  
  [1X2 [33X[0;0YWedderburn decomposition[133X[101X
  
  
  [1X2.1 [33X[0;0YWedderburn decomposition of a group algebra[133X[101X
  
  [1X2.1-1 WedderburnDecomposition[101X
  
  [33X[1;0Y[29X[2XWedderburnDecomposition[102X( [3XFG[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA list of simple algebras.[133X
  
  [33X[0;0YThe input [3XFG[103X should be a group algebra of a finite group [22XG[122X over the field [22XF[122X,
  where  [22XF[122X  is  either  an  abelian  number field (i.e. a subfield of a finite
  cyclotomic  extension  of the rationals) or a finite field of characteristic
  coprime with the order of [22XG[122X.[133X
  
  [33X[0;0YThe  function  returns  the  list  of all [13XWedderburn components[113X ([14X9.3[114X) of the
  group  algebra  [3XFG[103X.  If  [22XF[122X  is  an abelian number field then each Wedderburn
  component  is given as a matrix algebra of a [13Xcyclotomic algebra[113X ([14X9.11[114X). If [22XF[122X
  is  a  finite  field  then  the  Wedderburn  components  are given as matrix
  algebras over finite fields.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XWedderburnDecomposition( GroupRing( GF(5), DihedralGroup(16) ) );[127X[104X
    [4X[28X[ ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 1, 1 ] ),[128X[104X
    [4X[28X  ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 2, 2 ] ), ( GF(5^2)^[ 2, 2 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XWedderburnDecomposition( GroupRing( Rationals, DihedralGroup(16) ) );[127X[104X
    [4X[28X[ Rationals, Rationals, Rationals, Rationals, ( Rationals^[ 2, 2 ] ),[128X[104X
    [4X[28X  <crossed product with center NF(8,[ 1, 7 ]) over AsField( NF(8,[128X[104X
    [4X[28X    [ 1, 7 ]), CF(8) ) of a group of size 2> ][128X[104X
    [4X[25Xgap>[125X [27XWedderburnDecomposition( GroupRing( CF(5), DihedralGroup(16) ) );[127X[104X
    [4X[28X[ CF(5), CF(5), CF(5), CF(5), ( CF(5)^[ 2, 2 ] ),[128X[104X
    [4X[28X  <crossed product with center NF(40,[ 1, 31 ]) over AsField( NF(40,[128X[104X
    [4X[28X    [ 1, 31 ]), CF(40) ) of a group of size 2> ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  previous examples show that if [22XD_16[122X denotes the dihedral group of order
  [22X16[122X  then  the [13XWedderburn decomposition[113X ([14X9.3[114X) of [22XF_5 D_16[122X, [22Xℚ D_16[122X and [22Xℚ (ξ_5)
  D_16[122X are respectively[133X
  
  
  [24X[33X[0;6Y\mathbb  F_5  D_{16}  =  4 \mathbb F_5 \oplus M_2( \mathbb F_5 ) \oplus M_2(
  \mathbb F_{25} ),[133X
  
  [124X
  
  
  [24X[33X[0;6Yℚ D_{16} = 4 ℚ \oplus M_2( ℚ ) \oplus (K(\xi_8)/K,t),[133X
  
  [124X
  
  [33X[0;0Yand[133X
  
  
  [24X[33X[0;6Yℚ   (\xi_5)   D_{16}   =  4  ℚ  (\xi_5)  \oplus  M_2(  ℚ  (\xi_5)  )  \oplus
  (F(\xi_{40})/F,t),[133X
  
  [124X
  
  [33X[0;0Ywhere  [22X(K(ξ_8)/K,t)[122X  is a [13Xcyclotomic algebra[113X ([14X9.11[114X) with the centre [22XK=NF(8,[
  1,  7  ])=  ℚ (sqrt2)[122X, [22X(F(ξ_40)/F,t) = ℚ (sqrt2,ξ_5)[122X is a cyclotomic algebra
  with centre [22XF=NF(40,[ 1, 31 ])[122X and [22Xξ_n[122X denotes a [22Xn[122X-th root of unity.[133X
  
