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 \mathbb F_q the finite field of order q. For every positive integer k, we denote a complex k-th primitive root of unity by \xi_k and so &QQ;(\xi_k) is the k-th cyclotomic extension of &QQ;.

Group rings and group algebras
group ring
Given a group G and a ring R, the group ring RG over the group G with coefficients in R is the ring whose underlying additive group is a right R-module with basis G such that the product is defined by the following rule (gr)(hs)=(gh)(rs) for r,s \in R and g, h \in G, and extended to RG by linearity.

group algebra
A group algebra is a group ring in which the coefficient ring is a field.

Semisimple group algebras
semisimple ring

We say that a ring R is semisimple if it is a direct sum of simple left (alternatively right) ideals or equivalently if R 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 G is a finite group then the group algebra FG is semisimple if and only the characteristic of the coefficient field F does not divide the order of G.

In fact, an arbitrary group ring RG is semisimple if and only if the coefficient ring R is semisimple, the group G is finite and the order of G is invertible in R.

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.

Wedderburn decomposition
Wedderburn decomposition
Wedderburn components

If R is a semisimple ring () then the Wedderburn decomposition of R is the decomposition of R as a direct product of simple algebras. The factors of this Wedderburn decomposition are called Wedderburn components of R. Each Wedderburn component of R is of the form Re for e a primitive central idempotent () of R.

Let FG be a semisimple group algebra (). If F has positive characteristic, then the Wedderburn components of FG are matrix algebras over finite extensions of F. If F has zero characteristic then by the Brauer-Witt Theorem , the Wedderburn components of FG are Brauer equivalent () to cyclotomic algebras ().

The main functions of &Wedderga; compute the Wedderburn components of a semisimple group algebra FG, 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 A is Brauer equivalent () to a cyclotomic algebra () and therefore A 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,k,[d,\alpha,\beta]],
[n,K,k,[d_i,\alpha_i,\beta_i]_{i=1}^m, [\gamma_{i,j}]_{1\le i < j \le n} ],
where n is the matrix size, K is the centre of A (a finite field extension of F) and the remaining data are integers whose interpretation is explained in .

In some cases (for the zero characteristic coefficient field) the size n 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 only ensures that each Wedderburn component () of a semisimple group algebra is Brauer equivalent () to a cyclotomic algebra (), but not necessarily isomorphic to a full matrix algebra of a cyclotomic algebra. For example, a Wedderburn component D of a group algebra can be a division algebra but not a cyclotomic algebra. In this case M_n(D) is a cyclotomic algebra C for some n and therefore D can be described as M_{1/n}(C) (see last Example in ).

The main algorithm of &Wedderga; is based on a computational oriented proof of the Brauer-Witt Theorem due to Olteanu which uses previous work by Olivieri, del Río and Simón for rational group algebras of strongly monomial groups ().

Characters and primitive central idempotents
primitive central idempotent
field of character values

A primitive central idempotent of a ring R is a non-zero central idempotent e which cannot be written as the sum of two non-zero central idempotents of Re, or equivalently, such that Re is indecomposable as a direct product of two non-trivial two-sided ideals.

The Wedderburn components () of a semisimple ring R are the rings of the form Re for e running over the set of primitive central idempotents of R.

Let FG be a semisimple group algebra () and \chi an irreducible character of G (in an algebraic closure of F). Then there is a unique Wedderburn component A=A_F(\chi) of FG such that \chi(A)\ne 0. Let e_F(\chi) denote the unique primitive central idempotent of FG in A_F(\chi), that is the identity of A_F(\chi), i.e. A_F(\chi)=FGe_F(\chi).

The centre of A_F(\chi) is F(\chi)=F(\chi(g):g \in G), the field of character values of \chi over F.

The map \chi \mapsto A_F(\chi) defines a surjective map from the set of irreducible characters of G (in an algebraic closure of F) onto the set of Wedderburn components of FG.

Equivalently, the map \chi \mapsto e_F(\chi) defines a surjective map from the set of irreducible characters of G (in an algebraic closure of F) onto the set of primitive central idempontents of FG.

If the irreducible character \chi of G takes values in F then e_F(\chi) = e(\chi) = \frac{\chi(1)}{|G|} \sum_{g\in G} \chi(g^{-1}) g.

In general one has e_F(\chi) = \sum_{\sigma \in Gal(F(\chi)/F)} e(\sigma \circ \chi).

