#### Green Polynomials for B4 #### W-rep key: # x[1] = [[], [1, 1, 1, 1]] , orbit = [1, 1, 1, 1, 1, 1, 1, 1, 1] , A-rep = [] # x[2] = [[], [2, 1, 1]] , orbit = [2, 2, 1, 1, 1, 1, 1] , A-rep = [] # x[3] = [[], [2, 2]] , orbit = [2, 2, 2, 2, 1] , A-rep = [] # x[4] = [[1], [1, 1, 1]] , orbit = [3, 1, 1, 1, 1, 1, 1] , A-rep = [1] # x[5] = [[1, 1, 1, 1], []] , orbit = [3, 1, 1, 1, 1, 1, 1] , A-rep = [-1] # x[6] = [[1, 1], [1, 1]] , orbit = [3, 2, 2, 1, 1] , A-rep = [1] # x[7] = [[1, 1, 1], [1]] , orbit = [3, 2, 2, 1, 1] , A-rep = [-1] # x[8] = [[1], [2, 1]] , orbit = [3, 3, 1, 1, 1] , A-rep = [1] # x[9] = [[], [3, 1]] , orbit = [3, 3, 1, 1, 1] , A-rep = [-1] # x[10] = [[1, 1], [2]] , orbit = [3, 3, 3] , A-rep = [] # x[11] = [[1], [3]] , orbit = [4, 4, 1] , A-rep = [] # x[12] = [[2], [1, 1]] , orbit = [5, 1, 1, 1, 1] , A-rep = [1] # x[13] = [[2, 1, 1], []] , orbit = [5, 1, 1, 1, 1] , A-rep = [-1] # x[14] = [[2, 1], [1]] , orbit = [5, 2, 2] , A-rep = [] # x[15] = [[2], [2]] , orbit = [5, 3, 1] , A-rep = [1, 1] # x[16] = [[2, 2], []] , orbit = [5, 3, 1] , A-rep = [-1, 1] # x[17] = [[], [4]] , orbit = [5, 3, 1] , A-rep = [1, -1] # x[18] = [[3], [1]] , orbit = [7, 1, 1] , A-rep = [1] # x[19] = [[3, 1], []] , orbit = [7, 1, 1] , A-rep = [-1] # x[20] = [[4], []] , orbit = [9] , A-rep = [] ### Green Polynomials by Orbit orbit #1 : [1, 1, 1, 1, 1, 1, 1, 1, 1] dim = 0 A(O) = 1 , |A(O)_0| = 1 g_s = 36*V[0] Z_G(x)_0 = B4 # Green Polys by orbit reps #1.1 : x[1] : [1, 1, 1, 1, 1, 1, 1, 1, 1],[] : [[4], []] Qxi[B4,1,1] = (x[1])*q^16 + (x[4])*q^15 + (x[2]+x[6])*q^14 + (x[4]+x[7]+x[8])*q^13 + (x[2]+x[3]+x[5]+x[6]+x[10]+x[12])*q^12 + (x[4]+x[7]+2*x[8]+x[14])*q^11 + (x[2]+2*x[6]+x[9]+x[10]+x[12]+x[13]+x[15])*q^10 + (x[4]+x[7]+2*x[8]+x[11]+2*x[14])*q^9 + (x[3]+x[6]+x[9]+2*x[10]+2*x[12]+x[13]+x[15]+x[16])*q^8 + (x[7]+2*x[8]+x[11]+2*x[14]+x[18])*q^7 + (x[6]+x[9]+x[10]+x[12]+x[13]+2*x[15]+x[19])*q^6 + (x[8]+x[11]+2*x[14]+x[18])*q^5 + (x[10]+x[12]+x[15]+x[16]+x[17]+x[19])*q^4 + (x[11]+x[14]+x[18])*q^3 + (x[15]+x[19])*q^2 + (x[18])*q + x[20] # Green Polys by conj class in A(O) #1.1 : c = () |O_x_c^F| = 1 Qxc[B4,1,1] = (x[1])*q^16 + (x[4])*q^15 + (x[2]+x[6])*q^14 + (x[4]+x[7]+x[8])*q^13 + (x[2]+x[3]+x[5]+x[6]+x[10]+x[12])*q^12 + (x[4]+x[7]+2*x[8]+x[14])*q^11 + (x[2]+2*x[6]+x[9]+x[10]+x[12]+x[13]+x[15])*q^10 + (x[4]+x[7]+2*x[8]+x[11]+2*x[14])*q^9 + (x[3]+x[6]+x[9]+2*x[10]+2*x[12]+x[13]+x[15]+x[16])*q^8 + (x[7]+2*x[8]+x[11]+2*x[14]+x[18])*q^7 + (x[6]+x[9]+x[10]+x[12]+x[13]+2*x[15]+x[19])*q^6 + (x[8]+x[11]+2*x[14]+x[18])*q^5 + (x[10]+x[12]+x[15]+x[16]+x[17]+x[19])*q^4 + (x[11]+x[14]+x[18])*q^3 + (x[15]+x[19])*q^2 + (x[18])*q + x[20] orbit #2 : [2, 2, 1, 1, 1, 1, 1] dim = 12 A(O) = 1 , |A(O)_0| = 1 g_s = 10*V[1]+V[2]+13*V[0] Z_G(x)_0 = O5+Sp2 # Green Polys by orbit reps #2.1 : x[2] : [2, 2, 1, 1, 1, 1, 1],[] : [[], [2, 1, 1]] Qxi[B4,2,1] = (x[2])*q^10 + (x[4]+x[8])*q^9 + (x[3]+x[6]+x[9]+x[10]+x[12])*q^8 + (x[7]+2*x[8]+x[11]+x[14])*q^7 + (x[6]+x[9]+x[10]+x[12]+x[13]+2*x[15])*q^6 + (x[8]+x[11]+2*x[14]+x[18])*q^5 + (x[10]+x[12]+x[15]+x[16]+x[17]+x[19])*q^4 + (x[11]+x[14]+x[18])*q^3 + (x[15]+x[19])*q^2 + (x[18])*q + x[20] # Green Polys by conj class in A(O) #2.1 : c = () |O_x_c^F| = (q^2-q+1)*(q^2+q+1)*(q^8-1) Qxc[B4,2,1] = (x[2])*q^10 + (x[4]+x[8])*q^9 + (x[3]+x[6]+x[9]+x[10]+x[12])*q^8 + (x[7]+2*x[8]+x[11]+x[14])*q^7 + (x[6]+x[9]+x[10]+x[12]+x[13]+2*x[15])*q^6 + (x[8]+x[11]+2*x[14]+x[18])*q^5 + (x[10]+x[12]+x[15]+x[16]+x[17]+x[19])*q^4 + (x[11]+x[14]+x[18])*q^3 + (x[15]+x[19])*q^2 + (x[18])*q + x[20] orbit #3 : [2, 2, 2, 2, 1] dim = 16 A(O) = 1 , |A(O)_0| = 1 g_s = 6*V[2]+4*V[1]+10*V[0] Z_G(x)_0 = O1+Sp4 # Green Polys by orbit reps #3.1 : x[3] : [2, 2, 2, 2, 1],[] : [[], [2, 2]] Qxi[B4,3,1] = (x[3])*q^8 + (x[8])*q^7 + (x[6]+x[9]+x[15])*q^6 + (x[8]+x[11]+x[14])*q^5 + (x[10]+x[12]+x[15]+x[16]+x[17])*q^4 + (x[11]+x[14]+x[18])*q^3 + (x[15]+x[19])*q^2 + (x[18])*q + x[20] # Green Polys by conj class in A(O) #3.1 : c = () |O_x_c^F| = q^2*(q^6-1)*(q^8-1) Qxc[B4,3,1] = (x[3])*q^8 + (x[8])*q^7 + (x[6]+x[9]+x[15])*q^6 + (x[8]+x[11]+x[14])*q^5 + (x[10]+x[12]+x[15]+x[16]+x[17])*q^4 + (x[11]+x[14]+x[18])*q^3 + (x[15]+x[19])*q^2 + (x[18])*q + x[20] orbit #4 : [3, 1, 1, 1, 1, 1, 1] dim = 14 A(O) = Z2 , |A(O)_0| = 2 g_s = 7*V[2]+15*V[0] Z_G(x)_0 = O6+O1 # Green Polys by orbit reps #4.1 : x[4] : [3, 1, 1, 1, 1, 1, 1],[1] : [[1], [1, 1, 1]] Qxi[B4,4,1] = (x[4])*q^9 + (x[6]+x[12])*q^8 + (x[7]+x[8]+x[14])*q^7 + (x[6]+x[10]+x[12]+x[13]+x[15])*q^6 + (x[8]+2*x[14]+x[18])*q^5 + (x[10]+x[12]+x[15]+x[16]+x[19])*q^4 + (x[11]+x[14]+x[18])*q^3 + (x[15]+x[19])*q^2 + (x[18])*q + x[20] #4.2 : x[5] : [3, 1, 1, 1, 1, 1, 1],[-1] : [[1, 1, 1, 1], []] Qxi[B4,4,2] = (x[5])*q^9 + (x[7])*q^8 + (x[6]+x[13])*q^7 + (x[7]+x[14])*q^6 + (x[10]+x[13]+x[16])*q^5 + (x[14])*q^4 + (x[19])*q^3 # Green Polys by conj class in A(O) #4.1 : c = () |O_x_c^F| = 1/2*q^3*(q^3+1)*(q^8-1) Qxc[B4,4,1] = (x[4]+x[5])*q^9 + (x[6]+x[7]+x[12])*q^8 + (x[6]+x[7]+x[8]+x[13]+x[14])*q^7 + (x[6]+x[7]+x[10]+x[12]+x[13]+x[14]+x[15])*q^6 + (x[8]+x[10]+x[13]+2*x[14]+x[16]+x[18])*q^5 + (x[10]+x[12]+x[14]+x[15]+x[16]+x[19])*q^4 + (x[11]+x[14]+x[18]+x[19])*q^3 + (x[15]+x[19])*q^2 + (x[18])*q + x[20] #4.2 : c = (1) |O_x_c^F| = 1/2*q^3*(q^3-1)*(q^8-1) Qxc[B4,4,2] = (x[4]-x[5])*q^9 + (x[6]-x[7]+x[12])*q^8 + (-x[6]+x[7]+x[8]-x[13]+x[14])*q^7 + (x[6]-x[7]+x[10]+x[12]+x[13]-x[14]+x[15])*q^6 + (x[8]-x[10]-x[13]+2*x[14]-x[16]+x[18])*q^5 + (x[10]+x[12]-x[14]+x[15]+x[16]+x[19])*q^4 + (x[11]+x[14]+x[18]-x[19])*q^3 + (x[15]+x[19])*q^2 + (x[18])*q + x[20] orbit #5 : [3, 2, 2, 1, 1] dim = 20 A(O) = Z2 , |A(O)_0| = 2 g_s = 2*V[3]+4*V[2]+6*V[1]+4*V[0] Z_G(x)_0 = O2+Sp2+O1 # Green Polys by orbit reps #5.1 : x[6] : [3, 2, 2, 1, 1],[1] : [[1, 1], [1, 1]] Qxi[B4,5,1] = (x[6])*q^6 + (x[8]+x[14])*q^5 + (x[10]+x[12]+x[15]+x[16])*q^4 + (x[11]+x[14]+x[18])*q^3 + (x[15]+x[19])*q^2 + (x[18])*q + x[20] #5.2 : x[7] : [3, 2, 2, 1, 1],[-1] : [[1, 1, 1], [1]] Qxi[B4,5,2] = (x[7])*q^6 + (x[10]+x[13])*q^5 + (x[14])*q^4 + (x[19])*q^3 # Green Polys by conj class in A(O) #5.1 : c = () |O_x_c^F| = 1/2*q^3*(q+1)*(q^2+1)*(q^6-1)*(q^8-1) Qxc[B4,5,1] = (x[6]+x[7])*q^6 + (x[8]+x[10]+x[13]+x[14])*q^5 + (x[10]+x[12]+x[14]+x[15]+x[16])*q^4 + (x[11]+x[14]+x[18]+x[19])*q^3 + (x[15]+x[19])*q^2 + (x[18])*q + x[20] #5.2 : c = (1) |O_x_c^F| = 1/2*q^3*(q-1)*(q^2+1)*(q^6-1)*(q^8-1) Qxc[B4,5,2] = (x[6]-x[7])*q^6 + (x[8]-x[10]-x[13]+x[14])*q^5 + (x[10]+x[12]-x[14]+x[15]+x[16])*q^4 + (x[11]+x[14]+x[18]-x[19])*q^3 + (x[15]+x[19])*q^2 + (x[18])*q + x[20] orbit #6 : [3, 3, 1, 1, 1] dim = 22 A(O) = Z2 , |A(O)_0| = 2 g_s = V[4]+9*V[2]+4*V[0] Z_G(x)_0 = O3+O2 # Green Polys by orbit