7 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: ℚ denotes the field of rationals and 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 ℚ(xi_k) is the k-th cyclotomic
  extension of ℚ.
  
  
  7.1 Group rings and group algebras
  
  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.
  
  A group algebra is a group ring in which the coefficient ring is a field.
  
  
  7.2 Semisimple group algebras
  
  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.
  
  
  7.3 Wedderburn decomposition
  
  If  R  is  a semisimple ring (7.2) 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 (7.4) of R.
  
  Let   FG   be   a   semisimple  group  algebra  (7.2).  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  [Yam74],  the  Wedderburn  components of FG are Brauer
  equivalent (7.5) to cyclotomic algebras (7.11).
  
  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  (7.5)  to  a  cyclotomic  algebra  (7.11)  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 7.12.
  
  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
  [Yam74]  only  ensures  that each Wedderburn component (7.3) of a semisimple
  group algebra is Brauer equivalent (7.5) to a cyclotomic algebra (7.11), 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 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 (7.16).
  
  
  7.4 Characters and primitive central idempotents
  
  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 (7.3) 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 (7.2) 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 -> 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 -> 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).
  
  
  
  7.5 Central simple algebras and Brauer equivalence
  
  Let  K  be  a  field.  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.
  
  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).
  
  
  7.6 Crossed Products
  
  Let R be a ring and G a group.
  
  Intrinsic  definition.  A  crossed  product  [Pas89]  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  ->  Aut(R) and t:G x G -> 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-> 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 [Pas89] 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  : gin 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,sin R, and extended to R*_a^t G 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 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 -> u_1 r_0 r is a ring
  isomorphism R -> u_1R.
  
  Conversely, let R*G=bigoplus_gin G A_g be an (intrinsic) crossed product and
  select  for  each  gin  G a unit u_gin A_g of R*G. This is called a basis of
  units for the crossed product R*G. Then the maps a:G -> Aut(R) and t:Gx G ->
  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.
  
  
  7.7 Cyclic Crossed Products
  
  Let R*G=bigoplus_g in G A_g be a crossed product (7.6) and assume that G = <
  g  >  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 ] ]
  
  
  
  7.8 Abelian Crossed Products
  
  Let  R*G=bigoplus_g in G A_g be a crossed product (7.6) 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}
       ].
  
  
  
  7.9 Classical crossed products
  
  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
  (7.6)  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 [Rei03].
  
  
  7.10 Cyclic Algebras
  
  A cyclic algebra is a classical crossed product (7.9) (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  (7.6)  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 ].
  
  
  
  7.11 Cyclotomic algebras
  
  A   cyclotomic   algebra  over  F  is  a  classical  crossed  product  (7.9)
  (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  [Yam74]  asserts  that every Wedderburn component
  (7.3)  of  a group algebra is Brauer equivalent (7.5) (over its centre) to a
  cyclotomic algebra.
  
  
  7.12 Numerical description of cyclotomic algebras
  
  Let  A=(F(xi)/F,t)  be  a cyclotomic algebra (7.11), 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 (7.8).
  
  Then  the n x 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 7.8):
  
  
       [n,F,k,[d_i,\alpha_i,\beta_i]_{i=1}^m, [\gamma_{i,j}]_{1\le i < j
       \le m} ]
  
  
  representing the n x 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,...,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}.
  
  
  
  7.13 Idempotents given by subgroups
  
  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  widehatH  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)=widehatK.  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  (7.14) of G then there is a non-zero rational
  number a such that ae(G,K,H)) is a primitive central idempotent (7.4) of the
  rational group algebra ℚ G. If (K,H) is a strong Shoda pair (7.15) of G then
  e(G,K,H) is a primitive central idempotent of ℚ 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 7.17).
  
  
  7.14 Shoda pairs
  
  Let G be a finite group. 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 [Sho33] or [ORS04], (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.
  
  If  (K,H) is a Shoda pair and chi is a linear character of Kle G with kernel
  H  then  the  primitive  central  idempotent  (7.4) of ℚ G associated to the
  irreducible character chi^G is of the form e=e_ℚ (chi^G)=a e(G,K,H) for some
  a  in  ℚ  [ORS04] (see 7.13 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 ℚ G is realizable by a
  Shoda pair of G.
  
  
  7.15 Strong Shoda pairs
  
  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\ N , varepsilon(K,H)varepsilon(K,H)^g=0. (See 7.13 for
  the definition of varepsilon(K,H)).
  
  Let  (K,H) be a strong Shoda pair of G. Then (K,H) is a Shoda pair (7.14) 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  (7.4)  e_ℚ (chi) of ℚ G realized by (K,H) is
  e(G,K,H), see [ORS04].
  
  Two  strong  Shoda  pairs (7.15) (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 (7.11, see [ORS04] for F=ℚ and [Olt07] for the general case).
  
  More  precisely, ℚ Ge(G,K,H) is isomorphic to M_n(ℚ (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 ℚ
  (xi)*_a^t  N/K  is  a crossed product (see 7.6) with action a and twisting t
  given as follows:
  
  Let  x  be  a  fixed  generator  of K/H and varphi : N/K -> N/H a fixed left
  inverse of the canonical projection N/H-> 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 [ORS04]. 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 ℚ (chi), the field of character
  values over ℚ, and N/K is isomorphic to Gal(ℚ (xi)/ℚ (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_ℚ  (chi)A_ℚ (chi). Using the description given above of A_ℚ
  (chi)=ℚ    G    e(G,K,H)    one    can    easily    describe   A_F(chi)   as
  M_nd(F(xi)/F(chi),t'), where d=[ℚ (xi): ℚ(chi)]/[F(xi):F(chi)] and t' is the
  restriction  to  Gal(F(xi)/F(chi))  of  t  (a  cocycle of N/K = Gal(ℚ (xi)/ℚ
  (chi))).
  
  
  7.16 Strongly monomial characters and strongly monomial groups
  
  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
  (7.15)  (K,H) of G and a linear character theta of K of G with kernel H such
  that chi=theta^G.
  
  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
  [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 7.17). Finally, Olteanu [Olt07] shows how
  to  compute  the  Wedderburn  decomposition (7.3) 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.
  
  .
  
  
  7.17 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.
  
  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.
  
  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  (7.15)  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 (7.4) of FK.
  
  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 (7.4)
  of  FG  [BR07].  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 (7.16) then every primitive central idempotent of
  FG  is  realizable by some strong Shoda pair (7.15) of G and some cyclotomic
  class  C  [BR07].  As  in  the zero characteristic case, this explain how to
  compute  the  Wedderburn  decomposition  (7.3) of FG 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.