  
  [1X9 [33X[0;0YThe basic theory behind [5XWedderga[105X[101X[1X[133X[101X
  
  [33X[0;0YIn this chapter we describe the theory that is behind the algorithms used by
  [5XWedderga[105X.[133X
  
  [33X[0;0YAll  the  rings  considered  in  this  chapter  are  associative and have an
  identity.[133X
  
  [33X[0;0YWe  use the following notation: [22Xℚ[122X denotes the field of rationals and [22XF_q[122X the
  finite  field  of order [22Xq[122X. For every positive integer [22Xk[122X, we denote a complex
  [22Xk[122X-th  primitive  root  of  unity by [22Xξ_k[122X and so [22Xℚ(ξ_k)[122X is the [22Xk[122X-th cyclotomic
  extension of [22Xℚ[122X.[133X
  
  
  [1X9.1 [33X[0;0YGroup rings and group algebras[133X[101X
  
  [33X[0;0YGiven  a  group  [22XG[122X  and  a  ring  [22XR[122X, the [13Xgroup ring[113X [22XRG[122X over the group [22XG[122X with
  coefficients  in  [22XR[122X  is  the ring whose underlying additive group is a right
  [22XR-[122Xmodule with basis [22XG[122X such that the product is defined by the following rule[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y(gr)(hs)=(gh)(rs)[133X [124X[133X
  
  
  [33X[0;0Yfor [22Xr,s ∈ R[122X and [22Xg, h ∈ G[122X, and extended to [22XRG[122X by linearity.[133X
  
  [33X[0;0YA [13Xgroup algebra[113X is a group ring in which the coefficient ring is a field.[133X
  
  
  [1X9.2 [33X[0;0YSemisimple group algebras[133X[101X
  
  [33X[0;0YWe  say  that  a  ring  [22XR[122X is semisimple if it is a direct sum of simple left
  (alternatively  right) ideals or equivalently if [22XR[122X is isomorphic to a direct
  product  of  simple  algebras  each  one  isomorphic to a matrix ring over a
  division ring.[133X
  
  [33X[0;0YBy  Maschke's  Theorem,  if [22XG[122X is a finite group then the group algebra [22XFG[122X is
  semisimple  if  and  only the characteristic of the coefficient field [22XF[122X does
  not divide the order of [22XG[122X.[133X
  
  [33X[0;0YIn  fact,  an  arbitrary  group  ring  [22XRG[122X  is  semisimple if and only if the
  coefficient  ring  [22XR[122X is semisimple, the group [22XG[122X is finite and the order of [22XG[122X
  is invertible in [22XR[122X.[133X
  
  [33X[0;0YSome  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.[133X
  
  
  [1X9.3 [33X[0;0YWedderburn components[133X[101X
  
  [33X[0;0YIf  [22XR[122X  is  a [13Xsemisimple ring[113X ([14X9.2[114X) then the [13XWedderburn decomposition[113X of [22XR[122X is
  the  decomposition  of [22XR[122X as a direct product of simple algebras. The factors
  of this Wedderburn decomposition are called [13XWedderburn components[113X of [22XR[122X. Each
  Wedderburn  component  of  [22XR[122X  is  of  the  form [22XRe[122X for [22Xe[122X a [13Xprimitive central
  idempotent[113X ([14X9.4[114X) of [22XR[122X.[133X
  
  [33X[0;0YLet   [22XFG[122X   be   a   [13Xsemisimple  group  algebra[113X  ([14X9.2[114X).  If  [22XF[122X  has  positive
  characteristic,  then  the  Wedderburn  components of [22XFG[122X are matrix algebras
  over  finite  extensions  of  [22XF[122X.  If  [22XF[122X  has zero characteristic then by the
  [13XBrauer-Witt  Theorem[113X  [Yam74],  the  [13XWedderburn  components[113X of [22XFG[122X are [13XBrauer
  equivalent[113X ([14X9.5[114X) to [13Xcyclotomic algebras[113X ([14X9.11[114X).[133X
  
  [33X[0;0YThe  main  functions  of  [5XWedderga[105X  compute  the  Wedderburn components of a
  semisimple  group  algebra  [22XFG[122X, 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.[133X
  
  [33X[0;0YIn  the  zero  characteristic  case  each  Wedderburn  component [22XA[122X is [13XBrauer
  equivalent[113X  ([14X9.5[114X)  to  a  [13Xcyclotomic  algebra[113X  ([14X9.11[114X)  and  therefore [22XA[122X is a
  (possibly  fractional)  matrix  algebra  over  [13Xcyclotomic algebra[113X and can be
  described numerically in one of the following three forms:[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y[n,K],[133X [124X[133X
  
  
  
        [33X[1;6Y[24X[33X[0;0Y[n,K,k,[d,\alpha,\beta]],[133X [124X[133X
  
  
  
        [33X[1;6Y[24X[33X[0;0Y[n,K,k,[d_i,\alpha_i,\beta_i]_{i=1}^m, [\gamma_{i,j}]_{1\le i < j \le
        n} ],[133X [124X[133X
  
  
  [33X[0;0Ywhere  [22Xn[122X  is the matrix size, [22XK[122X is the centre of [22XA[122X (a finite field extension
  of  [22XF[122X) and the remaining data are integers whose interpretation is explained
  in [14X9.12[114X.[133X
  
  [33X[0;0YIn  some cases (for the zero characteristic coefficient field) the size [22Xn[122X of
  the  matrix  algebras  is  not  a  positive  integer but a positive rational
  number.  This  is  a  consequence  of  the fact that the [13XBrauer-Witt Theorem[113X
  [Yam74]  only  ensures  that each [13XWedderburn component[113X ([14X9.3[114X) of a semisimple
  group algebra is Brauer equivalent ([14X9.5[114X) to a [13Xcyclotomic algebra[113X ([14X9.11[114X), but
  not necessarily isomorphic to a full matrix algebra of a cyclotomic algebra.
  For  example,  a Wedderburn component [22XD[122X of a group algebra can be a division
  algebra  but  not  a cyclotomic algebra. In this case [22XM_n(D)[122X is a cyclotomic
  algebra  [22XC[122X for some [22Xn[122X and therefore [22XD[122X can be described as [22XM_1/n(C)[122X (see last
  Example in [2XWedderburnDecomposition[102X ([14X2.1-1[114X)).[133X
  
  [33X[0;0YThe main algorithm of [5XWedderga[105X 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 [13Xstrongly
  monomial groups[113X ([14X9.16[114X).[133X
  
  
  [1X9.4 [33X[0;0YCharacters and primitive central idempotents[133X[101X
  
  [33X[0;0YA  [13Xprimitive central idempotent[113X of a ring [22XR[122X is a non-zero central idempotent
  [22Xe[122X  which cannot be written as the sum of two non-zero central idempotents of
  [22XRe[122X,  or  equivalently, such that [22XRe[122X is indecomposable as a direct product of
  two non-trivial two-sided ideals.[133X
  
  [33X[0;0YThe  [13XWedderburn components[113X ([14X9.3[114X) of a semisimple ring [22XR[122X are the rings of the
  form [22XRe[122X for [22Xe[122X running over the set of primitive central idempotents of [22XR[122X.[133X
  
  [33X[0;0YLet [22XFG[122X be a [13Xsemisimple group algebra[113X ([14X9.2[114X) and [22Xχ[122X an irreducible character of
  [22XG[122X  (in  an  algebraic  closure  of  [22XF[122X).  Then  there  is a unique Wedderburn
  component  [22XA=A_F(χ)[122X  of  [22XFG[122X such that [22Xχ(A)ne 0[122X. Let [22Xe_F(χ)[122X denote the unique
  primitive  central  idempotent  of  [22XFG[122X  in  [22XA_F(χ)[122X,  that is the identity of
  [22XA_F(χ)[122X, i.e.[133X
  
