚG is realizable by a Shoda pair of GG.
9.15 Strong Shoda pairs of a group
A strong Shoda pair of GG is a pair (K,H)(K,H) of subgroups of GG satisfying the following conditions:
* HH is normal in KK and KK is normal in the normalizer NN of HH in GG,
* K/HK/H is cyclic and a maximal abelian subgroup of N/HN/H and
* for every g∈G∖Ng∈G∖N , ε(K,H)ε(K,H)g=0ε(K,H)ε(K,H)g=0. (See 9.13 for the definition of ε(K,H)ε(K,H)).
Let (K,H)(K,H) be a strong Shoda pair of GG. Then (K,H)(K,H) is a Shoda pair (9.14) of GG. Thus there is a linear character θθ of KK with kernel HH such that the induced character χ=χ(G,K,H)=θGχ=χ(G,K,H)=θG is irreducible. Moreover the primitive central idempotent (9.4) eℚ(χ)eℚ(χ) of ℚGℚG realized by (K,H)(K,H) is e(G,K,H)e(G,K,H), see [ORS04].
Two strong Shoda pairs (9.15) (K1,H1)(K1,H1) and (K2,H2)(K2,H2) of GG are said to be equivalent if the characters χ(G,K1,H1)χ(G,K1,H1) and χ(G,K2,H2)χ(G,K2,H2) are Galois conjugate, or equivalently if e(G,K1,H1)=e(G,K2,H2)e(G,K1,H1)=e(G,K2,H2).
The advantage of strong Shoda pairs over Shoda pairs is that one can describe the simple algebra FGeF(χ)FGeF(χ) as a matrix algebra of a cyclotomic algebra (9.11, see [ORS04] for F=ℚF=ℚ and [Olt07] for the general case).
More precisely, ℚGe(G,K,H)ℚGe(G,K,H) is isomorphic to Mn(ℚ(ξ)∗taN/K)Mn(ℚ(ξ)∗taN/K), where ξξ is a [K:H][K:H]-th root of unity, NN is the normalizer of HH in GG, n=[G:N]n=[G:N] and ℚ(ξ)∗taN/Kℚ(ξ)∗taN/K is a crossed product (see 9.6) with action aa and twisting tt given as follows:
Let xx be a fixed generator of K/HK/H and φ:N/K→N/Hφ:N/K→N/H a fixed left inverse of the canonical projection N/H→N/KN/H→N/K. Then
ξa(r)=ξi, if xφ(r)=xi
ξa(r)=ξi, if xφ(r)=xi
and
t(r,s)=ξj, if φ(rs)−1φ(r)φ(s)=xj,
t(r,s)=ξj, if φ(rs)−1φ(r)φ(s)=xj,
for r,s∈N/Kr,s∈N/K and integers ii and jj, see [ORS04]. Notice that the cocycle is the one given by the natural extension
1→K/H→N/H→N/K→1
1→K/H→N/H→N/K→1
where K/HK/H is identified with the multiplicative group generated by ξξ. Furthermore the centre of the algebra is ℚ(χ)ℚ(χ), the field of character values over ℚℚ, and N/KN/K is isomorphic to Gal(ℚ(ξ)/ℚ(χ))Gal(ℚ(ξ)/ℚ(χ)).
If the rational field is changed to an arbitrary ring FF of characteristic 00 then the Wedderburn component AF(χ)AF(χ), where χ=χ(G,K,H)χ=χ(G,K,H) is isomorphic to F(χ)⊗ℚ(χ)Aℚ(χ)F(χ)⊗ℚ(χ)Aℚ(χ). Using the description given above of Aℚ(χ)=ℚGe(G,K,H)Aℚ(χ)=ℚGe(G,K,H) one can easily describe AF(χ)AF(χ) as Mnd(F(ξ)/F(χ),t′)Mnd(F(ξ)/F(χ),t′), where d=[ℚ(ξ):ℚ(χ)]/[F(ξ):F(χ)]d=[ℚ(ξ):ℚ(χ)]/[F(ξ):F(χ)] and t′t′ is the restriction to Gal(F(ξ)/F(χ))Gal(F(ξ)/F(χ)) of tt (a cocycle of N/K=Gal(ℚ(ξ)/ℚ(χ))N/K=Gal(ℚ(ξ)/ℚ(χ))).
