  
  [1X2 [33X[0;0YNotation[133X[101X
  
  [33X[0;0YThis  chapter explains the notation of the package [5XWPE[105X, mainly influenced by
  the accompanying publication [BNRW22].[133X
  
  
  [1X2.1 [33X[0;0YWreath Products[133X[101X
  
  [33X[0;0YLet  [22XG  =  K ≀ H[122X be a wreath product of two groups, where [22XH[122X is a permutation
  group  of  degree [22Xm[122X. The wreath product is defined as the semidirect product
  of  the  function  space  [22XK^m[122X with [22XH[122X, where [22Xπ ∈ H[122X acts on [22Xf ∈ K^m[122X by setting
  [22Xf^{π} : {1, ..., m} → K, i ↦ [(i)π^{-1}]f[122X. Note that [22XG[122X naturally embeds into
  the [13Xparent wreath product[113X, that is [22XP = K ≀ Sym(m) ≥ G[122X.[133X
  
  [33X[0;0YFormally  we  can write an element of [22XG[122X as a tuple [22Xg = (f, π) ∈ G[122X, where [22Xf ∈
  K^m[122X is a function [22Xf : {1, ..., m} → K[122X and [22Xπ ∈ H ≤ Sym(m)[122X is a permutation on
  [22Xm[122X points. We call [22Xf[122X the [13Xbase component[113X and [22Xπ[122X the [13Xtop component[113X of [22Xg[122X.[133X
  
  [33X[0;0YWe  can naturally identify a map [22Xf ∈ K^m[122X with a tuple [22X(g_1, ..., g_m)[122X, where
  each  [22Xg_i  ∈ K[122X is the image of [22Xi ∈ {1, ..., m}[122X under [22Xf[122X. This yields a second
  useful  notation  for  elements in [22XG[122X by writing [22Xg = (g_1, ..., g_m; π)[122X. Note
  that  we  use  a  semicolon  to  seperate  the  base  component from the top
  component. Further we call the element [22Xg_i[122X the [13X[22Xi[122X-th base component[113X of [22Xg[122X.[133X
  
  [33X[0;0YAnalogously,  the subgroup [22XB = K^m × ⟨ 1_H ⟩ ≤ G[122X is called the [13Xbase group[113X of
  [22XG[122X and the subgroup [22XT = ⟨ 1_K ⟩^m × H ≤ G[122X is called the [13Xtop group[113X of [22XG[122X.[133X
  
  [33X[0;0YWith the above notation, the multiplication of two elements[133X
  
  
  [24X[33X[0;6Yg = (f, π) = (g_1, ..., g_m; π), h = (d, σ) = (h_1, ..., h_m; σ)[133X[124X
  
  [33X[0;0Yof [22XG = K ≀ H[122X, a wreath product of finite groups, can be written as[133X
  
  
  [24X[33X[0;6Yg ⋅ h = (f ⋅ d^(π^-1), π ⋅ σ) = (g_1 ⋅ h_1^π, ..., g_m ⋅ h_m^π; π ⋅ σ).[133X[124X
  
  
  [1X2.2 [33X[0;0YWreath Cycles[133X[101X
  
  [33X[0;0YIn   a  permutation  group  we  have  the  well-known  concept  of  a  cycle
  decomposition.  For  wreath products we have a similar concept called [13Xwreath
  cycle decomposition[113X that allows us to solve certain computational tasks more
  efficiently.[133X
  
  [33X[0;0YDetailed  information on [13Xwreath cycle decompositions[113X can be found in Chapter
  2  in [BNRW22]. Chapters 3-5 in [BNRW22] describe how these can be exploited
  for  finding  conjugating  elements,  conjugacy classes, and centralisers in
  wreath  products,  and  Chapter 6 in [BNRW22] contains a table of timings of
  sample computations done with [5XWPE[105X vs. native [5XGAP[105X.[133X
  
  [33X[0;0YWe  use  the  notation  from Section [14X2.1[114X in order to introduce the following
  concepts.[133X
  
  [33X[0;0Y[22XDefinition:[122X We define the [13Xterritory[113X of an element [22Xg = (g_1, ..., g_m; π) ∈ G[122X
  by  [22Xterr(g)  :=  supp(π)  ∪  {i : g_i ≠ 1}[122X, where [22Xsupp(π)[122X denotes the set of
  moved points of [22Xπ[122X.[133X
  
  [33X[0;0Y[22XDefinition:[122X  Two  elements  [22Xg,  h  ∈  G[122X  are  said  to  be [13Xdisjoint[113X if their
  territories are disjoint.[133X
  
  [33X[0;0Y[22XLemma:[122X Disjoint elements in [22XG[122X commute.[133X
  
  [33X[0;0Y[22XDefinition:[122X  An  element [22Xg = (g_1, ..., g_m; π) ∈ G[122X is called a [13Xwreath cycle[113X
  if either [22Xπ[122X is a cycle in [22XSym(n)[122X and [22Xterr(g) = supp(π)[122X, or [22X|terr(g)| = 1[122X.[133X
  
  [33X[0;0Y[22XExample:[122X For example, if we consider the wreath product [22XSym(4) ≀ Sym(5)[122X, the
  element[133X
  