  [33X[0;0YTwo more examples:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XWedderburnDecomposition( GroupRing( Rationals, SmallGroup(48,15) ) );[127X[104X
    [4X[28X[ Rationals, Rationals, Rationals, Rationals, [128X[104X
    [4X[28X  <crossed product with center Rationals over CF(3) of a group of size 2>, [128X[104X
    [4X[28X  <crossed product with center Rationals over GaussianRationals of a group of \[128X[104X
    [4X[28Xsize 2>, <crossed product with center Rationals over CF(3) of a group of size [128X[104X
    [4X[28X    2>, <crossed product with center NF(8,[ 1, 7 ]) over AsField( NF(8,[128X[104X
    [4X[28X    [ 1, 7 ]), CF(8) ) of a group of size 2>, ( CF(3)^[ 2, 2 ] ), [128X[104X
    [4X[28X  <crossed product with center Rationals over CF(12) of a group of size 4> ][128X[104X
    [4X[25Xgap>[125X [27XWedderburnDecomposition( GroupRing( CF(3), SmallGroup(48,15) ) );[127X[104X
    [4X[28X[ CF(3), CF(3), CF(3), CF(3), ( CF(3)^[ 2, 2 ] ), [128X[104X
    [4X[28X  <crossed product with center CF(3) over AsField( CF(3), CF([128X[104X
    [4X[28X    12) ) of a group of size 2>, ( CF(3)^[ 2, 2 ] ), [128X[104X
    [4X[28X  <crossed product with center NF(24,[ 1, 7 ]) over AsField( NF(24,[128X[104X
    [4X[28X    [ 1, 7 ]), CF(24) ) of a group of size 2>, ( CF(3)^[ 2, 2 ] ), [128X[104X
    [4X[28X  ( CF(3)^[ 2, 2 ] ), ( <crossed product with center CF(3) over AsField( CF([128X[104X
    [4X[28X    3), CF(12) ) of a group of size 2>^[ 2, 2 ] ) ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  some  cases,  in  characteristic  zero,  some  entries  of the output of
  [2XWedderburnDecomposition[102X   do   not  provide  full  matrix  algebras  over  a
  [13Xcyclotomic  algebra[113X  ([14X9.11[114X), but "fractional matrix algebras". That entry is
  not an algebra that can be used as a [5XGAP[105X object. Instead it is a pair formed
  by  a  rational giving the "size" of the matrices and a crossed product. See
  [14X9.3[114X for a theoretical explanation of this phenomenon. In this case a warning
  message is displayed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XQG:=GroupRing(Rationals,SmallGroup(240,89));[127X[104X
    [4X[28X<algebra-with-one over Rationals, with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XWedderburnDecomposition(QG);[127X[104X
    [4X[28XWedderga: Warning!!![128X[104X
    [4X[28XSome of the Wedderburn components displayed are FRACTIONAL MATRIX ALGEBRAS!!![128X[104X
    [4X[28X[128X[104X
    [4X[28X[ Rationals, Rationals, <crossed product with center Rationals over CF([128X[104X
    [4X[28X    5) of a group of size 4>, ( Rationals^[ 4, 4 ] ), ( Rationals^[ 4, 4 ] ),[128X[104X
    [4X[28X  ( Rationals^[ 5, 5 ] ), ( Rationals^[ 5, 5 ] ), ( Rationals^[ 6, 6 ] ),[128X[104X
    [4X[28X  <crossed product with center NF(12,[ 1, 11 ]) over AsField( NF(12,[128X[104X
    [4X[28X    [ 1, 11 ]), NF(60,[ 1, 11 ]) ) of a group of size 4>,[128X[104X
    [4X[28X  [ 3/2, <crossed product with center NF(8,[ 1, 7 ]) over AsField( NF(8,[128X[104X
    [4X[28X        [ 1, 7 ]), NF(40,[ 1, 31 ]) ) of a group of size 4> ] ]  [128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X2.1-2 WedderburnDecompositionInfo[101X
  
  [33X[1;0Y[29X[2XWedderburnDecompositionInfo[102X( [3XFG[103X ) [32X attribute[133X
  [6XReturns:[106X  [33X[0;10YA  list  with  each  entry a numerical description of a [13Xcyclotomic
            algebra[113X ([14X9.11[114X).[133X
  
  [33X[0;0YThe input [3XFG[103X should be a group algebra of a finite group [22XG[122X over the field [22XF[122X,
  where  [22XF[122X  is  either  an  abelian  number field (i.e. a subfield of a finite
  cyclotomic  extension  of the rationals) or a finite field of characteristic
  coprime to the order of [22XG[122X.[133X
  
  [33X[0;0YThis function is a numerical counterpart of [2XWedderburnDecomposition[102X ([14X2.1-1[114X).[133X
  