9.16 Strongly monomial characters and strongly monomial groups
Let GG be a finite group an χχ an irreducible character of GG.
One says that χχ is strongly monomial if there is a strong Shoda pair (9.15) (K,H)(K,H) of GG and a linear character θθ of KK of GG with kernel HH such that χ=θGχ=θG.
The group GG is strongly monomial if every irreducible character of GG is strongly monomial.
Strong Shoda pairs where firstly introduced by Olivieri, del Río and
Simón who proved that every abelian-by-supersolvable group is strongly
monomial [ORS04].
The algorithm to compute the Wedderburn decomposition of rational group
algebras for strongly monomial groups was explained in [OR03]. This method was extended for semisimple finite group algebras by Broche Cristo and del Río in [BR07] (see Section 9.17). Finally, Olteanu [Olt07] shows how to compute the Wedderburn decomposition (9.3) of an arbitrary semisimple group ring by making use of not only the strong Shoda pairs of GG but also the strong Shoda pairs of the subgroups of GG.
.
9.17 Cyclotomic Classes and Strong Shoda Pairs
Let GG be a finite group and FF a finite field of order qq, coprime to the order of GG.
Given a positive integer nn, coprime to qq, the qq-cyclotomic classes modulo nn are the set of residue classes module nn of the form
{i,iq,iq2,iq3,…}
{i,iq,iq2,iq3,…}
The qq-cyclotomic classes module nn form a partition of the set of residue classes module nn.
A generating cyclotomic class module nn is a cyclotomic class containing a generator of the additive group of residue classes module nn, or equivalently formed by integers coprime to nn.
Let (K,H)(K,H) be a strong Shoda pair (9.15) of GG and set n=[K:H]n=[K:H]. Fix a primitive nn-th root of unity ξξ in some extension of FF and an element gg of KK such that gHgH is a generator of K/HK/H. Let CC be a generating qq-cyclotomic class modulo nn. Then set
εC(K,H)=[K:H]−1ˆHn−1∑i=0tr(ξ−ci)gi,
εC(K,H)=[K:H]−1Hˆ∑i=0n−1tr(ξ−ci)gi,
where cc is an arbitrary element of CC and trtr is the trace map of the field extension F(ξ)/FF(ξ)/F. Then εC(K,H)εC(K,H) does not depend on the choice of c∈Cc∈C and is a primitive central idempotent (9.4) of FKFK.
Finally, let eC(G,K,H)eC(G,K,H) denote the sum of the different GG-conjugates of εC(K,H)εC(K,H). Then eC(G,K,H)eC(G,K,H) is a primitive central idempotent (9.4) of FGFG [BR07]. We say that eC(G,K,H)eC(G,K,H) is the primitive central idempotent realized by the strong Shoda pair (K,H)(K,H) of the group GG and the cyclotomic class CC.
If GG is strongly monomial (9.16) then every primitive central idempotent of FGFG is realizable by some strong Shoda pair (9.15) of GG and some cyclotomic class CC [BR07]. As in the zero characteristic case, this explain how to compute the Wedderburn decomposition (9.3) of FGFG for a finite semisimple algebra of a strongly monomial group (see [BR07]
for details). For non strongly monomial groups the algorithm to compute
the Wedderburn decomposition just uses the Brauer characters.
.
9.18 Theory for Local Schur Index and Division Algebra Part Calculations
(By Allen Herman, May 2013. Updated October 2014.)
The division algebra parts of simple algebras in the Wedderburn
Decomposition of the group algebra of a finite group over an abelian
number field FF correspond to elements of the Schur Subgroup S(F)S(F) of the Brauer group of FF. Like all classes in the Brauer group of an algebraic number field FF, the division algebra part of a representative of a given Brauer class is determined up to FF-algebra isomorphism by its list of local Hasse invariants at all primes (i.e. places) of FF. The local invariant at a prime PP of FF is a lowest terms fraction r/mPr/mP whose denominator is the local Schur index mPmP of the simple algebra at the prime qq (see [Rei03]). For division algebras whose Brauer class lies in the Schur Subgroup of an abelian number field FF, the local indices at any of the primes PP lying over the same rational prime pp are equal to the same positive integer mpmp, and the numerator of the local invariants among these primes are uniformly distributed among the integers rr coprime to mpmp [BS72].