  
        [33X[1;6Y[24X[33X[0;0YA_F(\chi)=FGe_F(\chi).[133X [124X[133X
  
  
  [33X[0;0YThe centre of [22XA_F(χ)[122X is [22XF(χ)=F(χ(g):g ∈ G)[122X, the [13Xfield of character values[113X of
  [22Xχ[122X over [22XF[122X.[133X
  
  [33X[0;0YThe  map  [22Xχ  ↦  A_F(χ)[122X  defines a surjective map from the set of irreducible
  characters  of  [22XG[122X  (in an algebraic closure of [22XF[122X) onto the set of Wedderburn
  components of [22XFG[122X.[133X
  
  [33X[0;0YEquivalently,  the  map  [22Xχ ↦ e_F(χ)[122X defines a surjective map from the set of
  irreducible  characters  of [22XG[122X (in an algebraic closure of [22XF[122X) onto the set of
  primitive central idempontents of [22XFG[122X.[133X
  
  [33X[0;0YIf the irreducible character [22Xχ[122X of [22XG[122X takes values in [22XF[122X then[133X
  
  
        [33X[1;6Y[24X[33X[0;0Ye_F(\chi) = e(\chi) = \frac{\chi(1)}{|G|} \sum_{g\in G} \chi(g^{-1})
        g.[133X [124X[133X
  
  
  [33X[0;0YIn general one has[133X
  
  
        [33X[1;6Y[24X[33X[0;0Ye_F(\chi) = \sum_{\sigma \in Gal(F(\chi)/F)} e(\sigma \circ \chi).[133X [124X[133X
  
  
  
  [1X9.5 [33X[0;0YCentral simple algebras and Brauer equivalence[133X[101X
  
  [33X[0;0YLet  [22XK[122X  be  a  field.  A  [13Xcentral  simple  [22XK[122X-algebra[113X is a finite dimensional
  [22XK[122X-algebra  with  center  [22XK[122X  which  has  no  non-trivial proper ideals. Every
  central simple [22XK[122X-algebra is isomorphic to a matrix algebra [22XM_n(D)[122X where [22XD[122X is
  a  division  algebra  (which is finite-dimensional over [22XK[122X and has centre [22XK[122X).
  The division algebra [22XD[122X is unique up to [22XK[122X-isomorphisms.[133X
  
  [33X[0;0YTwo  central  simple [22XK[122X-algebras [22XA[122X and [22XB[122X are said to be [13XBrauer equivalent[113X, or
  simply  [13Xequivalent[113X,  if  there  is  a  division  algebra  [22XD[122X and two positive
  integers  [22Xm[122X and [22Xn[122X such that [22XA[122X is isomorphic to [22XM_m(D)[122X and [22XB[122X is isomorphic to
  [22XM_n(D)[122X.[133X
  
  
  [1X9.6 [33X[0;0YCrossed Products[133X[101X
  
  [33X[0;0YLet [22XR[122X be a ring and [22XG[122X a group.[133X
  
  [33X[0;0Y[12XIntrinsic  definition[112X.  A  [13Xcrossed  product[113X  [Pas89]  of  [22XG[122X  over [22XR[122X (or with
  coefficients  in  [22XR[122X) is a ring [22XR*G[122X with a decomposition into a direct sum of
  additive subgroups[133X
  
  
        [33X[1;6Y[24X[33X[0;0YR*G = \bigoplus_{g \in G} A_g[133X [124X[133X
  
  
  [33X[0;0Ysuch that for each [22Xg,h[122X in [22XG[122X one has:[133X
  
  [33X[0;0Y* [22XA_1=R[122X (here [22X1[122X denotes the identity of [22XG[122X),[133X
  
  [33X[0;0Y* [22XA_g A_h = A_gh[122X and[133X
  
  [33X[0;0Y* [22XA_g[122X has a unit of [22XR*G[122X.[133X
  
  [33X[0;0Y[12XExtrinsic  definition[112X. Let [22XAut(R)[122X denote the group of automorphisms of [22XR[122X and
  let [22XR^*[122X denote the group of units of [22XR[122X.[133X
  
  [33X[0;0YLet  [22Xa:G  →  Aut(R)[122X  and  [22Xt:G × G → R^*[122X be mappings satisfying the following
  conditions for every [22Xg[122X, [22Xh[122X and [22Xk[122X in [22XG[122X:[133X
  
  [33X[0;0Y(1)  [22Xa(gh)^-1  a(g)  a(h)[122X  is  the inner automorphism of [22XR[122X induced by [22Xt(g,h)[122X
  (i.e. the automorphism [22Xx↦ t(g,h)^-1 x t(g,h)[122X) and[133X
  
  [33X[0;0Y(2)  [22Xt(gh,k)  t(g,h)^k = t(g,hk) t(h,k)[122X, where for [22Xg ∈ G[122X and [22Xx ∈ R[122X we denote
  [22Xa(g)(x)[122X by [22Xx^g[122X.[133X
  
  [33X[0;0YThe  [13Xcrossed product[113X [Pas89] of [22XG[122X over [22XR[122X (or with coefficients in [22XR[122X), action
  [22Xa[122X and twisting [22Xt[122X is the ring[133X
  
  
        [33X[1;6Y[24X[33X[0;0YR*_a^t G = \bigoplus_{g\in G} u_g R[133X [124X[133X
  
  
  [33X[0;0Ywhere [22X{u_g : g∈ G }[122X is a set of symbols in one-to-one correspondence with [22XG[122X,
  with addition and multiplication defined by[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y(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[133X [124X[133X
  
  
  [33X[0;0Yfor [22Xg,h ∈ G[122X and [22Xr,s∈ R[122X, and extended to [22XR*_a^t G[122X by linearity.[133X
  
  [33X[0;0YThe  associativity of the product defined is a consequence of conditions (1)
  and (2) [Pas89].[133X
  
  [33X[0;0Y[12XEquivalence  of the two definitions[112X. Obviously the crossed product of [22XG[122X over
  [22XR[122X  defined using the extrinsic definition is a crossed product of [22XG[122X over [22Xu_1
  R[122X  in  the sense of the first definition. Moreover, there is [22Xr_0[122X in [22XR^*[122X such
  that  [22Xu_1r_0[122X is the identity of [22XR*_a^t G[122X and the map [22Xr ↦ u_1 r_0 r[122X is a ring
  isomorphism [22XR → u_1R[122X.[133X
  
  [33X[0;0YConversely,  let [22XR*G=⨁_g∈ G A_g[122X be an (intrinsic) crossed product and select
  for  each  [22Xg∈  G[122X a unit [22Xu_g∈ A_g[122X of [22XR*G[122X. This is called a [13Xbasis of units for
  the  crossed  product[113X [22XR*G[122X. Then the maps [22Xa:G → Aut(R)[122X and [22Xt:G× G → R^*[122X given
  by[133X
  
  
        [33X[1;6Y[24X[33X[0;0Yr^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)[133X [124X[133X
  
  
  [33X[0;0Ysatisfy conditions (1) and (2) and [22XR*G = R*_a^t G[122X.[133X
  
  [33X[0;0YThe  choice  of  a  basis  of  units  [22Xu_g  ∈ A_g[122X determines the action [22Xa[122X and
  twisting [22Xt[122X. If [22X{u_g ∈ A_g : g ∈ G }[122X and [22X{v_g ∈ A_g : g ∈ G }[122X are two sets of
  units  of [22XR*G[122X then [22Xv_g = u_g r_g[122X for some units [22Xr_g[122X of [22XR[122X. 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.[133X
  