  [33X[0;0YIt returns a list formed by lists of lengths 2, 4 or 5.[133X
  
  [33X[0;0YThe lists of length 2 are of the form[133X
  
  
  [24X[33X[0;6Y[n,F],[133X
  
  [124X
  
  [33X[0;0Ywhere  [22Xn[122X  is  a  positive  integer  and [22XF[122X is a field. It represents the [22Xn× n[122X
  matrix algebra [22XM_n(F)[122X over the field [22XF[122X.[133X
  
  [33X[0;0YThe lists of length 4 are of the form[133X
  
  
  [24X[33X[0;6Y[n,F,k,[d,\alpha,\beta]],[133X
  
  [124X
  
  [33X[0;0Ywhere  [22XF[122X  is a field and [22Xn,k,d,α,β[122X are non-negative integers, satisfying the
  conditions  mentioned in Section [14X9.12[114X. It represents the [22Xn× n[122X matrix algebra
  [22XM_n(A)[122X over the cyclic algebra[133X
  
  
  [24X[33X[0;6YA=F(\xi_k)[u | \xi_k^u = \xi_k^{\alpha}, u^d = \xi_k^{\beta}],[133X
  
  [124X
  
  [33X[0;0Ywhere [22Xξ_k[122X is a primitive [22Xk[122X-th root of unity.[133X
  
  [33X[0;0YThe lists of length 5 are of the form[133X
  
  
  [24X[33X[0;6Y[n,F,k,[d_i,\alpha_i,\beta_i]_{i=1}^m, [\gamma_{i,j}]_{1\le i < j \le m} ],[133X
  
  [124X
  
  [33X[0;0Ywhere  [22XF[122X  is a field and [22Xn,k,d_i,α_i,β_i,γ_i,j[122X are non-negative integers. It
  represents the [22Xn× n[122X matrix algebra [22XM_n(A)[122X over the [13Xcyclotomic algebra[113X ([14X9.11[114X)[133X
  
  
  [24X[33X[0;6YA   =   F(\xi_k)[g_1,\ldots,g_m   \mid   \xi_k^{g_i}   =   \xi_k^{\alpha_i},
  g_i^{d_i}=\xi_k^{\beta_i}, g_jg_i=\xi_k^{\gamma_{ij}} g_i g_j],[133X
  
  [124X
  
  [33X[0;0Ywhere [22Xξ_k[122X is a primitive [22Xk[122X-th root of unity (see [14X9.12[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XWedderburnDecompositionInfo( GroupRing( Rationals, DihedralGroup(16) ) );[127X[104X
    [4X[28X[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ],[128X[104X
    [4X[28X  [ 2, Rationals ], [ 1, NF(8,[ 1, 7 ]), 8, [ 2, 7, 0 ] ] ][128X[104X
    [4X[25Xgap>[125X [27XWedderburnDecompositionInfo( GroupRing( CF(5), DihedralGroup(16) ) );[127X[104X
    [4X[28X[ [ 1, CF(5) ], [ 1, CF(5) ], [ 1, CF(5) ], [ 1, CF(5) ], [ 2, CF(5) ],[128X[104X
    [4X[28X  [ 1, NF(40,[ 1, 31 ]), 8, [ 2, 7, 0 ] ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  interpretation  of  the  previous  example  gives rise to the following
  [13XWedderburn  decompositions[113X  ([14X9.3[114X), where [22XD_16[122X is the dihedral group of order
  16 and [22Xξ_5[122X is a primitive [22X5[122X-th root of unity.[133X
  
  
  [24X[33X[0;6Yℚ D_{16} = 4 ℚ \oplus M_2( ℚ ) \oplus M_2( ℚ (\sqrt{2})).[133X
  
  [124X
  
  
  [24X[33X[0;6Yℚ  (\xi_5)  D_{16}  =  4  ℚ  (\xi_5)  \oplus  M_2(  ℚ (\xi_5)) \oplus M_2( ℚ
  (\xi_5,\sqrt{2})).[133X
  