reps #6.1 : x[8] : [3, 3, 1, 1, 1],[1] : [[1], [2, 1]] Qxi[B4,6,1] = (x[8])*q^5 + (x[10]+x[12]+x[15])*q^4 + (x[11]+x[14]+x[18])*q^3 + (x[15]+x[19])*q^2 + (x[18])*q + x[20] #6.2 : x[9] : [3, 3, 1, 1, 1],[-1] : [[], [3, 1]] Qxi[B4,6,2] = (x[9])*q^5 + (x[11])*q^4 + (x[17])*q^3 # Green Polys by conj class in A(O) #6.1 : c = () |O_x_c^F| = 1/2*q^5*(q+1)*(q^2+1)*(q^6-1)*(q^8-1) Qxc[B4,6,1] = (x[8]+x[9])*q^5 + (x[10]+x[11]+x[12]+x[15])*q^4 + (x[11]+x[14]+x[17]+x[18])*q^3 + (x[15]+x[19])*q^2 + (x[18])*q + x[20] #6.2 : c = (1) |O_x_c^F| = 1/2*q^5*(q-1)*(q^2+1)*(q^6-1)*(q^8-1) Qxc[B4,6,2] = (x[8]-x[9])*q^5 + (x[10]-x[11]+x[12]+x[15])*q^4 + (x[11]+x[14]-x[17]+x[18])*q^3 + (x[15]+x[19])*q^2 + (x[18])*q + x[20] orbit #7 : [3, 3, 3] dim = 24 A(O) = 1 , |A(O)_0| = 1 g_s = 3*V[4]+6*V[2]+3*V[0] Z_G(x)_0 = O3 # Green Polys by orbit reps #7.1 : x[10] : [3, 3, 3],[] : [[1, 1], [2]] Qxi[B4,7,1] = (x[10])*q^4 + (x[11]+x[14])*q^3 + (x[15]+x[19])*q^2 + (x[18])*q + x[20] # Green Polys by conj class in A(O) #7.1 : c = () |O_x_c^F| = q^6*(q^4-1)*(q^6-1)*(q^8-1) Qxc[B4,7,1] = (x[10])*q^4 + (x[11]+x[14])*q^3 + (x[15]+x[19])*q^2 + (x[18])*q + x[20] orbit #8 : [4, 4, 1] dim = 26 A(O) = 1 , |A(O)_0| = 1 g_s = V[6]+3*V[4]+2*V[3]+V[2]+3*V[0] Z_G(x)_0 = O1+Sp2 # Green Polys by orbit reps #8.1 : x[11] : [4, 4, 1],[] : [[1], [3]] Qxi[B4,8,1] = (x[11])*q^3 + (x[15]+x[17])*q^2 + (x[18])*q + x[20] # Green Polys by conj class in A(O) #8.1 : c = () |O_x_c^F| = q^8*(q^4-1)*(q^6-1)*(q^8-1) Qxc[B4,8,1] = (x[11])*q^3 + (x[15]+x[17])*q^2 + (x[18])*q + x[20] orbit #9 : [5, 1, 1, 1, 1] dim = 24 A(O) = Z2 , |A(O)_0| = 2 g_s = V[6]+4*V[4]+V[2]+6*V[0] Z_G(x)_0 = O4+O1 # Green Polys by orbit reps #9.1 : x[12] : [5, 1, 1, 1, 1],[1] : [[2], [1, 1]] Qxi[B4,9,1] = (x[12])*q^4 + (x[14]+x[18])*q^3 + (x[15]+x[19])*q^2 + (x[18])*q + x[20] #9.2 : x[13] : [5, 1, 1, 1, 1],[-1] : [[2, 1, 1], []] Qxi[B4,9,2] = (x[13])*q^4 + (x[14])*q^3 + (x[16]+x[19])*q^2 # Green Polys by conj class in A(O) #9.1 : c = () |O_x_c^F| = 1/2*q^8*(q^2+1)*(q^6-1)*(q^8-1) Qxc[B4,9,1] = (x[12]+x[13])*q^4 + (2*x[14]+x[18])*q^3 + (x[15]+x[16]+2*x[19])*q^2 + (x[18])*q + x[20] #9.2 : c = (1) |O_x_c^F| = 1/2*q^8*(q^2-1)*(q^6-1)*(q^8-1) Qxc[B4,9,2] = (x[12]-x[13])*q^4 + (x[18])*q^3 + (x[15]-x[16])*q^2 + (x[18])*q + x[20] orbit #10 : [5, 2, 2] dim = 26 A(O) = 1 , |A(O)_0| = 1 g_s = V[6]+2*V[5]+2*V[3]+2*V[2]+3*V[0] Z_G(x)_0 = O1+Sp2 # Green Polys by orbit reps #10.1 : x[14] : [5, 2, 2],[] : [[2, 1], [1]] Qxi[B4,10,1] = (x[14])*q^3 + (x[15]+x[16]+x[19])*q^2 + (x[18])*q + x[20] # Green Polys by conj class in A(O) #10.1 : c = () |O_x_c^F| = q^8*(q^4-1)*(q^6-1)*(q^8-1) Qxc[B4,10,1] = (x[14])*q^3 + (x[15]+x[16]+x[19])*q^2 + (x[18])*q + x[20] orbit #11 : [5, 3, 1] dim = 28 A(O) = Z2^2 , |A(O)_0| = 3 g_s = 2*V[6]+2*V[4]+4*V[2] Z_G(x)_0 = 3*O1 # Green Polys by orbit reps #11.1 : x[15] : [5, 3, 1],[1, 1] : [[2], [2]] Qxi[B4,11,1] = (x[15])*q^2 + (x[18])*q + x[20] #11.2 : x[16] : [5, 3, 1],[-1, 1] : [[2, 2], []] Qxi[B4,11,2] = (x[16])*q^2 #11.3 : x[17] : [5, 3, 1],[1, -1] : [[], [4]] Qxi[B4,11,3] = (x[17])*q^2 # Green Polys by conj class in A(O) #11.1 : c = () |O_x_c^F| = 1/4*q^8*(q^2-1)*(q^4-1)*(q^6-1)*(q^8-1) Qxc[B4,11,1] = (x[15]+x[16]+x[17])*q^2 + (x[18])*q + x[20] #11.2 : c = (1) |O_x_c^F| = 1/4*q^8*(q^2-1)*(q^4-1)*(q^6-1)*(q^8-1) Qxc[B4,11,2] = (x[15]-x[16]+x[17])*q^2 + (x[18])*q + x[20] #11.3 : c = (2) |O_x_c^F| = 1/4*q^8*(q^2-1)*(q^4-1)*(q^6-1)*(q^8-1) Qxc[B4,11,3] = (x[15]+x[16]-x[17])*q^2 + (x[18])*q + x[20] #11.4 : c = (12) |O_x_c^F| = 1/4*q^8*(q^2-1)*(q^4-1)*(q^6-1)*(q^8-1) Qxc[B4,11,4] = (x[15]-x[16]-x[17])*q^2 + (x[18])*q + x[20] orbit #12 : [7, 1, 1] dim = 30 A(O) = Z2 , |A(O)_0| = 2 g_s = V[10]+3*V[6]+V[2]+V[0] Z_G(x)_0 = O2+O1 # Green Polys by orbit reps #12.1 : x[18] : [7, 1, 1],[1] : [[3], [1]] Qxi[B4,12,1] = (x[18])*q + x[20] #12.2 : x[19] : [7, 1, 1],[-1] : [[3, 1], []] Qxi[B4,12,2] = (x[19])*q # Green Polys by conj class in A(O) #12.1 : c = () |O_x_c^F| = 1/2*q^11*(q+1)*(q^4-1)*(q^6-1)*(q^8-1) Qxc[B4,12,1] = (x[18]+x[19])*q + x[20] #12.2 : c = (1) |O_x_c^F| = 1/2*q^11*(q-1)*(q^4-1)*(q^6-1)*(q^8-1) Qxc[B4,12,2] = (x[18]-x[19])*q + x[20] orbit #13 : [9] dim = 32 A(O) = 1 , |A(O)_0| = 1 g_s = V[14]+V[10]+V[6]+V[2] Z_G(x)_0 = O1 # Green Polys by orbit reps #13.1 : x[20] : [9],[] : [[4], []] Qxi[B4,13,1] = x[20] # Green Polys by conj class in A(O) #13.1 : c = () |O_x_c^F| = q^12*(q^2-1)*(q^4-1)*(q^6-1)*(q^8-1) Qxc[B4,13,1] = x[20]