  [33X[0;0YIt  is customary to select [22Xu_1=1[122X. In that case [22Xa(1)[122X is the identity map of [22XR[122X
  and [22Xt(1,g)=t(g,1)=1[122X for each [22Xg[122X in [22XG[122X.[133X
  
  
  [1X9.7 [33X[0;0YCyclic Crossed Products[133X[101X
  
  [33X[0;0YLet  [22XR*G=⨁_g ∈ G A_g[122X be a [13Xcrossed product[113X ([14X9.6[114X) and assume that [22XG = ⟨ g ⟩[122X is
  cyclic.  Then  the  crossed  product  can be given using a particularly nice
  description.[133X
  
  [33X[0;0YSelect  a  unit  [22Xu[122X in [22XA_g[122X, and let [22Xa[122X be the automorphism of [22XR[122X given by [22Xr^a =
  u^-1 r u[122X.[133X
  
  [33X[0;0YIf [22XG[122X is infinite then set [22Xu_g^k = u^k[122X for every integer [22Xk[122X. Then[133X
  
  
        [33X[1;6Y[24X[33X[0;0YR*G = R[ u | ru = u r^a ],[133X [124X[133X
  
  
  [33X[0;0Ya skew polynomial ring. Therefore in this case [22XR*G[122X is determined by[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y[ R, a ].[133X [124X[133X
  
  
  [33X[0;0YIf  [22XG[122X is finite of order [22Xd[122X then set [22Xu_g^k = u^k[122X for [22X0 le k < d[122X. Then [22Xb = u^d
  ∈ R[122X and[133X
  
  
        [33X[1;6Y[24X[33X[0;0YR*G = R[ u | ru = u r^a, u^d = b ][133X [124X[133X
  
  
  [33X[0;0YTherefore, [22XR*G[122X is completely determined by the following data:[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y[ R , [ d , a , b ] ][133X [124X[133X
  
  
  
  [1X9.8 [33X[0;0YAbelian Crossed Products[133X[101X
  
  [33X[0;0YLet [22XR*G=⨁_g ∈ G A_g[122X be a [13Xcrossed product[113X ([14X9.6[114X) and assume that [22XG[122X is abelian.
  Then the crossed product can be given using a simple description.[133X
  
  [33X[0;0YExpress [22XG[122X as a direct sum of cyclic groups:[133X
  
  
        [33X[1;6Y[24X[33X[0;0YG = \langle g_1 \rangle \times \cdots \times \langle g_n \rangle[133X [124X[133X
  
  
  [33X[0;0Yand for each [22Xi=1,dots,n[122X select a unit [22Xu_i[122X in [22XA_g_i[122X.[133X
  
  [33X[0;0YEach element [22Xg[122X of [22XG[122X has a unique expression[133X
  
  
        [33X[1;6Y[24X[33X[0;0Yg = g_1^{k_1} \cdots g_n^{k_n},[133X [124X[133X
  
  
  [33X[0;0Ywhere [22Xk_i[122X is an arbitrary integer, if [22Xg_i[122X has infinite order, and [22X0 le k_i <
  d_i[122X,  if  [22Xg_i[122X has finite order [22Xd_i[122X. Then one selects a basis for the crossed
  product by taking[133X
  
  
        [33X[1;6Y[24X[33X[0;0Yu_g = u_{g_1^{k_1} \cdots g_n^{k_n}} = u_1^{k_1} \cdots u_n^{k_n}.[133X [124X[133X
  
  
  [33X[0;0Y*  For  each  [22Xi=1,dots, n[122X, let [22Xa_i[122X be the automorphism of [22XR[122X given by [22Xr^a_i =
  u_i^-1 r u_i[122X.[133X
  
  [33X[0;0Y* For each [22X1 le i < j le n[122X, let [22Xt_i,j = u_j^-1 u_i^-1 u_j u_i ∈ R[122X.[133X
  
  [33X[0;0Y* If [22Xg_i[122X has finite order [22Xd_i[122X, let [22Xb_i=u_i^d_i ∈ R[122X.[133X
  
  [33X[0;0YThen[133X
  
  
        [33X[1;6Y[24X[33X[0;0YR*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) ],[133X [124X[133X
  
  
  [33X[0;0Ywhere the last relation vanishes if [22Xg_i[122X has infinite order.[133X
  
  [33X[0;0YTherefore [22XR*G[122X is completely determined by the following data:[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y[ R , [ d_i , a_i , b_i ]_{i=1}^n, [ t_{i,j} ]_{1 \le i < j \le n} ].[133X
        [124X[133X
  
  
  
  [1X9.9 [33X[0;0YClassical crossed products[133X[101X
  
  [33X[0;0YA  [13Xclassical  crossed  product[113X is a crossed product [22XL*_a^t G[122X, where [22XL/K[122X is a
  finite  Galois extension, [22XG=Gal(L/K)[122X is the Galois group of [22XL/K[122X and [22Xa[122X is the
  natural  action  of  [22XG[122X  on  [22XL[122X. Then [22Xt[122X is a [22X2[122X-cocycle and the [13Xcrossed product[113X
  ([14X9.6[114X)  [22XL*_a^t  G[122X is denoted by [22X(L/K,t)[122X. The crossed product [22X(L/K,t)[122X is known
  to be a central simple [22XK[122X-algebra [Rei03].[133X
  
  
  [1X9.10 [33X[0;0YCyclic Algebras[133X[101X
  
  [33X[0;0YA [13Xcyclic algebra[113X is a [13Xclassical crossed product[113X ([14X9.9[114X) [22X(L/K,t)[122X where [22XL/K[122X is a
  finite cyclic field extension. The cyclic algebras have a very simple form.[133X
  
  [33X[0;0YAssume  that  [22XGal(L/K)[122X  is  generated by [22Xg[122X and has order [22Xd[122X. Let [22Xu=u_g[122X be the
  basis  unit  ([14X9.6[114X)  of  the  crossed product corresponding to [22Xg[122X and take the
  remaining  basis  units for the crossed product by setting [22Xu_g^i = u^i[122X, ([22Xi =
  0, 1, dots, d-1[122X). Then [22Xa = u^n ∈ K[122X. The cyclic algebra is usually denoted by
  [22X(L/K,a)[122X and one has the following description of [22X(L/K,t)[122X[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y(L/K,t) = (L/K,a) = L[u| r u = u r^g, u^d = a ].[133X [124X[133X
  
  
  
  [1X9.11 [33X[0;0YCyclotomic algebras[133X[101X
  
  [33X[0;0YA [13Xcyclotomic algebra[113X over [22XF[122X is a [13Xclassical crossed product[113X ([14X9.9[114X) [22X(F(ξ)/F,t)[122X,
  where  [22XF[122X is a field, [22Xξ[122X is a root of unity in an extension of [22XF[122X and [22Xt(g,h)[122X is
  a root of unity for every [22Xg[122X and [22Xh[122X in [22XGal(F(ξ)/F)[122X.[133X
  
  [33X[0;0YThe  [13XBrauer-Witt  Theorem[113X  [Yam74]  asserts  that every [13XWedderburn component[113X
  ([14X9.3[114X)  of  a group algebra is [13XBrauer equivalent[113X ([14X9.5[114X) (over its centre) to a
  cyclotomic algebra.[133X
  