  [124X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XF:=FreeGroup("a","b");;a:=F.1;;b:=F.2;;rel:=[a^8,a^4*b^2,b^-1*a*b*a];;[127X[104X
    [4X[25Xgap>[125X [27XQ16:=F/rel;; QQ16:=GroupRing( Rationals, Q16 );;[127X[104X
    [4X[25Xgap>[125X [27XQS4:=GroupRing( Rationals, SymmetricGroup(4) );;[127X[104X
    [4X[25Xgap>[125X [27XWedderburnDecomposition(QQ16);[127X[104X
    [4X[28X[ Rationals, Rationals, Rationals, Rationals, ( Rationals^[ 2, 2 ] ),[128X[104X
    [4X[28X  <crossed product with center NF(8,[ 1, 7 ]) over AsField( NF(8,[128X[104X
    [4X[28X    [ 1, 7 ]), CF(8) ) of a group of size 2> ][128X[104X
    [4X[25Xgap>[125X [27XWedderburnDecomposition( QS4 );[127X[104X
    [4X[28X[ Rationals, Rationals, <crossed product with center Rationals over CF([128X[104X
    [4X[28X    3) of a group of size 2>, ( Rationals^[ 3, 3 ] ), ( Rationals^[ 3, 3 ] ) ][128X[104X
    [4X[25Xgap>[125X [27XWedderburnDecompositionInfo(QQ16);[127X[104X
    [4X[28X[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [128X[104X
    [4X[28X  [ 2, Rationals ], [ 1, NF(8,[ 1, 7 ]), 8, [ 2, 7, 4 ] ] ][128X[104X
    [4X[25Xgap>[125X [27XWedderburnDecompositionInfo(QS4);  [127X[104X
    [4X[28X[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals, 3, [ 2, 2, 0 ] ], [128X[104X
    [4X[28X  [ 3, Rationals ], [ 3, Rationals ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  the  previous  example  we  computed the Wedderburn decomposition of the
  rational  group  algebra  [22Xℚ Q_16[122X of the quaternion group of order [22X16[122X and the
  rational group algebra [22Xℚ S_4[122X of the symmetric group on four letters. For the
  two   group  algebras  we  used  both  [2XWedderburnDecomposition[102X  ([14X2.1-1[114X)  and
  [2XWedderburnDecompositionInfo[102X.[133X
  
  [33X[0;0YThe output of [2XWedderburnDecomposition[102X ([14X2.1-1[114X) shows that[133X
  
  
  [24X[33X[0;6Yℚ Q_{16} = 4 ℚ \oplus M_2( ℚ ) \oplus A,[133X
  
  [124X
  
  
  [24X[33X[0;6Yℚ S_{4} = 2 ℚ \oplus 2 M_3( ℚ ) \oplus B,[133X
  
  [124X
  
  [33X[0;0Ywhere [22XA[122X and [22XB[122X are [13Xcrossed products[113X ([14X9.6[114X) with coefficients in the cyclotomic
  fields  [22Xℚ  (ξ_8)[122X  and [22Xℚ (ξ_3)[122X respectively. This output can be used as a [5XGAP[105X
  object,  but  it  does  not  give  clear information on the structure of the
  algebras [22XA[122X and [22XB[122X.[133X
  
  [33X[0;0YThe  numerical  information  displayed  by [2XWedderburnDecompositionInfo[102X means
  that[133X
  
  
  [24X[33X[0;6YA = ℚ (\xi|\xi^8=1)[g | \xi^g = \xi^7 = \xi^{-1}, g^2 = \xi^4 = -1],[133X
  
  [124X
  
  
  [24X[33X[0;6YB = ℚ (\xi|\xi^3=1)[g | \xi^g = \xi^2 = \xi^{-1}, g^2 = 1].[133X
  
  [124X
  
  [33X[0;0YBoth [22XA[122X and [22XB[122X are quaternion algebras over its centre which is [22Xℚ (ξ+ξ^-1)[122X and
  the former is equal to [22Xℚ (sqrt2)[122X and [22Xℚ[122X respectively.[133X
  
  [33X[0;0YIn  [22XB[122X,  one  has  [22X(g+1)(g-1)=0[122X, while [22Xg[122X is neither [22X1[122X nor [22X-1[122X. This shows that
  [22XB=M_2( ℚ )[122X. However the relation [22Xg^2=-1[122X in [22XA[122X shows that[133X
  
  
  [24X[33X[0;6YA=ℚ (\sqrt{2})[i,g|i^2=g^2=-1,ig=-gi][133X
  
  [124X
  
  [33X[0;0Yand  so [22XA[122X is a division algebra with centre [22Xℚ (sqrt2)[122X, which is a subalgebra
  of  the algebra of Hamiltonian quaternions. This could be deduced also using
  well known methods on cyclic algebras (see e.g. [Rei03]).[133X
  