Central simple algebras and Brauer equivalence

Let K be a field. central simple algebra A central simple K-algebra is a finite dimensional K-algebra with center K which has no non-trivial proper ideals. Every central simple K-algebra is isomorphic to a matrix algebra M_n(D) where D is a division algebra (which is finite-dimensional over K and has centre K). The division algebra D is unique up to K-isomorphisms.

(Brauer) equivalence
equivalence (Brauer)
Two central simple K-algebras A and B are said to be Brauer equivalent, or simply equivalent, if there is a division algebra D and two positive integers m and n such that A is isomorphic to M_m(D) and B is isomorphic to M_n(D).

Crossed Products
Crossed Product
Let R be a ring and G a group.

Intrinsic definition.

A crossed product of G over R (or with coefficients in R) is a ring R*G with a decomposition into a direct sum of additive subgroups R*G = \bigoplus_{g \in G} A_g such that for each g,h in G one has:

* A_1=R (here 1 denotes the identity of G),

* A_g A_h = A_{gh} and

* A_g has a unit of R*G.

Extrinsic definition.

Let Aut(R) denote the group of automorphisms of R and let R^* denote the group of units of R.

Let a:G \rightarrow Aut(R) and t:G \times G \rightarrow R^* be mappings satisfying the following conditions for every g, h and k in G:

(1) a(gh)^{-1} a(g) a(h) is the inner automorphism of R induced by t(g,h) (i.e. the automorphism x\mapsto t(g,h)^{-1} x t(g,h)) and

(2) t(gh,k) t(g,h)^k = t(g,hk) t(h,k), where for g \in G and x \in R we denote a(g)(x) by x^g.

The crossed product of G over R (or with coefficients in R), action a and twisting t is the ring R*_a^t G = \bigoplus_{g\in G} u_g R where \{u_g : g\in G \} is a set of symbols in one-to-one correspondence with G, with addition and multiplication defined by (u_g r) + (u_g s) = u_g(r+s), \quad (u_g r)(u_h s) = u_{gh} t(g,h) r^h s for g,h \in G and r,s\in R, and extended to R*_a^t G by linearity.

The associativity of the product defined is a consequence of conditions (1) and (2) .

Equivalence of the two definitions.

Obviously the crossed product of G over R defined using the extrinsic definition is a crossed product of G over u_1 R in the sense of the first definition. Moreover, there is r_0 in R^* such that u_1r_0 is the identity of R*_a^t G and the map r \mapsto u_1 r_0 r is a ring isomorphism R \rightarrow u_1R .

Basis of units (for crossed product)
Conversely, let R*G=\bigoplus_{g\in G} A_g be an (intrinsic) crossed product and select for each g\in G a unit u_g\in A_g of R*G. This is called a basis of units for the crossed product R*G. Then the maps a:G \rightarrow Aut(R) and t:G\times G \rightarrow R^* given by r^g = u_g^{-1} r u_g, \quad t(g,h) = u_{gh}^{-1} u_g u_h \quad (g,h \in G, r \in R) satisfy conditions (1) and (2) and R*G = R*_a^t G.

The choice of a basis of units u_g \in A_g determines the action a and twisting t. If \{u_g \in A_g : g \in G \} and \{v_g \in A_g : g \in G \} are two sets of units of R*G then v_g = u_g r_g for some units r_g of R. 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 u_1=1. In that case a(1) is the identity map of R and t(1,g)=t(g,1)=1 for each g in G.

Cyclic Crossed Products
Cyclic Crossed Product
Let R*G=\bigoplus_{g \in G} A_g be a crossed product () and assume that G = \langle g \rangle is cyclic. Then the crossed product can be given using a particularly nice description.

Select a unit u in A_{g}, and let a be the automorphism of R given by r^a = u^{-1} r u.

If G is infinite then set u_{g^k} = u^k for every integer k. Then R*G = R[ u | ru = u r^a ], a skew polynomial ring. Therefore in this case R*G is determined by [ R, a ].

If G is finite of order d then set u_{g^k} = u^k for 0 \le k < d. Then b = u^d \in R and R*G = R[ u | ru = u r^a, u^d = b ] Therefore, R*G is completely determined by the following data: [ R , [ d , a , b ] ]

Abelian Crossed Products
Abelian Crossed Product
Let R*G=\bigoplus_{g \in G} A_g be a crossed product () and assume that G is abelian. Then the crossed product can be given using a simple description.