  
  [1X9.12 [33X[0;0YNumerical description of cyclotomic algebras[133X[101X
  
  [33X[0;0YLet  [22XA=(F(ξ)/F,t)[122X be a [13Xcyclotomic algebra[113X ([14X9.11[114X), where [22Xξ=ξ_k[122X is a [22Xk[122X-th root
  of  unity.  Then the Galois group [22XG=Gal(F(ξ)/F)[122X is abelian and therefore one
  can  obtain  a simplified form for the description of cyclotomic algebras as
  for any [13Xabelian crossed product[113X ([14X9.8[114X).[133X
  
  [33X[0;0YThen  the [22Xn × n[122X matrix algebra [22XM_n(A)[122X can be described numerically in one of
  the following forms:[133X
  
  [33X[0;0Y*  If  [22XF(ξ)=F[122X,  (i.e.  [22XG=1[122X)  then  [22XA=M_n(F)[122X and thus the only data needed to
  describe [22XA[122X are the matrix size [22Xn[122X and the field [22XF[122X:[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y[n,F][133X [124X[133X
  
  
  [33X[0;0Y*  If [22XG[122X is cyclic (but not trivial) of order [22Xd[122X then [22XA[122X is a cyclic cyclotomic
  algebra[133X
  
  
        [33X[1;6Y[24X[33X[0;0YA = F(\xi) [ u | \xi u = u \xi^\alpha, u^d = \xi^\beta ][133X [124X[133X
  
  
  [33X[0;0Yand so [22XM_n(A)[122X can be described with the following data[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y[n,F,k,[d,\alpha,\beta]],[133X [124X[133X
  
  
  [33X[0;0Ywhere the integers [22Xk[122X, [22Xd[122X, [22Xα[122X and [22Xβ[122X satisfy the following conditions:[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y\alpha^d \equiv 1 \; mod \; k, \quad \beta(\alpha-1) \equiv 0 \; mod
        \; k.[133X [124X[133X
  
  
  [33X[0;0Y*  If  [22XG[122X  is  abelian  but  not cyclic then [22XM_n(A)[122X can be described with the
  following data (see [14X9.8[114X):[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y[n,F,k,[d_i,\alpha_i,\beta_i]_{i=1}^m, [\gamma_{i,j}]_{1\le i < j \le
        m} ][133X [124X[133X
  
  
  [33X[0;0Yrepresenting the [22Xn × n[122X matrix ring over the following algebra:[133X
  
  
        [33X[1;6Y[24X[33X[0;0YA = 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 ][133X [124X[133X
  
  
  [33X[0;0Ywhere[133X
  
  [33X[0;0Y* [22X{g_1,...,g_m}[122X is an independent set of generators of [22XG[122X,[133X
  
  [33X[0;0Y* [22Xd_i[122X is the order of [22Xg_i[122X,[133X
  
  [33X[0;0Y* [22Xα_i[122X, [22Xβ_i[122X and [22Xγ_rs[122X are integers, and[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y\xi^{g_i} = \xi^{\alpha_i}.[133X [124X[133X
  
  
  
  [1X9.13 [33X[0;0YIdempotents given by subgroups[133X[101X
  
  [33X[0;0YLet  [22XG[122X  be a finite group and [22XF[122X a field whose characteristic does not divide
  the order of [22XG[122X. If [22XH[122X is a subgroup of [22XG[122X then set[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y\widehat{H} = |H|^{-1}\sum_{x \in H} x.[133X [124X[133X
  
  
  [33X[0;0YThe  element  [22XwidehatH[122X  is an idempotent of [22XFG[122X which is central in [22XFG[122X if and
  only if [22XH[122X is normal in [22XG[122X.[133X
  
  [33X[0;0YIf [22XH[122X is a proper normal subgroup of a subgroup [22XK[122X of [22XG[122X then set[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y\varepsilon(K,H) = \prod_{L} (\widehat{N}-\widehat{L})[133X [124X[133X
  
  
  [33X[0;0Ywhere [22XL[122X runs on the normal subgroups of [22XK[122X which are minimal among the normal
  subgroups  of  [22XK[122X  containing [22XN[122X properly. By convention, [22Xε(K,K)=widehatK[122X. The
  element [22Xε(K,H)[122X is an idempotent of [22XFG[122X.[133X
  
  [33X[0;0YIf  [22XH[122X  and  [22XK[122X  are  subgroups  of [22XG[122X such that [22XH[122X is normal in [22XK[122X then [22Xe(G,K,H)[122X
  denotes  the  sum  of  all  different  [22XG[122X-conjugates  of  [22Xε(K,H)[122X. The element
  [22Xe(G,K,H)[122X  is  central  in  [22XFG[122X. In general it is not an idempotent but if the
  different  conjugates  of  [22Xε(K,H)[122X  are orthogonal then [22Xe(G,K,H)[122X is a central
  idempotent of [22XFG[122X.[133X
  
  [33X[0;0YIf  [22X(K,H)[122X  is  a  Shoda  Pair  ([14X9.14[114X) of [22XG[122X then there is a non-zero rational
  number [22Xa[122X such that [22Xae(G,K,H))[122X is a [13Xprimitive central idempotent[113X ([14X9.4[114X) of the
  rational group algebra [22Xℚ G[122X. If [22X(K,H)[122X is a strong Shoda pair ([14X9.15[114X) of [22XG[122X then
  [22Xe(G,K,H)[122X is a primitive central idempotent of [22Xℚ G[122X.[133X
  
  [33X[0;0YAssume now that [22XF[122X is a finite field of order [22Xq[122X, [22X(K,H)[122X is a strong Shoda pair
  of  [22XG[122X and [22XC[122X is a cyclotomic class of [22XK/H[122X containing a generator of [22XK/H[122X. Then
  [22Xe_C(G,K,H)[122X is a primitive central idempotent of [22XFG[122X (see [14X9.17[114X).[133X
  
  
  [1X9.14 [33X[0;0YShoda pairs of a group[133X[101X
  
  [33X[0;0YLet [22XG[122X be a finite group. A [13XShoda pair[113X of [22XG[122X is a pair [22X(K,H)[122X of subgroups of [22XG[122X
  for  which  there  is  a linear character [22Xχ[122X of [22XK[122X with kernel [22XH[122X such that the
  induced character [22Xχ^G[122X in [22XG[122X is irreducible. By [Sho33] or [ORS04], [22X(K,H)[122X is a
  Shoda pair if and only if the following conditions hold:[133X
  
  [33X[0;0Y* [22XH[122X is normal in [22XK[122X,[133X
  
  [33X[0;0Y* [22XK/H[122X is cyclic and[133X
  
  [33X[0;0Y* if [22XK^g ∩ K ⊆ H[122X for some [22Xg ∈ G[122X then [22Xg ∈ K[122X.[133X
  
  [33X[0;0YIf  [22X(K,H)[122X is a Shoda pair and [22Xχ[122X is a linear character of [22XKle G[122X with kernel [22XH[122X
  then  the  [13Xprimitive  central  idempotent[113X  ([14X9.4[114X)  of  [22Xℚ  G[122X associated to the
  irreducible character [22Xχ^G[122X is of the form [22Xe=e_ℚ (χ^G)=a e(G,K,H)[122X for some [22Xa ∈
  ℚ[122X  [ORS04]  (see  [14X9.13[114X  for the definition of [22Xe(G,K,H)[122X). In that case we say
  that  [22Xe[122X is the [13Xprimitive central idempotent realized by the Shoda pair[113X [22X(K,H)[122X
  of [22XG[122X.[133X
  
