#### Green Polynomials for D3

####  W-rep key:
#  x[1] = [[], [1, 1, 1]]   ,   orbit = [1, 1, 1, 1, 1, 1]  ,   A-rep = []    
#  x[2] = [[1], [1, 1]]   ,   orbit = [2, 2, 1, 1]  ,   A-rep = []    
#  x[3] = [[], [2, 1]]   ,   orbit = [3, 1, 1, 1]  ,   A-rep = []    
#  x[4] = [[1], [2]]   ,   orbit = [3, 3]    ,   A-rep = []    
#  x[5] = [[], [3]]    ,   orbit = [5, 1]    ,   A-rep = []    


###   Green Polynomials by Orbit

orbit #1 : [1, 1, 1, 1, 1, 1]
  dim = 0
  A(O) = 1  ,  |A(O)_0| = 1
  g_s = 15*V[0]
  Z_G(x)_0 = D3

  # Green Polys by orbit reps
    #1.1 : x[1]  :  [1, 1, 1, 1, 1, 1],[]  : [[], [3]]
        Qxi[D3,1,1] = (x[1])*q^6
                    + (x[2])*q^5
                    + (x[2]+x[3])*q^4
                    + (x[2]+x[4])*q^3
                    + (x[3]+x[4])*q^2
                    + (x[4])*q
                    + x[5]

  # Green Polys by conj class in A(O)
    #1.1  :  c = ()
        |O_x_c^F| = 1
        Qxc[D3,1,1] = (x[1])*q^6
                     + (x[2])*q^5
                     + (x[2]+x[3])*q^4
                     + (x[2]+x[4])*q^3
                     + (x[3]+x[4])*q^2
                     + (x[4])*q
                     + x[5]


orbit #2 : [2, 2, 1, 1]
  dim = 6
  A(O) = 1  ,  |A(O)_0| = 1
  g_s = 4*V[1]+V[2]+4*V[0]
  Z_G(x)_0 = Sp2+O2

  # Green Polys by orbit reps
    #2.1 : x[2]  :  [2, 2, 1, 1],[]  : [[1], [1, 1]]
        Qxi[D3,2,1] = (x[2])*q^3 
                      + (x[3]+x[4])*q^2 
                      + (x[4])*q 
                      + x[5]

  # Green Polys by conj class in A(O)
    #2.1  :  c = ()
    
        |O_x_c^F| = (q^2+q+1)*(q^4-1)
        Qxc[D3,2,1] = (x[2])*q^3 
                      + (x[3]+x[4])*q^2 
                      + (x[4])*q 
                      + x[5]


orbit #3 : [3, 1, 1, 1] 
  dim = 8
  A(O) = 1 , |A(O)_0| = 1
  g_s = 4*V[2]+3*V[0]
  Z_G(x)_0 = O3+O1:

  # Green Polys by orbit reps
    #3.1 : x[3]  :  [3, 1, 1, 1],[]  : [[], [2, 1]]
       Qxi[D3,3,1] := (x[3])*q^2 
                      + (x[4])*q 
                      + x[5]

  # Green Polys by conj class in A(O)
    #3.1  :  c = ()
        |O_x_c^F| = q*(q^3-1)*(q^4-1)
        Qxc[D3,3,1] = (x[3])*q^2 
                      + (x[4])*q 
                      + x[5]


orbit #4 : [3, 3] 
  dim = 10 
  A(O) = 1 , |A(O)_0| = 1
  g_s = V[4]+3*V[2]+V[0]
  Z_G(x)_0 := O2:

  # Green Polys by W-reps
    #4.1 : x[4]  :  [3, 3],[]  : [[1], [2]]
        Qxi[D3,4,1] = (x[4])*q 
                      + x[5]

  # Green Polys by conj class in A(O)
    #4.1  :  c = ()
        |O_x_c^F| = q^2*(q+1)*(q^3-1)*(q^4-1)
        Qxc[D3,4,1] = (x[4])*q 
                      + x[5]


orbit #5 : [5, 1] 
  dim = 12 
  A(O) = 1 , |A(O)_0| = 1
  g_s = V[6]+V[4]+V[2]
  Z_G(x)_0 = 2*O1

  # Green Polys by orbit reps
    #5.1 : x[5]  :  [5, 1],[]  : [[], [3]]
        Qxi[D3,5,1] = x[5]

  # Green Polys by conj class in A(O)
    #5.1  :  c = ()
        |O_x_c^F| = q^3*(q^2-1)*(q^3-1)*(q^4-1)
        Qxc[D3,5,1] = x[5]