Express G as a direct sum of cyclic groups: G = \langle g_1 \rangle \times \cdots \times \langle g_n \rangle and for each i=1,\dots,n select a unit u_i in A_{g_i}.

Each element g of G has a unique expression g = g_1^{k_1} \cdots g_n^{k_n}, where k_i is an arbitrary integer, if g_i has infinite order, and 0 \le k_i < d_i, if g_i has finite order d_i. Then one selects a basis for the crossed product by taking u_g = u_{g_1^{k_1} \cdots g_n^{k_n}} = u_1^{k_1} \cdots u_n^{k_n}.

* For each i=1,\dots, n, let a_i be the automorphism of R given by r^{a_i} = u_i^{-1} r u_i.

* For each 1 \le i < j \le n, let t_{i,j} = u_j^{-1} u_i^{-1} u_j u_i \in R.

* If g_i has finite order d_i, let b_i=u_i^{d_i} \in R.

Then R*G = R[u_1,\dots,u_n | ru_i = u_i r^{a_i}, u_j u_i = t_{ij} u_i u_j, u_i^{d_i} = b_i (1 \le i < j \le n) ], where the last relation vanishes if g_i has infinite order.

Therefore R*G is completely determined by the following data: [ R , [ d_i , a_i , b_i ]_{i=1}^n, [ t_{i,j} ]_{1 \le i < j \le n} ].

Classical crossed products
Classical Crossed Product
A classical crossed product is a crossed product L*_a^t G, where L/K is a finite Galois extension, G=Gal(L/K) is the Galois group of L/K and a is the natural action of G on L.

Then t is a 2-cocycle and the crossed product () L*_a^t G is denoted by (L/K,t). The crossed product (L/K,t) is known to be a central simple K-algebra .

Cyclic Algebras
Cyclic Algebra
A cyclic algebra is a classical crossed product () (L/K,t) where L/K is a finite cyclic field extension. The cyclic algebras have a very simple form.

Assume that Gal(L/K) is generated by g and has order d. Let u=u_g be the basis unit () of the crossed product corresponding to g and take the remaining basis units for the crossed product by setting u_{g^i} = u^i, ( i = 0, 1, \dots, d-1 ). Then a = u^n \in K. The cyclic algebra is usually denoted by (L/K,a) and one has the following description of (L/K,t) (L/K,t) = (L/K,a) = L[u| r u = u r^g, u^d = a ].

Cyclotomic algebras
Cyclotomic algebra
A cyclotomic algebra over F is a classical crossed product () (F(\xi)/F,t), where F is a field, \xi is a root of unity in an extension of F and t(g,h) is a root of unity for every g and h in Gal(F(\xi)/F).

The Brauer-Witt Theorem asserts that every Wedderburn component () of a group algebra is Brauer equivalent () (over its centre) to a cyclotomic algebra.

Numerical description of cyclotomic algebras

Let A=(F(\xi)/F,t) be a cyclotomic algebra (), where \xi=\xi_k is a k-th root of unity. Then the Galois group G=Gal(F(\xi)/F) is abelian and therefore one can obtain a simplified form for the description of cyclotomic algebras as for any abelian crossed product ().

Then the n \times n matrix algebra M_n(A) can be described numerically in one of the following forms:

* If F(\xi)=F, (i.e. G=1) then A=M_n(F) and thus the only data needed to describe A are the matrix size n and the field F: [n,F]

* If G is cyclic (but not trivial) of order d then A is a cyclic cyclotomic algebra A = F(\xi) [ u | \xi u = u \xi^\alpha, u^d = \xi^\beta ] and so M_n(A) can be described with the following data [n,F,k,[d,\alpha,\beta]], where the integers k, d, \alpha and \beta satisfy the following conditions: \alpha^d \equiv 1 \; mod \; k, \quad \beta(\alpha-1) \equiv 0 \; mod \; k.

* If G is abelian but not cyclic then M_n(A) can be described with the following data (see ): [n,F,k,[d_i,\alpha_i,\beta_i]_{i=1}^m, [\gamma_{i,j}]_{1\le i < j \le m} ] representing the n \times n matrix ring over the following algebra: A = F(\xi)[ u_1, \ldots, u_m \mid \xi u_i = u_i \xi^{\alpha_i}, \quad u_i^{d_i}=\xi^{\beta_i}, \quad u_s u_r = \xi^{\gamma_{rs}} u_r u_s, \quad i = 1, \ldots, m, \quad 0 \le r < s \le m ] where

* \{g_1,\ldots,g_m\} is an independent set of generators of G,

* d_i is the order of g_i,

* \alpha_i, \beta_i and \gamma_{rs} are integers, and \xi^{g_i} = \xi^{\alpha_i}.