  [33X[0;0YA group [22XG[122X is monomial, that is every irreducible character of [22XG[122X is monomial,
  if  and only if every primitive central idempotent of [22Xℚ G[122X is realizable by a
  Shoda pair of [22XG[122X.[133X
  
  
  [1X9.15 [33X[0;0YStrong Shoda pairs of a group[133X[101X
  
  [33X[0;0YA  [13Xstrong  Shoda  pair[113X of [22XG[122X is a pair [22X(K,H)[122X of subgroups of [22XG[122X satisfying the
  following conditions:[133X
  
  [33X[0;0Y* [22XH[122X is normal in [22XK[122X and [22XK[122X is normal in the normalizer [22XN[122X of [22XH[122X in [22XG[122X,[133X
  
  [33X[0;0Y* [22XK/H[122X is cyclic and a maximal abelian subgroup of [22XN/H[122X and[133X
  
  [33X[0;0Y*  for  every  [22Xg  ∈ G∖ N[122X , [22Xε(K,H)ε(K,H)^g=0[122X. (See [14X9.13[114X for the definition of
  [22Xε(K,H)[122X).[133X
  
  [33X[0;0YLet  [22X(K,H)[122X be a strong Shoda pair of [22XG[122X. Then [22X(K,H)[122X is a Shoda pair ([14X9.14[114X) of
  [22XG[122X.  Thus  there  is  a  linear  character [22Xθ[122X of [22XK[122X with kernel [22XH[122X such that the
  induced  character  [22Xχ=χ(G,K,H)=θ^G[122X  is  irreducible.  Moreover the [13Xprimitive
  central  idempotent[113X  ([14X9.4[114X) [22Xe_ℚ (χ)[122X of [22Xℚ G[122X realized by [22X(K,H)[122X is [22Xe(G,K,H)[122X, see
  [ORS04].[133X
  
  [33X[0;0YTwo  [13Xstrong  Shoda  pairs[113X ([14X9.15[114X) [22X(K_1,H_1)[122X and [22X(K_2,H_2)[122X of [22XG[122X are said to be
  [13Xequivalent[113X  if  the  characters  [22Xχ(G,K_1,H_1)[122X  and  [22Xχ(G,K_2,H_2)[122X  are Galois
  conjugate, or equivalently if [22Xe(G,K_1,H_1)=e(G,K_2,H_2)[122X.[133X
  
  [33X[0;0YThe  advantage  of  strong  Shoda  pairs  over  Shoda  pairs is that one can
  describe  the  simple  algebra  [22XFGe_F(χ)[122X as a matrix algebra of a [13Xcyclotomic
  algebra[113X ([14X9.11[114X, see [ORS04] for [22XF=ℚ[122X and [Olt07] for the general case).[133X
  
  [33X[0;0YMore precisely, [22Xℚ Ge(G,K,H)[122X is isomorphic to [22XM_n(ℚ (ξ)*_a^t N/K)[122X, where [22Xξ[122X is
  a  [22X[K:H][122X-th  root  of  unity,  [22XN[122X  is the normalizer of [22XH[122X in [22XG[122X, [22Xn=[G:N][122X and [22Xℚ
  (ξ)*_a^t  N/K[122X  is  a  [13Xcrossed product[113X (see [14X9.6[114X) with action [22Xa[122X and twisting [22Xt[122X
  given as follows:[133X
  
  [33X[0;0YLet  [22Xx[122X be a fixed generator of [22XK/H[122X and [22Xφ : N/K → N/H[122X a fixed left inverse of
  the canonical projection [22XN/H→ N/K[122X. Then[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y\xi^{a(r)} = \xi^i, \mbox{ if } x^{\varphi(r)}= x^i[133X [124X[133X
  
  
  [33X[0;0Yand[133X
  
  
        [33X[1;6Y[24X[33X[0;0Yt(r,s) = \xi^j, \mbox{ if } \varphi(rs)^{-1} \varphi(r)\varphi(s) =
        x^j,[133X [124X[133X
  
  
  [33X[0;0Yfor  [22Xr,s ∈ N/K[122X and integers [22Xi[122X and [22Xj[122X, see [ORS04]. Notice that the cocycle is
  the one given by the natural extension[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y1 \rightarrow K/H \rightarrow N/H \rightarrow N/K \rightarrow 1[133X [124X[133X
  
  
  [33X[0;0Ywhere  [22XK/H[122X  is  identified  with  the  multiplicative  group generated by [22Xξ[122X.
  Furthermore  the  centre  of  the  algebra  is [22Xℚ (χ)[122X, the field of character
  values over [22Xℚ[122X, and [22XN/K[122X is isomorphic to [22XGal(ℚ (ξ)/ℚ (χ))[122X.[133X
  
  [33X[0;0YIf  the rational field is changed to an arbitrary ring [22XF[122X of characteristic [22X0[122X
  then  the  Wedderburn  component [22XA_F(χ)[122X, where [22Xχ = χ(G,K,H)[122X is isomorphic to
  [22XF(χ)⊗_ℚ  (χ)A_ℚ  (χ)[122X.  Using  the  description  given  above  of [22XA_ℚ (χ)=ℚ G
  e(G,K,H)[122X  one  can  easily describe [22XA_F(χ)[122X as [22XM_nd(F(ξ)/F(χ),t')[122X, where [22Xd=[ℚ
  (ξ):  ℚ(χ)]/[F(ξ):F(χ)][122X  and [22Xt'[122X is the restriction to [22XGal(F(ξ)/F(χ))[122X of [22Xt[122X (a
  cocycle of [22XN/K = Gal(ℚ (ξ)/ℚ (χ))[122X).[133X
  
  
  [1X9.16 [33X[0;0YStrongly monomial characters and strongly monomial groups[133X[101X
  
  [33X[0;0YLet [22XG[122X be a finite group an [22Xχ[122X an irreducible character of [22XG[122X.[133X
  
  [33X[0;0YOne  says that [22Xχ[122X is [13Xstrongly monomial[113X if there is a [13Xstrong Shoda pair[113X ([14X9.15[114X)
  [22X(K,H)[122X of [22XG[122X and a linear character [22Xθ[122X of [22XK[122X of [22XG[122X with kernel [22XH[122X such that [22Xχ=θ^G[122X.[133X
  
  [33X[0;0YThe  group  [22XG[122X  is  [13Xstrongly  monomial[113X if every irreducible character of [22XG[122X is
  strongly monomial.[133X
  
  [33X[0;0YStrong  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 [14X9.17[114X). Finally, Olteanu [Olt07] shows how
  to  compute  the  [13XWedderburn  decomposition[113X ([14X9.3[114X) of an arbitrary semisimple
  group  ring  by  making use of not only the strong Shoda pairs of [22XG[122X but also
  the strong Shoda pairs of the subgroups of [22XG[122X.[133X
  
  [33X[0;0Y.[133X
  
  
  [1X9.17 [33X[0;0YCyclotomic Classes and Strong Shoda Pairs[133X[101X
  
  [33X[0;0YLet  [22XG[122X  be  a  finite  group and [22XF[122X a finite field of order [22Xq[122X, coprime to the
  order of [22XG[122X.[133X
  
  [33X[0;0YGiven  a positive integer [22Xn[122X, coprime to [22Xq[122X, the [22Xq[122X-[13Xcyclotomic classes[113X modulo [22Xn[122X
  are the set of residue classes module [22Xn[122X of the form[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y\{i,iq,iq^2,iq^3, \dots \}[133X [124X[133X
  