  [33X[0;0YThe next example shows the output of [10XWedderburnDecompositionInfo[110X for [22Xℚ G[122X and
  [22Xℚ  (ξ_3)  G[122X,  where  [22XG=SmallGroup(48,15)[122X.  The  user can compare it with the
  output of [2XWedderburnDecomposition[102X ([14X2.1-1[114X) for the same group in the previous
  section. Notice that the last entry of the [13XWedderburn decomposition[113X ([14X9.3[114X) of
  [22Xℚ  G[122X  is  not  given  as  a matrix algebra of a cyclic algebra. However, the
  corresponding entry of [22Xℚ (ξ_3) G[122X is a matrix algebra of a cyclic algebra.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XWedderburnDecompositionInfo( GroupRing( Rationals, SmallGroup(48,15) ) );[127X[104X
    [4X[28X[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals ], [128X[104X
    [4X[28X  [ 1, Rationals, 3, [ 2, 2, 0 ] ], [ 1, Rationals, 4, [ 2, 3, 0 ] ], [128X[104X
    [4X[28X  [ 1, Rationals, 6, [ 2, 5, 0 ] ], [ 1, NF(8,[ 1, 7 ]), 8, [ 2, 7, 0 ] ], [128X[104X
    [4X[28X  [ 2, CF(3) ], [ 1, Rationals, 12, [ [ 2, 5, 3 ], [ 2, 7, 0 ] ], [ [ 3 ] ] ] [128X[104X
    [4X[28X ][128X[104X
    [4X[25Xgap>[125X [27XWedderburnDecompositionInfo( GroupRing( CF(3), SmallGroup(48,15) ) );[127X[104X
    [4X[28X[ [ 1, CF(3) ], [ 1, CF(3) ], [ 1, CF(3) ], [ 1, CF(3) ], [128X[104X
    [4X[28X  [ 2, CF(3), 3, [ 1, 1, 0 ] ], [ 1, CF(3), 4, [ 2, 3, 0 ] ], [128X[104X
    [4X[28X  [ 2, CF(3), 6, [ 1, 1, 0 ] ], [ 1, NF(24,[ 1, 7 ]), 8, [ 2, 7, 0 ] ], [128X[104X
    [4X[28X  [ 2, CF(3) ], [ 2, CF(3) ], [ 2, CF(3), 12, [ 2, 7, 0 ] ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn   some   cases   some   of   the   first   entries   of   the  output  of
  [2XWedderburnDecompositionInfo[102X   are  not  integers  and  so  the  correspoding
  [13XWedderburn  components[113X  ([14X9.3[114X)  are  given as "fractional matrix algebras" of
  [13Xcyclotomic  algebras[113X  ([14X9.11[114X).  See [14X9.3[114X for a theoretical explanation of this
  phenomenon.  In  that  case  a  warning message will be displayed during the
  first call of [10XWedderburnDecompositionInfo[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XQG:=GroupRing(Rationals,SmallGroup(240,89));[127X[104X
    [4X[28X<algebra-with-one over Rationals, with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XWedderburnDecompositionInfo(QG);[127X[104X
    [4X[28XWedderga: Warning!!! [128X[104X
    [4X[28XSome of the Wedderburn components displayed are FRACTIONAL MATRIX ALGEBRAS!!![128X[104X
    [4X[28X[128X[104X
    [4X[28X[ [ 1, Rationals ], [ 1, Rationals ], [ 1, Rationals, 10, [ 4, 3, 5 ] ],[128X[104X
    [4X[28X  [ 4, Rationals ], [ 4, Rationals ], [ 5, Rationals ], [ 5, Rationals ],[128X[104X
    [4X[28X  [ 6, Rationals ], [ 1, NF(12,[ 1, 11 ]), 10, [ 4, 3, 5 ] ],[128X[104X
    [4X[28X  [ 3/2, NF(8,[ 1, 7 ]), 10, [ 4, 3, 5 ] ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  interpretation  of the output in the previous example gives rise to the
  following  [13XWedderburn  decomposition[113X  ([14X9.3[114X)  of  [22Xℚ  G[122X  for [22XG[122X the small group
  [22X[240,89][122X:[133X
  
  
  [24X[33X[0;6Yℚ  G  =  2  ℚ  \oplus  2 M_4( ℚ ) \oplus 2 M_5( ℚ ) \oplus M_6( ℚ ) \oplus A
  \oplus B \oplus C[133X
  
  [124X
  
  [33X[0;0Ywhere[133X
  
  
  [24X[33X[0;6YA = ℚ (\xi_{10})[u|\xi_{10}^u = \xi_{10}^3, u^4 = -1],[133X
  
  [124X
  
  [33X[0;0Y[22XB[122X is an algebra of degree [22X(4*2 )/2 = 4[122X which is [13XBrauer equivalent[113X ([14X9.5[114X) to[133X
  