Idempotents given by subgroups
\varepsilon(K,H)
e(G,K,H)
e_C(G,K,H)

Let G be a finite group and F a field whose characteristic does not divide the order of G. If H is a subgroup of G then set \widehat{H} = |H|^{-1}\sum_{x \in H} x. The element \widehat{H} is an idempotent of FG which is central in FG if and only if H is normal in G.

If H is a proper normal subgroup of a subgroup K of G then set \varepsilon(K,H) = \prod_{L} (\widehat{N}-\widehat{L}) where L runs on the normal subgroups of K which are minimal among the normal subgroups of K containing N properly. By convention, \varepsilon(K,K)=\widehat{K}. The element \varepsilon(K,H) is an idempotent of FG.

If H and K are subgroups of G such that H is normal in K then e(G,K,H) denotes the sum of all different G-conjugates of \varepsilon(K,H). The element e(G,K,H) is central in FG. In general it is not an idempotent but if the different conjugates of \varepsilon(K,H) are orthogonal then e(G,K,H) is a central idempotent of FG.

If (K,H) is a Shoda Pair () of G then there is a non-zero rational number a such that ae(G,K,H)) is a primitive central idempotent () of the rational group algebra &QQ; G. If (K,H) is a strong Shoda pair () of G then e(G,K,H) is a primitive central idempotent of &QQ; G.

Assume now that F is a finite field of order q, (K,H) is a strong Shoda pair of G and C is a cyclotomic class of K/H containing a generator of K/H. Then e_C(G,K,H) is a primitive central idempotent of FG (see ).

Shoda pairs

Let G be a finite group. Shoda pair A Shoda pair of G is a pair (K,H) of subgroups of G for which there is a linear character \chi of K with kernel H such that the induced character \chi^G in G is irreducible. By or , (K,H) is a Shoda pair if and only if the following conditions hold:

* H is normal in K,

* K/H is cyclic and

* if K^g \cap K \subseteq H for some g \in G then g \in K.

primitive central idempotent realized by a Shoda pair
If (K,H) is a Shoda pair and \chi is a linear character of K\le G with kernel H then the primitive central idempotent () of &QQ; G associated to the irreducible character \chi^G is of the form e=e_&QQ; (\chi^G)=a e(G,K,H) for some a \in &QQ; (see for the definition of e(G,K,H)). In that case we say that e is the primitive central idempotent realized by the Shoda pair (K,H) of G.

A group G is monomial, that is every irreducible character of G is monomial, if and only if every primitive central idempotent of &QQ; G is realizable by a Shoda pair of G.

Strong Shoda pairs
strong Shoda pair
A strong Shoda pair of G is a pair (K,H) of subgroups of G satisfying the following conditions:

* H is normal in K and K is normal in the normalizer N of H in G,

* K/H is cyclic and a maximal abelian subgroup of N/H and

* for every g \in G\setminus N , \varepsilon(K,H)\varepsilon(K,H)^g=0. (See for the definition of \varepsilon(K,H)).

Let (K,H) be a strong Shoda pair of G. Then (K,H) is a Shoda pair () of G. Thus there is a linear character \theta of K with kernel H such that the induced character \chi=\chi(G,K,H)=\theta^G is irreducible. Moreover the primitive central idempotent () e_{&QQ; }(\chi) of &QQ; G realized by (K,H) is e(G,K,H), see .

equivalent strong Shoda pairs
Two strong Shoda pairs () (K_1,H_1) and (K_2,H_2) of G are said to be equivalent if the characters \chi(G,K_1,H_1) and \chi(G,K_2,H_2) are Galois conjugate, or equivalently if e(G,K_1,H_1)=e(G,K_2,H_2).

The advantage of strong Shoda pairs over Shoda pairs is that one can describe the simple algebra FGe_F(\chi) as a matrix algebra of a cyclotomic algebra (, see for F=&QQ; and for the general case).