  
  [33X[0;0YThe  [22Xq[122X-cyclotomic  classes  module  [22Xn[122X form a partition of the set of residue
  classes module [22Xn[122X.[133X
  
  [33X[0;0YA  [13Xgenerating  cyclotomic class [113X module [22Xn[122X is a cyclotomic class containing a
  generator of the additive group of residue classes module [22Xn[122X, or equivalently
  formed by integers coprime to [22Xn[122X.[133X
  
  [33X[0;0YLet  [22X(K,H)[122X  be  a  strong  Shoda  pair  ([14X9.15[114X)  of  [22XG[122X and set [22Xn=[K:H][122X. Fix a
  primitive  [22Xn[122X-th root of unity [22Xξ[122X in some extension of [22XF[122X and an element [22Xg[122X of [22XK[122X
  such that [22XgH[122X is a generator of [22XK/H[122X. Let [22XC[122X be a generating [22Xq[122X-cyclotomic class
  modulo [22Xn[122X. Then set[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y\varepsilon_C(K,H) = [K:H]^{-1} \widehat{H} \sum_{i=0}^{n-1}
        tr(\xi^{-ci})g^i,[133X [124X[133X
  
  
  [33X[0;0Ywhere  [22Xc[122X  is  an arbitrary element of [22XC[122X and [22Xtr[122X is the trace map of the field
  extension  [22XF(ξ)/F[122X.  Then [22Xε_C(K,H)[122X does not depend on the choice of [22Xc ∈ C[122X and
  is a [13Xprimitive central idempotent[113X ([14X9.4[114X) of [22XFK[122X.[133X
  
  [33X[0;0YFinally,  let  [22Xe_C(G,K,H)[122X  denote  the  sum of the different [22XG[122X-conjugates of
  [22Xε_C(K,H)[122X.  Then  [22Xe_C(G,K,H)[122X  is  a  [13Xprimitive central idempotent[113X ([14X9.4[114X) of [22XFG[122X
  [BR07].  We say that [22Xe_C(G,K,H)[122X is the primitive central idempotent realized
  by the strong Shoda pair [22X(K,H)[122X of the group [22XG[122X and the cyclotomic class [22XC[122X.[133X
  
  [33X[0;0YIf  [22XG[122X is [13Xstrongly monomial[113X ([14X9.16[114X) then every primitive central idempotent of
  [22XFG[122X  is  realizable by some [13Xstrong Shoda pair[113X ([14X9.15[114X) of [22XG[122X and some cyclotomic
  class  [22XC[122X  [BR07].  As  in  the zero characteristic case, this explain how to
  compute  the  [13XWedderburn  decomposition[113X  ([14X9.3[114X) of [22XFG[122X 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.[133X
  
  [33X[0;0Y.[133X
  
  
  [1X9.18 [33X[0;0YTheory for Local Schur Index and Division Algebra Part Calculations[133X[101X
  
  [33X[0;0Y(By Allen Herman, May 2013. Updated October 2014.)[133X
  
  [33X[0;0YThe   division   algebra   parts   of  simple  algebras  in  the  Wedderburn
  Decomposition  of the group algebra of a finite group over an abelian number
  field  [22XF[122X  correspond  to  elements  of the Schur Subgroup [22XS(F)[122X of the Brauer
  group  of  [22XF[122X.  Like  all  classes in the Brauer group of an algebraic number
  field  [22XF[122X,  the  division  algebra part of a representative of a given Brauer
  class  is  determined up to [22XF[122X-algebra isomorphism by its list of local Hasse
  invariants  at all primes (i.e. places) of [22XF[122X. The local invariant at a prime
  [22XP[122X of [22XF[122X is a lowest terms fraction [22Xr/m_P[122X whose denominator is the local Schur
  index  [22Xm_P[122X  of the simple algebra at the prime [22Xq[122X (see [Rei03]). For division
  algebras  whose Brauer class lies in the Schur Subgroup of an abelian number
  field  [22XF[122X,  the  local  indices  at  any  of the primes [22XP[122X lying over the same
  rational  prime  [22Xp[122X  are  equal  to  the  same  positive integer [22Xm_p[122X, and the
  numerator   of  the  local  invariants  among  these  primes  are  uniformly
  distributed among the integers [22Xr[122X coprime to [22Xm_p[122X [BS72].[133X
  
  [33X[0;0YThe 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 [22XF[122X is the least
  common  multiple [22Xm[122X of these local indices, and the dimension of the division
  algebra  part  of  the simple algebra over [22XF[122X is [22Xm^2[122X. 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.[133X
  
  [33X[0;0YWedderga'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 [22Xp[122X-local index for an ordinary irreducible character [22Xχ[122X of a
  [22Xp[122X-solvable  group using the values of an irreducible Brauer character in the
  same [22Xp[122X-block in cases where the [22Xp[122X-defect group of [22Xχ[122X 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  [22X(a,b)[122X  over [22XQ[122X at the infinite prime will be [22X2[122X if both [22Xa[122X
  and  [22Xb[122X  are  negative,  and  otherwise  [22X1[122X.  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 [22XQ[122X where the local index is [22X2[122X, it determines the local index
  at  the  prime  [22X2[122X.  For generalized quaternion algebras over number fields [22XF[122X
  other than [22XQ[122X, 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.[133X
  
  [33X[0;0YThere  are  three shortcut functions used to compute local indices of cyclic
  cyclotomic  algebras,  which  wedderga's -Info functions produce in the form
  [22X[r,F,n,[a,b,c]][122X. 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, [22XF[122X must be a real subfield, [22Xn[122X must
  be  strictly  greater than [22X2[122X, and [22XE(n)^c[122X (which has to be a root of unity in
  [22XF[122X)  must  be [22X-1[122X. 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  [22XE_p/F_p[122X is a Galois extension of [22Xp[122X-local fields with
  ramification index [22Xe[122X, and [22Xz[122X is a root of unity with order prime to [22Xp[122X, then [22Xz[122X
  is  a norm in [22XE_p/F_p[122X if and only if it is the [22Xe[122X-th power of a root of unity
  in  [22XF[122X.  ([Jan75],  pg. 535). It follows that in order to calculate the local
  index  at  [22Xp[122X  of  a  cyclic  cyclotomic  algebra  [22X[r,F,n,[a,b,c]][122X,  we first
  determine  the splitting degree, residue degree, and ramification index [22Xe[122X of
  the  extension  [22XF(ζ_n)/F[122X  at  [22Xp[122X.  Comparing  the  behaviour  of  the  Galois
  automorphism  [22Xσ_b[122X to the behaviour of the Frobenius automorphism at [22Xp[122X allows
  us  to determine the order of the largest root of unity [22Xz[122X with order coprime
  to [22Xp[122X in the [22Xp[122X-completion [22XF_p[122X. The local index [22Xm_p[122X is then the least power of
  [22XE(n)^c[122X that lies in the group generated by [22Xz^e[122X.[133X
  
  [33X[0;0YCalculation  of  the  local  index at the prime [22X2[122X makes use of the following
  consequence   of   ([Jan75],   Theorem   5):  A  cyclic  cyclotomic  algebra
  [22X[r,F_2,n,[a,b,c]][122X  over  a  [22X2[122X-local  field  [22XF_2[122X  that  is  a  subfield  of a
  cyclotomic  extension  of  the rational [22X2[122X-local field [22XQ_2[122X has Schur index at
  most [22X2[122X. It has Schur index [22X2[122X if and only if [22X4[122X divides [22Xn[122X, [22XF_2(ζ_4)[122X is totally
  ramified  of  degree  2, the Galois automorphism [22Xσ_b[122X of [22XF_2(ζ_n)/F_2[122X inverts
  all  [22X2[122X-power  roots  of unity in [22XF_2(ζ_n)[122X, the order of [22XE(n)^c[122X is 2 times an
  odd   number,  and  [22X(F_2:Q_2)[122X  is  odd.  The  same  approach  to  cyclotomic
  reciprocity  makes  it  possible  to  check  all  of these conditions in the
  [22X2[122X-local situation.[133X
  