The local Schur index functions in wedderga produce a list of the
nontrivial local indices of the division algebra part of the simple
algebra at all rational primes. The Schur index of the simple algebra
over FF is the least common multiple mm of these local indices, and the dimension of the division algebra part of the simple algebra over FF is m2m2.
While not sufficient to identify these division algebras up to ring
isomorphism in general, this list of local indices does identify the
division algebra up to ring isomorphism whenever there is no pair of
local indices at odd primes that are greater than 2. (This is at least
the case for groups of order less than 3^2*7*13.) So it gives the
information desired in most basic situations, and allows one to
distinguish almost all pairs of simple components of group algebras.
Wedderga's functions compute local indices for generalized quaternion
algebras defined over the rationals and cyclotomic algebras defined
over any abelian number field. Special shortcut functions are available
for cyclic cyclotomic algebras. There are also versions of the functions
that compute the local and global Schur index of a character of a
finite group over a given abelian number field. The steps in the general
character- theoretic method involve 1) a Brauer-Witt reduction to a
cyclic-by-abelian group, 2) use of the Frobenius-Schur indicator to
compute the local index at infinity, 3) computing the pp-local index for an ordinary irreducible character χχ of a pp-solvable group using the values of an irreducible Brauer character in the same pp-block in cases where the pp-defect group of χχ is cyclic, and 4) use of Riese and Schmid's characterization of dyadic Schur groups ([Sch94] and [RS96])
to handle the exceptional cases where step 3) is not available. Our
approach to rational quaternion algebras is the standard one given, for
example, in [Pie82].
The Legendre symbol operation in GAP is used to determine the local
index at odd primes. The local index of the generalized quaternion
algebra (a,b)(a,b) over QQ at the infinite prime will be 22 if both aa and bb are negative, and otherwise 11. We avoid the complicated case of quadratic reciprocity when working over Q by using the Hasse-Brauer-Albert-Noether Theorem ([Rei03], pg. 276): since we know the other primes of QQ where the local index is 22, it determines the local index at the prime 22. For generalized quaternion algebras over number fields FF other than QQ,
we have to convert to cyclic or cyclic cyclotomic algebras and use the
other local index functions, or appeal to a number theory system outside
of GAP that can solve norm equations.
There are three shortcut functions used to compute local indices of
cyclic cyclotomic algebras, which wedderga's -Info functions produce in
the form [r,F,n,[a,b,c]][r,F,n,[a,b,c]].
The local index at infinity is calculated by determining if the real
completion of the corresponding algebra will produce a real quaternion
algebra. In order to do this, FF must be a real subfield, nn must be strictly greater than 22, and E(n)cE(n)c (which has to be a root of unity in FF) must be −1−1.
These facts can be checked directly, so this is faster than calculating
the character table of the group and checking the value of a
Frobenius-Schur indicator. The shortcut to calculate the local index of a
cyclic cyclotomic algebra at an odd prime makes direct use of the
following lemma of Janusz: If Ep/FpEp/Fp is a Galois extension of pp-local fields with ramification index ee, and zz is a root of unity with order prime to pp, then zz is a norm in Ep/FpEp/Fp if and only if it is the ee-th power of a root of unity in FF. ([Jan75], pg. 535). It follows that in order to calculate the local index at pp of a cyclic cyclotomic algebra [r,F,n,[a,b,c]][r,F,n,[a,b,c]], we first determine the splitting degree, residue degree, and ramification index ee of the extension F(ζn)/FF(ζn)/F at pp. Comparing the behaviour of the Galois automorphism σbσb to the behaviour of the Frobenius automorphism at pp allows us to determine the order of the largest root of unity zz with order coprime to pp in the pp-completion FpFp. The local index mpmp is then the least power of E(n)cE(n)c that lies in the group generated by zeze.