  
  [24X[33X[0;6Y( (), (1,2,3), (), (1,2), (); (1,2,4) )[133X[124X
  
  [33X[0;0Yis a wreath cycle as described in the first case and the element[133X
  
  
  [24X[33X[0;6Y( (), (), (1,3), (), (); () )[133X[124X
  
  [33X[0;0Yis  a wreath cycle as described in the second case. Moreover, these elements
  are disjoint and thus commute.[133X
  
  [33X[0;0Y[22XTheorem:[122X  Every  element of [22XG[122X can be written as a finite product of disjoint
  wreath  cycles  in  [22XP[122X.  This  decomposition  is unique up to ordering of the
  factors. We call such a decomposition a [13Xwreath cycle decomposition[113X.[133X
  
  
  [1X2.3 [33X[0;0YSparse Wreath Cycles[133X[101X
  
  [33X[0;0YWe  use  the  notation  from Section [14X2.1[114X in order to introduce the following
  concepts.[133X
  
  [33X[0;0YThe  main  motivation for introducing the concept of [13Xsparse wreath cycles[113X is
  the efficient computation of centralisers of wreath product elements. Simply
  put, we compute the centraliser [22XC_G(g)[122X of an arbitrary element [22Xg ∈ P[122X in [22XG[122X by
  conjugating  it  in  [22XP[122X to a restricted representative [22Xh = g^c ∈ P[122X, computing
  the  centraliser  of  [22Xh[122X  in [22XG[122X and then conjugating it back. The wreath cycle
  decomposition of the representative [22Xh[122X consists only of sparse wreath cycles.[133X
  
  [33X[0;0YMore  information on [13Xsparse wreath cycles[113X and centralisers of wreath product
  elements can be found in Chapter 5 in [BNRW22].[133X
  
  [33X[0;0Y[22XDefinition:[122X  We  say  that  a  wreath  cycle [22Xg = (g_1, ..., g_m; π) ∈ G[122X is a
  [13Xsparse  wreath  cycle[113X,  if there exists an [22Xi_0[122X such that [22Xg_i = 1[122X for all [22Xi ≠
  i_0[122X.[133X
  
  [33X[0;0Y[22XExample:[122X For example, if we consider the wreath product [22XSym(4) ≀ Sym(5)[122X, the
  element[133X
  
  
  [24X[33X[0;6Y( (), (1,2,3), (), (), (); (1,2,4) )[133X[124X
  
  [33X[0;0Yis a sparse wreath cycle, as well as the element[133X
  
  
  [24X[33X[0;6Y( (), (), (1,3), (), (); () ) .[133X[124X
  
  [33X[0;0YA very important invariant under conjugation is the [13Xyade[113X of a wreath cycle.[133X
  
  [33X[0;0Y[22XDefinition:[122X  For  a  wreath  cycle [22Xg = (f, π) ∈ G[122X and a point [22Xi ∈ terr(g)[122X we
  define the [13Xyade[113X of [22Xg[122X in [22Xi[122X as[133X
  
  
  [24X[33X[0;6Y[(i)π^0]f ⋅ [(i)π^1]f ⋯ [(i)π^|π| - 1]f .[133X[124X
  
  [33X[0;0Y[22XExample:[122X Consider the wreath product [22XSym(4) ≀ Sym(5)[122X, and the wreath cycle[133X
  
  
  [24X[33X[0;6Yg = (f, π) = ( (), (1,2,3), (), (1,2), (); (1,2,4) ).[133X[124X
  
  [33X[0;0YThe yade evaluated at [22Xi = 1[122X is given by[133X
  
  
  [24X[33X[0;6Y[(1)π^0]f ⋅ [(1)π^1]f ⋅ [(1)π^2]f = [1]f ⋅ [2]f ⋅ [4]f = () ⋅ (1,2,3) ⋅ (1,2) = (2,3)[133X[124X
  
  [33X[0;0Yand the yade evaluated at [22Xj = 4[122X is given by[133X
  
  
  [24X[33X[0;6Y[(4)π^0]f ⋅ [(4)π^1]f ⋅ [(4)π^2]f = [4]f ⋅ [1]f ⋅ [2]f = (1,2) ⋅ () ⋅ (1,2,3) = (1,3) .[133X[124X
  
  [33X[0;0YUp  to  conjugacy, the yade is independent under the chosen evaluation point
  [22Xi[122X.  Moreover,  wreath  cycles  are  conjugate  over [22XG[122X if and only if the top
  components  are  conjugate  over  [22XH[122X and the yades are conjugate over [22XK[122X. More
  specific,  we can conjugate a wreath cycle [22Xg[122X to a sparse wreath cycle [22Xh[122X such
  that the [22Xi[122X-th base component of [22Xh[122X contains the yade of [22Xg[122X in [22Xi[122X. This leads to
  the following result.[133X
  
  [33X[0;0Y[22XTheorem:[122X Every element [22Xg ∈ P[122X can be conjugated by some [22Xc ∈ K^m × ⟨ 1_H ⟩ ≤ P[122X
  to  an  element  [22Xh  =  g^c ∈ P[122X such that the wreath cycle decomposition of [22Xh[122X
  consists only of sparse wreath cycles.[133X
  