  [33X[0;0YThe  wedderga  function  that  computes  the  [22Xp[122X-local  index  of an ordinary
  irreducible character [22Xχ[122X of a finite non-nilpotent cyclic-by- abelian group [22XG[122X
  is  based  directly on a theorem of Benard [Ben76] that applies whenever the
  [22Xp[122X-defect  group of [22Xχ[122X is cyclic. We have to restrict our application of it to
  groups whose orders are small because the [5XGAP[105X 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 [22Xχ[122X. 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  [5XGAP[105X'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 [5XGAP[105X. The current algorithm we use is
  based on an approach suggested by Ladisch.[133X
  
  
  [1X9.19  [33X[0;0YObtaining Algebras with structure constants as terms of the Wedderburn[101X
  [1Xdecomposition[133X[101X
  
  [33X[0;0YSome  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 [5XGAP[105X are designed
  for  algebras  with  structure constants. We have provided such an algorithm
  that   converts  the  output  of  [2XWedderburnDecompositionInfo[102X  ([14X2.1-2[114X)  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 [5XWedderga[105X operations, then
  uses               [2XIrreducibleRepresentationsDixon[102X               ([14XReference:
  IrreducibleRepresentationsDixon[114X)  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    [2XIsomorphismSCAlgebra[102X    ([14XReference:
  IsomorphismSCAlgebra (w.r.t. a given basis)[114X) to this algebra.[133X
  
  
  [1X9.20 [33X[0;0YA complete set of orthogonal primitive idempotents[133X[101X
  
  [33X[0;0YWhen  [22XR[122X  is  a  semisimple ring, then every left ideal [22XL[122X of [22XR[122X is of the form
  [22XL=Re[122X,  where  [22Xe[122X is an idempotent of [22XR[122X. Therefore, we can use the idempotents
  to  characterize  the  decompositions of semisimple rings as a direct sum of
  minimal  left ideals. In particular, let [22XR=⊕_i=1^t L_i[122X be a decomposition of
  a semisimple ring as a direct sum of minimal left ideals. Then, there exists
  a  family  [22X{e_1,dots,e_t}[122X  of  elements  of  [22XR[122X  such that: each [22Xe_i≠ 0[122X is an
  idempotent  element, if [22Xi≠ j[122X, then [22Xe_ie_j=0[122X, [22X1=e_1+⋯+e_t[122X and each [22Xe_i[122X cannot
  be  written  as  [22Xe_i=e_i'+e_i''[122X,  where [22Xe_i',e_i''[122X are idempotents such that
  [22Xe_i',e_i''≠  0[122X  and [22Xe_i'e_i''=0[122X, [22X1≤ i≤[122X. Conversely, if there exists a family
  of  idempotents  [22X{e_1,dots,e_t}[122X satisfying the previous conditions, then the
  left  ideals  [22XL_i=Re_i[122X  are  minimal  and  [22XR=⊕_i=1^t  L_i[122X.  Such  a  set  of
  idempotents  is called a [13Xcomplete set of orthogonal primitive idempotents[113X of
  the ring [22XR[122X. Such a set is not uniquely determined.[133X
  
  [33X[0;0YLet  [22XF[122X  be  a  finite  field and [22XG[122X a finite nilpotent group such that [22XF G[122X is
  semisimple.  Let  [22X(H,K)[122X be a strong Shoda pair of [22XG[122X, [22XC∈mathcalC(H/K)[122X and set
  [22Xe_C=e_C(G,H,K)[122X,  [22Xε_C=ε_C(H,K)[122X,  [22XH/K=⟨overlinea⟩[122X,  [22XE=E_G(H/K)[122X.  Let [22XE_2/K[122X and
  [22XH_2/K=⟨overlinea_2⟩[122X  (respectively [22XE_2'/K[122X and [22XH_2'/K=⟨overlinea_2'}⟩[122X) denote
  the  2-parts  (respectively  2'-parts)  of  [22XE/K[122X  and  [22XH/K[122X respectively. Then
  [22X⟨overlinea_2'}⟩[122X has a cyclic complement [22X⟨overlineb_2'}⟩[122X in [22XE_2'/K[122X. Using the
  description   of  the  primitive  central  idempotents  and  the  Wedderburn
  components  of  a semisimple finite group algebra [22XF G[122X ([14X9.17[114X), a complete set
  of  orthogonal  primitive idempotents of [22XF Ge_C[122X is described (see [OG11]) as
  the  set  of  conjugates  of  [22Xβ_e_C=widetildeb_2'}β_2ε_C[122X  by the elements of
  [22XT_e_C=T_2'T_2T_E[122X,  where  [22XT_2'={1,a_2',a_2'^2,dots,a_2'^[E_2':H_2']-1}[122X,  [22XT_E[122X
  denotes a right transversal of [22XE[122X in [22XG[122X and [22Xβ_2[122X and [22XT_2[122X are given according to
  the cases below.[133X
  
  [31X1[131X   [33X[0;6YIf  [22XH_2/K[122X  has  a  complement  [22XM_2/K[122X  in  [22XE_2/K[122X then [22Xβ_2=widetildeM_2[122X.
        Moreover,  if  [22XM_2/K[122X  is  cyclic, then there exists [22Xb_2∈ E_2[122X such that
        [22XE_2/K[122X is given by the following presentation[133X
  
  
              [33X[1;12Y[24X [33X[0;6Y\langle \overline{a_2},\overline{b_2}\mid
              \overline{a_2}\hspace{1pt}^{2^n}=\overline{b_2}\hspace{1pt}^{2^k}=1,
              \overline{a_2}\hspace{1pt}^{\overline{b_2}}=\overline{a_2}\hspace{1pt}^r
              \rangle,[133X [124X[133X
  
  
        [33X[0;6Yand  if  [22XM_2/K[122X  is not cyclic, then there exist [22Xb_2,c_2∈ E_2[122X such that
        [22XE_2/K[122X is given by the following presentation[133X
  
  
              [33X[1;12Y[24X [33X[0;6Y\langle \overline{a_2},\overline{b_2},\overline{c_2}\mid
              \overline{a_2}\hspace{1pt}^{2^n}=
              \overline{b_2}\hspace{1pt}^{2^k}=\overline{c_2}\hspace{1pt}^2=1,
              \overline{a_2}\hspace{1pt}^{\overline{b_2}}=\overline{a_2}\hspace{1pt}^r,
              \overline{a_2}\hspace{1pt}^{\overline{c_2}}=\overline{a_2}\hspace{1pt}^{-1},
              [\overline{b_2},\overline{c_2}]=1 \rangle,[133X [124X[133X
  
  
        [33X[0;6Ywith  [22Xr  ≡  1  mod  4[122X  (or equivalently [22Xoverlinea_2hspace1pt^2^n-2}[122X is
        central in [22XE_2/K[122X). Then[133X
  
        [31X1[131X   [33X[0;12Y[22XT_2={1,a_2,a_2^2,dots,               a_2^2^k-1}[122X,              if
              [22Xoverlinea_2hspace1pt^2^n-2}[122X  is  central  in [22XE_2/K[122X (unless [22Xn≤ 1[122X)
              and [22XM_2/K[122X is cyclic; and[133X
  