Calculation of the local index at the prime 2 makes use of the following consequence of ([Jan75], Theorem 5): A cyclic cyclotomic algebra [r,F2,n,[a,b,c]] over a 2-local field F2 that is a subfield of a cyclotomic extension of the rational 2-local field Q2 has Schur index at most 2. It has Schur index 2 if and only if 4 divides n, F2(ζ4) is totally ramified of degree 2, the Galois automorphism σb of F2(ζn)/F2 inverts all 2-power roots of unity in F2(ζn), the order of E(n)c is 2 times an odd number, and (F2:Q2) is odd. The same approach to cyclotomic reciprocity makes it possible to check all of these conditions in the 2-local situation.
The wedderga function that computes the p-local index of an ordinary irreducible character χ of a finite non-nilpotent cyclic-by- abelian group G is based directly on a theorem of Benard [Ben76] that applies whenever the p-defect group of χ is cyclic. We have to restrict our application of it to groups whose orders are small because the GAP
records for irreducible Brauer characters are only available in these
cases. In order to use this approach effectively, we developed a
function that computes the defect group of the block containing a given
ordinary irreducible character χ. This function makes use of the Min half of Brauer's Min-Max theorem (see Theorem 4.4 of [Nav98]),
and thus is able to find the defect group directly from the ordinary
character table. It is thus available for nonsolvable groups, even in
cases where GAP's Brauer character records
are not available. We are indebted to Michael Geline and Friederich
Ladisch for discussions concerning the calculation of defect groups in GAP. The current algorithm we use is based on an approach suggested by Ladisch.
9.19 Obtaining Algebras with structure constants as terms of the Wedderburn decomposition
Some users may find it desirable to have an alternative description
for the components of the Wedderburn decomposition of a group ring as
algebras with structure constants, because the operations for algebras
in GAP are designed for algebras with structure constants. We have provided such an algorithm that converts the output of WedderburnDecompositionInfo
(2.1-2)
into algebras with structure constants. Matrix rings over fields are
converted directly. For components that are cyclotomic algebras, it
calculates their defining group and defining character using those Wedderga operations, then uses IrreducibleRepresentationsDixon
(Reference: IrreducibleRepresentationsDixon)
to obtain matrix generators of an algebra isomorphic to the simple
component corresponding to the character over a suitable field. An
algebra with structure constants version of this is finally obtained by
applying IsomorphismSCAlgebra
(Reference: IsomorphismSCAlgebra (w.r.t. a given basis)) to this algebra.
9.20 A complete set of orthogonal primitive idempotents
When R is a semisimple ring, then every left ideal L of R is of the form L=Re, where e is an idempotent of R.
Therefore, we can use the idempotents to characterize the
decompositions of semisimple rings as a direct sum of minimal left
ideals. In particular, let R=⊕ti=1Li be a decomposition of a semisimple ring as a direct sum of minimal left ideals. Then, there exists a family {e1,…,et} of elements of R such that: each ei≠0 is an idempotent element, if i≠j, then eiej=0, 1=e1+⋯+et and each ei cannot be written as ei=e′i+e″i, where e′i,e″i are idempotents such that e′i,e″i≠0 and e′ie″i=0, 1≤i≤. Conversely, if there exists a family of idempotents {e1,…,et} satisfying the previous conditions, then the left ideals Li=Rei are minimal and R=⊕ti=1Li. Such a set of idempotents is called a complete set of orthogonal primitive idempotents of the ring R. Such a set is not uniquely determined.
Let F be a finite field and G a finite nilpotent group such that FG is semisimple. Let (H,K) be a strong Shoda pair of G, C∈C(H/K) and set eC=eC(G,H,K), εC=εC(H,K), H/K=⟨¯a⟩, E=EG(H/K). Let E2/K and H2/K=⟨¯a2⟩ (respectively E2′/K and H2′/K=⟨¯a2′⟩) denote the 2-parts (respectively 2'-parts) of E/K and H/K respectively. Then ⟨¯a2′⟩ has a cyclic complement ⟨¯b2′⟩ in E2′/K. Using the description of the primitive central idempotents and the Wedderburn components of a semisimple finite group algebra FG (9.17), a complete set of orthogonal primitive idempotents of FGeC is described (see [OG11]) as the set of conjugates of βeC=~b2′β2εC by the elements of TeC=T2′T2TE, where T2′={1,a2′,a22′,…,a[E2′:H2′]−12′}, TE denotes a right transversal of E in G and β2 and T2 are given according to the cases below.