More precisely, &QQ; Ge(G,K,H) is isomorphic to M_n(&QQ; (\xi)*_a^t N/K), where \xi is a [K:H]-th root of unity, N is the normalizer of H in G, n=[G:N] and &QQ; (\xi)*_a^t N/K is a crossed product (see ) with action a and twisting t given as follows:

Let x be a fixed generator of K/H and \varphi : N/K \rightarrow N/H a fixed left inverse of the canonical projection N/H\rightarrow N/K. Then \xi^{a(r)} = \xi^i, \mbox{ if } x^{\varphi(r)}= x^i and t(r,s) = \xi^j, \mbox{ if } \varphi(rs)^{-1} \varphi(r)\varphi(s) = x^j, for r,s \in N/K and integers i and j, see . Notice that the cocycle is the one given by the natural extension 1 \rightarrow K/H \rightarrow N/H \rightarrow N/K \rightarrow 1 where K/H is identified with the multiplicative group generated by \xi. Furthermore the centre of the algebra is &QQ; (\chi), the field of character values over &QQ; , and N/K is isomorphic to Gal(&QQ; (\xi)/&QQ; (\chi)).

If the rational field is changed to an arbitrary ring F of characteristic 0 then the Wedderburn component A_F(\chi), where \chi = \chi(G,K,H) is isomorphic to F(\chi)\otimes_{&QQ; (\chi)}A_&QQ; (\chi). Using the description given above of A_&QQ; (\chi)=&QQ; G e(G,K,H) one can easily describe A_F(\chi) as M_{nd}(F(\xi)/F(\chi),t'), where d=[&QQ; (\xi): &QQ;(\chi)]/[F(\xi):F(\chi)] and t' is the restriction to Gal(F(\xi)/F(\chi)) of t (a cocycle of N/K = Gal(&QQ; (\xi)/&QQ; (\chi))).

Strongly monomial characters and strongly monomial groups
strongly monomial character
Let G be a finite group an \chi an irreducible character of G.

One says that \chi is strongly monomial if there is a strong Shoda pair () (K,H) of G and a linear character \theta of K of G with kernel H such that \chi=\theta^G.

strongly monomial group
The group G is strongly monomial if every irreducible character of G 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 . The algorithm to compute the Wedderburn decomposition of rational group algebras for strongly monomial groups was explained in . This method was extended for semisimple finite group algebras by Broche Cristo and del Río in (see Section ). Finally, Olteanu shows how to compute the Wedderburn decomposition () of an arbitrary semisimple group ring by making use of not only the strong Shoda pairs of G but also the strong Shoda pairs of the subgroups of G.

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Cyclotomic Classes and Strong Shoda Pairs

Let G be a finite group and F a finite field of order q, coprime to the order of G.

cyclotomic class
Given a positive integer n, coprime to q, the q-cyclotomic classes modulo n are the set of residue classes module n of the form \{i,iq,iq^2,iq^3, \dots \} The q-cyclotomic classes module n form a partition of the set of residue classes module n.

generating cyclotomic class
A generating cyclotomic class module n is a cyclotomic class containing a generator of the additive group of residue classes module n, or equivalently formed by integers coprime to n.

Let (K,H) be a strong Shoda pair () of G and set n=[K:H]. Fix a primitive n-th root of unity \xi in some extension of F and an element g of K such that gH is a generator of K/H. Let C be a generating q-cyclotomic class modulo n. Then set \varepsilon_C(K,H) = [K:H]^{-1} \widehat{H} \sum_{i=0}^{n-1} tr(\xi^{-ci})g^i, where c is an arbitrary element of C and tr is the trace map of the field extension F(\xi)/F. Then \varepsilon_C(K,H) does not depend on the choice of c \in C and is a primitive central idempotent () of FK.

primitive central idempotent realized by a strong Shoda pair and a cyclotomic class
Finally, let e_C(G,K,H) denote the sum of the different G-conjugates of \varepsilon_C(K,H). Then e_C(G,K,H) is a primitive central idempotent () of FG . We say that e_C(G,K,H) is the primitive central idempotent realized by the strong Shoda pair (K,H) of the group G and the cyclotomic class C.

If G is strongly monomial () then every primitive central idempotent of FG is realizable by some strong Shoda pair () of G and some cyclotomic class C . As in the zero characteristic case, this explain how to compute the Wedderburn decomposition () of FG for a finite semisimple algebra of a strongly monomial group (see for details).

For non strongly monomial groups the algorithm to compute the Wedderburn decomposition just uses the Brauer characters.