        [31X2[131X   [33X[0;12Y[22XT_2={1,a_2,a_2^2,dots,a_2^d/2-1,a_2^2^n-2},a_2^2^n-2+1,dots,a_2^2^n-2+d/2-1}[122X,
              where [22Xd=[E_2:H_2][122X, otherwise.[133X
  
  [31X2[131X   [33X[0;6YIf  [22XH_2/K[122X  has  no  complement in [22XE_2/K[122X, then there exist [22Xb_2,c_2∈ E_2[122X
        such that [22XE_2/K[122X is given by the following presentation[133X
  
  
              [33X[1;12Y[24X [33X[0;6Y\langle \overline{a_2},\overline{b_2},\overline{c_2}\mid
              \overline{a_2}\hspace{1pt}^{2^n}=
              \overline{b_2}\hspace{1pt}^{2^k}=1,
              \overline{c_2}\hspace{1pt}^2=\overline{a_2}\hspace{1pt}^{2^{n-1}},
              \overline{a_2}\hspace{1pt}^{\overline{b_2}}=\overline{a_2}\hspace{1pt}^r,\\
              \overline{a_2}\hspace{1pt}^{\overline{c_2}}=\overline{a_2}\hspace{1pt}^{-1},[\overline{b_2},\overline{c_2}]=1
              \rangle,[133X [124X[133X
  
  
        [33X[0;6Ywith       [22Xr≡       1       mod       4[122X.       In      this      case,
        [22Xβ_2=widetildeb_2frac1+xa_2^2^n-2}+ya_2^2^n-2}c_22[122X and[133X
  
  
              [33X[1;12Y[24X [33X[0;6YT_2=\{1,a_2,a_2^2,\dots,
              a_2^{2^k-1},c_2,c_2a_2,c_2a_2^2,\dots,c_2a_2^{2^k-1}\},[133X [124X[133X
  
  
        [33X[0;6Ywith [22Xx,y∈ F[122X, satisfying [22Xx^2+y^2=-1[122X and [22Xy≠ 0[122X.[133X
  
  [33X[0;0YWhen  [22XG[122X is not nilpotent, we can still use the following description in some
  specific cases. Let [22XG[122X be a finite group and [22XF[122X a finite field of order [22Xs[122X such
  that  [22Xs[122X  is coprime to the order of [22XG[122X. Let [22X(H,K)[122X be a strong Shoda pair of [22XG[122X
  such that [22Xτ(gH,g'H)=1[122X for all [22Xg,g'∈ E=E_G(H/K)[122X, and let [22XC∈mathcalC(H/K)[122X. Let
  [22Xε=ε_C(H,K)[122X  and  [22Xe=e_C(G,H,K)[122X  ([14X9.17[114X).  Let  [22Xw[122X be a normal element of [22XF_s^o/
  F_s^o/[E:H]}[122X  (with  [22Xo[122X the multiplicative order of [22Xs[122X modulo [22X[H:K][122X) and [22XB[122X the
  normal basis determined by [22Xw[122X. Let [22Xψ[122X be the isomorphism between [22XF E ε[122X and the
  matrix  algebra [22XM_[E:H]( F_s^o/[E:H]})[122X with respect to the basis [22XB[122X as stated
  in  Corollary  29.8  in  [Rei03].  Let  [22XP,A∈  M_[E:H](  F_s^o/[E:H]})[122X be the
  matrices[133X
  
  
        [33X[1;6Y[24X[33X[0;0YP= \left( \begin{array}{rrrrrr} 1 & 1 & 1 & \cdots & 1 & 1\\ 1 & -1 &
        0 & \cdots & 0 & 0\\ 1 & 0 & -1 & \cdots & 0 & 0\\ \vdots & \vdots &
        \vdots & \ddots & \vdots & \vdots\\ 1 & 0 & 0 & \cdots & -1 & 0\\ 1 &
        0 & 0 & \cdots & 0 & -1\\ \end{array} \right) \quad {\rm and } \quad
        A= \left( \begin{array}{ccccc} 0 & 0 & \cdots & 0 & 1\\ 1 & 0 & \cdots
        & 0 & 0\\ 0 & 1 & \cdots & 0 & 0\\ \vdots & \vdots & \ddots & \vdots &
        \vdots\\ 0 & 0 & \cdots & 0 & 0\\ 0 & 0 & \cdots & 1 & 0\\ \end{array}
        \right).[133X [124X[133X
  
  
  [33X[0;0YThen[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y\{x\widehat{T_1}\varepsilon x^{-1} \mid x\in T_2\langle{x_e}\rangle\}[133X
        [124X[133X
  
  
  [33X[0;0Yis  a  complete  set  of  orthogonal  primitive  idempotents  of [22XF G e[122X where
  [22Xx_e=ψ^-1(PAP^-1)[122X,  [22XT_1[122X  is  a  transversal  of  [22XH[122X  in  [22XE[122X  and [22XT_2[122X is a right
  transversal  of  [22XE[122X  in  [22XG[122X  ([OGnt]).  By  [22XwidehatT_1[122X  we  denote the element
  [22Xfrac1|T_1|∑_t∈ T_1t[122X in [22XF G[122X.[133X
  
  
  [1X9.21 [33X[0;0YApplications to coding theory[133X[101X
  
  [33X[0;0YA [13Xlinear code[113X of length [22Xn[122X and rank [22Xk[122X is a linear subspace [22XC[122X with dimension [22Xk[122X
  of  the  vector  space  [22XF_q^n[122X.  The  standard  basis  of [22XF_q^n[122X is denoted by
  [22XE={e_1,...,e_n}[122X.  The  vectors in [22XC[122X are called codewords, the size of a code
  is  the  number  of  codewords and equals [22Xq^k[122X. The distance of a code is the
  minimum  distance between distinct codewords, i.e. the number of elements in
  which they differ.[133X
  
  [33X[0;0YFor  any  group  [22XG[122X,  we  denote  by  [22XF_q  G[122X  the  group  algebra over [22XG[122X with
  coefficients  in  [22XF_q[122X.  If  [22XG[122X is a group of order [22Xn[122X and [22XC⊆ F_q^n[122X is a linear
  code,  then  we say that [22XC[122X is a left [22XG[122X-code (respectively a [22XG[122X-code) if there
  is  a bijection [22Xϕ:E→ G[122X such that the linear extension of [22Xϕ[122X to an isomorphism
  [22Xϕ:  F_q^n→  F_q G[122X maps [22XC[122X to a left ideal (respectively a two-sided ideal) of
  [22XF_q  G[122X. A left [13Xgroup code[113X (respectively a group code) is a linear code which
  is a left [22XG[122X-code (respectively a [22XG[122X-code) for some group [22XG[122X.[133X
  
  [33X[0;0YSince  left ideals in [22XF_q G[122X are generated by idempotents, there is a one-one
  relation  between  (sums of) primitive idempotents of [22XF_q G[122X and left [22XG[122X-codes
  over [22XF_q[122X.[133X
  
  [33X[0;0YNote that each element [22Xc[122X in [22XF_q G[122X is of the form [22Xc=∑_i=1^n f_i g_i[122X, where we
  fix  an ordering [22X{g_1,g_2,...,g_n }[122X of the group elements of [22XG[122X and [22Xf_i∈ F_q[122X.
  If one looks at [22Xc[122X as a codeword, one writes [22X[f_1 f_2 ... f_n][122X.[133X
  