If H2/K has a complement M2/K in E2/K then β2=~M2. Moreover, if M2/K is cyclic, then there exists b2∈E2 such that E2/K is given by the following presentation
⟨¯a2,¯b2∣¯a22n=¯b22k=1,¯a2¯b2=¯a2r⟩,
and if M2/K is not cyclic, then there exist b2,c2∈E2 such that E2/K is given by the following presentation
⟨¯a2,¯b2,¯c2∣¯a22n=¯b22k=¯c22=1,¯a2¯b2=¯a2r,¯a2¯c2=¯a2−1,[¯b2,¯c2]=1⟩,
with r≡1mod4 (or equivalently ¯a22n−2 is central in E2/K). Then
T2={1,a2,a22,…,a2k−12}, if ¯a22n−2 is central in E2/K (unless n≤1) and M2/K is cyclic; and
T2={1,a2,a22,…,ad/2−12,a2n−22,a2n−2+12,…,a2n−2+d/2−12}, where d=[E2:H2], otherwise.
If H2/K has no complement in E2/K, then there exist b2,c2∈E2 such that E2/K is given by the following presentation
⟨¯a2,¯b2,¯c2∣¯a22n=¯b22k=1,¯c22=¯a22n−1,¯a2¯b2=¯a2r,¯a2¯c2=¯a2−1,[¯b2,¯c2]=1⟩,
with r≡1mod4. In this case, β2=~b21+xa2n−22+ya2n−22c22 and
T2={1,a2,a22,…,a2k−12,c2,c2a2,c2a22,…,c2a2k−12},
with x,y∈F, satisfying x2+y2=−1 and y≠0.
When G is not nilpotent, we can still use the following description in some specific cases. Let G be a finite group and F a finite field of order s such that s is coprime to the order of G. Let (H,K) be a strong Shoda pair of G such that τ(gH,g′H)=1 for all g,g′∈E=EG(H/K), and let C∈C(H/K). Let ε=εC(H,K) and e=eC(G,H,K) (9.17). Let w be a normal element of Fso/Fso/[E:H] (with o the multiplicative order of s modulo [H:K]) and B the normal basis determined by w. Let ψ be the isomorphism between FEε and the matrix algebra M[E:H](Fso/[E:H]) with respect to the basis B as stated in Corollary 29.8 in [Rei03]. Let P,A∈M[E:H](Fso/[E:H]) be the matrices
P=(111⋯111−10⋯0010−1⋯00⋮⋮⋮⋱⋮⋮100⋯−10100⋯0−1)andA=(00⋯0110⋯0001⋯00⋮⋮⋱⋮⋮00⋯0000⋯10).
Then
{x^T1εx−1∣x∈T2⟨xe⟩}
is a complete set of orthogonal primitive idempotents of FGe where xe=ψ−1(PAP−1), T1 is a transversal of H in E and T2 is a right transversal of E in G ([OGnt]). By ^T1 we denote the element 1|T1|∑t∈T1t in FG.
9.21 Applications to coding theory
A linear code of length n and rank k is a linear subspace C with dimension k of the vector space Fnq. The standard basis of Fnq is denoted by E={e1,...,en}. The vectors in C are called codewords, the size of a code is the number of codewords and equals qk. The distance of a code is the minimum distance between distinct codewords, i.e. the number of elements in which they differ.
For any group G, we denote by FqG the group algebra over G with coefficients in Fq. If G is a group of order n and C⊆Fnq is a linear code, then we say that C is a left G-code (respectively a G-code) if there is a bijection ϕ:E→G such that the linear extension of ϕ to an isomorphism ϕ:Fnq→FqG maps C to a left ideal (respectively a two-sided ideal) of FqG. A left group code (respectively a group code) is a linear code which is a left G-code (respectively a G-code) for some group G.
Since left ideals in FqG are generated by idempotents, there is a one-one relation between (sums of) primitive idempotents of FqG and left G-codes over Fq.
Note that each element c in FqG is of the form c=∑ni=1figi, where we fix an ordering {g1,g2,...,gn} of the group elements of G and fi∈Fq. If one looks at c as a codeword, one writes [f1f2...fn].