  
  [24X[33X[0;6YB_1  =  ℚ  (\xi_{60})[u,v|\xi_{60}^u  =  \xi_{60}^{13},  u^4  =  \xi_{60}^5,
  \xi_{60}^v = \xi_{60}^{11}, v^2 = 1, vu=uv],[133X
  
  [124X
  
  [33X[0;0Yand [22XC[122X is an algebra of degree [22X(4*2)*3/4 = 6[122X which is [13XBrauer equivalent[113X ([14X9.5[114X)
  to[133X
  
  
  [24X[33X[0;6YC_1 = ℚ (\xi_{60})[u,v|\xi_{60}^u = \xi_{60}^7, u^4 = \xi_{60}^5, \xi_{60}^v
  = \xi_{60}^{31}, v^2 = 1, vu=uv].[133X
  
  [124X
  
  [33X[0;0YThe precise description of [22XB[122X and [22XC[122X requires the usage of "ad hoc" arguments.[133X
  
  
  [1X2.2 [33X[0;0YSimple quotients[133X[101X
  
  [1X2.2-1 SimpleAlgebraByCharacter[101X
  
  [33X[1;0Y[29X[2XSimpleAlgebraByCharacter[102X( [3XFG[103X, [3Xchi[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA simple algebra.[133X
  
  [33X[0;0YThe  first input [3XFG[103X should be a [13Xsemisimple group algebra[113X ([14X9.2[114X) over a finite
  group [22XG[122X and the second input should be an irreducible character of [22XG[122X.[133X
  
  [33X[0;0YThe  output  is  a  matrix  algebra of a [13Xcyclotomic algebras[113X ([14X9.11[114X) which is
  isomorphic to the unique [13XWedderburn component[113X ([14X9.3[114X) [22XA[122X of [3XFG[103X such that [22Xχ(A)ne
  0[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XA5 := AlternatingGroup(5);[127X[104X
    [4X[28XAlt( [ 1 .. 5 ] )[128X[104X
    [4X[25Xgap>[125X [27XSimpleAlgebraByCharacter( GroupRing( Rationals , A5 ) , Irr( A5 ) [3] );[127X[104X
    [4X[28X( NF(5,[ 1, 4 ])^[ 3, 3 ] )[128X[104X
    [4X[25Xgap>[125X [27XSimpleAlgebraByCharacter( GroupRing( GF(7) , A5 ) , Irr( A5 ) [3] );[127X[104X
    [4X[28X( GF(7^2)^[ 3, 3 ] )[128X[104X
    [4X[25Xgap>[125X [27XG:=SmallGroup(128,100);               [127X[104X
    [4X[28X<pc group of size 128 with 7 generators>[128X[104X
    [4X[25Xgap>[125X [27Xchi4:=Filtered(Irr(G),x->Degree(x)=4);;[127X[104X
    [4X[25Xgap>[125X [27XList(chi4,x->SimpleAlgebraByCharacter(GroupRing(Rationals,G),x));[127X[104X
    [4X[28X[ ( <crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8,[128X[104X
    [4X[28X    [ 1, 3 ]), CF(8) ) of a group of size 2>^[ 2, 2 ] ), [128X[104X
    [4X[28X  ( <crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8,[128X[104X
    [4X[28X    [ 1, 3 ]), CF(8) ) of a group of size 2>^[ 2, 2 ] ), [128X[104X
    [4X[28X  ( <crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8,[128X[104X
    [4X[28X    [ 1, 3 ]), CF(8) ) of a group of size 2>^[ 2, 2 ] ), [128X[104X
    [4X[28X  ( <crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8,[128X[104X
    [4X[28X    [ 1, 3 ]), CF(8) ) of a group of size 2>^[ 2, 2 ] ) ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X2.2-2 SimpleAlgebraByCharacterInfo[101X
  
  [33X[1;0Y[29X[2XSimpleAlgebraByCharacterInfo[102X( [3XFG[103X, [3Xchi[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YThe     numerical     description     of     the     output     of
            [2XSimpleAlgebraByCharacter[102X ([14X2.2-1[114X).[133X
  
  [33X[0;0YThe first input [3XFG[103X is a [13Xsemisimple group algebra[113X ([14X9.2[114X) over a finite group [22XG[122X
  and the second input is an irreducible character of [22XG[122X.[133X
  
  [33X[0;0YThe  output  is  the  numerical  description  [14X9.12[114X of the [13Xcyclotomic algebra[113X
  ([14X9.11[114X)  which is isomorphic to the unique [13XWedderburn component[113X ([14X9.3[114X) [22XA[122X of [3XFG[103X
  such that [22Xχ(A)ne 0[122X.[133X
  
  [33X[0;0YSee  [14X9.12[114X  for  the interpretation of the numerical information given by the
  output.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XG:=SmallGroup(128,100);[127X[104X
    [4X[28X<pc group of size 128 with 7 generators>[128X[104X
    [4X[25Xgap>[125X [27XQG:=GroupRing(Rationals,G);[127X[104X
    [4X[28X<algebra-with-one over Rationals, with 7 generators>[128X[104X
    [4X[25Xgap>[125X [27Xchi4:=Filtered(Irr(G),x->Degree(x)=4);;[127X[104X
    [4X[25Xgap>[125X [27XList(chi4,x->SimpleAlgebraByCharacterInfo(QG,x));[127X[104X
    [4X[28X[ [ 2, NF(8,[ 1, 3 ]), 8, [ 2, 3, 0 ] ], [ 2, NF(8,[ 1, 3 ]), 8, [ 2, 3, 0 ] ], [128X[104X
    [4X[28X  [ 2, NF(8,[ 1, 3 ]), 8, [ 2, 3, 4 ] ], [ 2, NF(8,[ 1, 3 ]), 8, [ 2, 3, 4 ] ] ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X2.2-3 SimpleAlgebraByStrongSP[101X
  
  [33X[1;0Y[29X[2XSimpleAlgebraByStrongSP[102X( [3XQG[103X, [3XK[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSimpleAlgebraByStrongSPNC[102X( [3XQG[103X, [3XK[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSimpleAlgebraByStrongSP[102X( [3XFG[103X, [3XK[103X, [3XH[103X, [3XC[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSimpleAlgebraByStrongSPNC[102X( [3XFG[103X, [3XK[103X, [3XH[103X, [3XC[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA simple algebra.[133X
  
  [33X[0;0YIn  the  three-argument  version  the  input  must be formed by a [13Xsemisimple
  rational  group  algebra[113X  [3XQG[103X  (see [14X9.2[114X) and two subgroups [3XK[103X and [3XH[103X of [22XG[122X which
  form a [13Xstrong Shoda pair[113X ([14X9.15[114X) of [22XG[122X.[133X
  
  [33X[0;0YThe  three-argument  version  returns  the Wedderburn component ([14X9.3[114X) of the
  rational group algebra [3XQG[103X realized by the strong Shoda pair ([3XK[103X,[3XH[103X).[133X
  
  [33X[0;0YIn the four-argument version the first argument is a semisimple finite group
  algebra  [3XFG[103X,  [3X(K,H)[103X is a strong Shoda pair of [22XG[122X and the fourth input data is
  either  a  generating  [22Xq[122X-cyclotomic  class  modulo  the index of [3XH[103X in [3XK[103X or a
  representative of a generating [22Xq[122X-cyclotomic class modulo the index of [3XH[103X in [3XK[103X
  (see [14X9.19[114X).[133X
  
  [33X[0;0YThe  four-argument  version  returns  the  Wedderburn component ([14X9.3[114X) of the
  finite  group  algebra  [3XFG[103X  realized  by the strong Shoda pair ([3XK[103X,[3XH[103X) and the
  cyclotomic class [3XC[103X (or the cyclotomic class containing [3XC[103X).[133X
  
  [33X[0;0YThe versions ending in NC do not check if ([3XK[103X,[3XH[103X) is a strong Shoda pair of [22XG[122X.
  In  the  four-argument  version it is also not checked whether [3XC[103X is either a
  generating  [22Xq[122X-cyclotomic  class  modulo  the  index  of [3XH[103X in [3XK[103X or an integer
  coprime to the index of [3XH[103X in [3XK[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XF:=FreeGroup("a","b");; a:=F.1;; b:=F.2;;[127X[104X
    [4X[25Xgap>[125X [27XG:=F/[ a^16, b^2*a^8, b^-1*a*b*a^9 ];; a:=G.1;; b:=G.2;;[127X[104X
    [4X[25Xgap>[125X [27XK:=Subgroup(G,[a]);; H:=Subgroup(G,[]);;[127X[104X
    [4X[25Xgap>[125X [27XQG:=GroupRing( Rationals, G );;[127X[104X
    [4X[25Xgap>[125X [27XFG:=GroupRing( GF(7), G );;[127X[104X
    [4X[25Xgap>[125X [27XSimpleAlgebraByStrongSP( QG, K, H );[127X[104X
    [4X[28X<crossed product over CF(16) of a group of size 2>[128X[104X
    [4X[25Xgap>[125X [27XSimpleAlgebraByStrongSP( FG, K, H, [1,7] );[127X[104X
    [4X[28X( GF(7)^[ 2, 2 ] )[128X[104X
    [4X[25Xgap>[125X [27XSimpleAlgebraByStrongSP( FG, K, H, 1 );[127X[104X
    [4X[28X( GF(7)^[ 2, 2 ] )[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
  [1X2.2-4 SimpleAlgebraByStrongSPInfo[101X
  
  [33X[1;0Y[29X[2XSimpleAlgebraByStrongSPInfo[102X( [3XQG[103X, [3XK[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSimpleAlgebraByStrongSPInfoNC[102X( [3XQG[103X, [3XK[103X, [3XH[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSimpleAlgebraByStrongSPInfo[102X( [3XFG[103X, [3XK[103X, [3XH[103X, [3XC[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XSimpleAlgebraByStrongSPInfoNC[102X( [3XFG[103X, [3XK[103X, [3XH[103X, [3XC[103X ) [32X operation[133X
  [6XReturns:[106X  [33X[0;10YA numerical description of one simple algebra.[133X
  
  [33X[0;0YIn  the  three-argument  version  the  input  must be formed by a [13Xsemisimple
  rational  group algebra[113X ([14X9.2[114X) [3XQG[103X and two subgroups [3XK[103X and [3XH[103X of [22XG[122X which form a
  [13Xstrong  Shoda  pair[113X  ([14X9.15[114X)  of  [22XG[122X.  It  returns  the  numerical information
  describing  the Wedderburn component ([14X9.12[114X) of the rational group algebra [3XQG[103X
  realized by a the strong Shoda pair ([3XK[103X,[3XH[103X).[133X
  
  [33X[0;0YIn  the  four-argument  version the first input is a semisimple finite group
  algebra  [3XFG[103X,  [3X(K,H)[103X is a strong Shoda pair of [22XG[122X and the fourth input data is
  either  a  generating  [22Xq[122X-cyclotomic  class  modulo  the index of [3XH[103X in [3XK[103X or a
  representative of a generating [22Xq[122X-cyclotomic class modulo the index of [3XH[103X in [3XK[103X
  ([14X9.19[114X).  It returns a pair of positive integers [22X[n,r][122X which represent the [22Xn×
  n[122X  matrix  algebra  over  the  field  of  order [22Xr[122X which is isomorphic to the
  Wedderburn component of [3XFG[103X realized by a the strong Shoda pair ([3XK[103X,[3XH[103X) and the
  cyclotomic class [3XC[103X (or the cyclotomic class containing the integer [3XC[103X).[133X
  
  [33X[0;0YThe versions ending in NC do not check if ([3XK[103X,[3XH[103X) is a strong Shoda pair of [22XG[122X.
  In  the  four-argument  version it is also not checked whether [3XC[103X is either a
  generating  [22Xq[122X-cyclotomic  class  modulo  the  index  of [3XH[103X in [3XK[103X or an integer
  coprime with the index of [3XH[103X in [3XK[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XF:=FreeGroup("a","b");; a:=F.1;; b:=F.2;;[127X[104X
    [4X[25Xgap>[125X [27XG:=F/[ a^16, b^2*a^8, b^-1*a*b*a^9 ];; a:=G.1;; b:=G.2;;[127X[104X
    [4X[25Xgap>[125X [27XK:=Subgroup(G,[a]);; H:=Subgroup(G,[]);; [127X[104X
    [4X[25Xgap>[125X [27XQG:=GroupRing( Rationals, G );;[127X[104X
    [4X[25Xgap>[125X [27XFG:=GroupRing( GF(7), G );;[127X[104X
    [4X[25Xgap>[125X [27XSimpleAlgebraByStrongSP( QG, K, H );[127X[104X
    [4X[28X<crossed product over CF(16) of a group of size 2>[128X[104X
    [4X[25Xgap>[125X [27XSimpleAlgebraByStrongSPInfo( QG, K, H );[127X[104X
    [4X[28X[ 1, NF(16,[ 1, 7 ]), 16, [ [ 2, 7, 8 ] ], [  ] ][128X[104X
    [4X[25Xgap>[125X [27XSimpleAlgebraByStrongSPInfo( FG, K, H, [1,7] );[127X[104X
    [4X[28X[ 2, 7 ][128X[104X
    [4X[25Xgap>[125X [27XSimpleAlgebraByStrongSPInfo( FG, K, H, 1 );[127X[104X
    [4X[28X[ 2, 7 ][128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
