#### Green Polynomials for D4 #### W-rep key: # x[1] = [[], [1, 1, 1, 1]] , orbit = [1, 1, 1, 1, 1, 1, 1, 1] , A-rep = [] # x[2] = [[1], [1, 1, 1]] , orbit = [2, 2, 1, 1, 1, 1] , A-rep = [] # x[3] = [[1, 1], [1, 1], 2] , orbit = [2, 2, 2, 2, 0] , A-rep = [] # x[4] = [[1, 1], [1, 1], 1] , orbit = [2, 2, 2, 2] , A-rep = [] # x[5] = [[], [2, 1, 1]] , orbit = [3, 1, 1, 1, 1, 1] , A-rep = [] # x[6] = [[], [2, 2]] , orbit = [3, 2, 2, 1] , A-rep = [] # x[7] = [[1], [2, 1]] , orbit = [3, 3, 1, 1] , A-rep = [1] # x[8] = [[2], [1, 1]] , orbit = [3, 3, 1, 1] , A-rep = [-1] # x[9] = [[2], [2], 2] , orbit = [4, 4, 0] , A-rep = [] # x[10] = [[2], [2], 1] , orbit = [4, 4] , A-rep = [] # x[11] = [[], [3, 1]] , orbit = [5, 1, 1, 1] , A-rep = [] # x[12] = [[1], [3]] , orbit = [5, 3] , A-rep = [] # x[13] = [[], [4]] , orbit = [7, 1] , A-rep = [] ### Green Polynomials by Orbit orbit #1 : [1, 1, 1, 1, 1, 1, 1, 1] dim = 0 A(O) = 1 , |A(O)_0| = 1 g_s = 28*V[0] Z_G(x)_0 = D4 # Green Polys by orbit reps #1.1 : x[1] : [1, 1, 1, 1, 1, 1, 1, 1],[] : [[], [4]] Qxi[D4,1,1] = (x[1])*q^12 + (x[2])*q^11 + (x[3]+x[4]+x[5])*q^10 + (2*x[2]+x[7])*q^9 + (x[3]+x[4]+x[5]+x[6]+2*x[8])*q^8 + (x[2]+3*x[7])*q^7 + (x[3]+x[4]+x[5]+2*x[8]+x[9]+x[10]+x[11])*q^6 + (3*x[7]+x[12])*q^5 + (x[6]+2*x[8]+x[9]+x[10]+x[11])*q^4 + (x[7]+2*x[12])*q^3 + (x[9]+x[10]+x[11])*q^2 + (x[12])*q + x[13] # Green Polys by conj class in A(O) #1.1 : c = () |O_x_c^F| = 1 Qxc[D4,1,1] = (x[1])*q^12 + (x[2])*q^11 + (x[3]+x[4]+x[5])*q^10 + (2*x[2]+x[7])*q^9 + (x[3]+x[4]+x[5]+x[6]+2*x[8])*q^8 + (x[2]+3*x[7])*q^7 + (x[3]+x[4]+x[5]+2*x[8]+x[9]+x[10]+x[11])*q^6 + (3*x[7]+x[12])*q^5 + (x[6]+2*x[8]+x[9]+x[10]+x[11])*q^4 + (x[7]+2*x[12])*q^3 + (x[9]+x[10]+x[11])*q^2 + (x[12])*q + x[13] orbit #2 : [2, 2, 1, 1, 1, 1] dim = 10 A(O) = 1 , |A(O)_0| = 1 g_s = 8*V[1]+V[2]+9*V[0] Z_G(x)_0 = Sp2+O4 # Green Polys by orbit reps #2.1 : x[2] : [2, 2, 1, 1, 1, 1],[] : [[1], [1, 1, 1]] Qxi[D4,2,1] = (x[2])*q^7 + (x[3]+x[4]+x[5]+x[8])*q^6 + (3*x[7])*q^5 + (x[6]+2*x[8]+x[9]+x[10]+x[11])*q^4 + (x[7]+2*x[12])*q^3 + (x[9]+x[10]+x[11])*q^2 + (x[12])*q + x[13] # Green Polys by conj class in A(O) #2.1 : c = () |O_x_c^F| = (q^2+1)^2*(q^6-1) Qxc[D4,2,1] = (x[2])*q^7 + (x[3]+x[4]+x[5]+x[8])*q^6 + (3*x[7])*q^5 + (x[6]+2*x[8]+x[9]+x[10]+x[11])*q^4 + (x[7]+2*x[12])*q^3 + (x[9]+x[10]+x[11])*q^2 + (x[12])*q + x[13] orbit #3 : [2, 2, 2, 2, 0] dim = 12 A(O) = 1 , |A(O)_0| = 1 g_s = 6*V[2]+10*V[0] Z_G(x)_0 = Sp4 # Green Polys by orbit reps #3.1 : x[3] : [2, 2, 2, 2, 0],[] : [[1, 1], [1, 1], 2] Qxi[D4,3,1] = (x[3])*q^6 + (x[7])*q^5 + (x[6]+x[8]+x[9])*q^4 + (x[7]+x[12])*q^3 + (x[9]+x[10]+x[11])*q^2 + (x[12])*q + x[13] # Green Polys by conj class in A(O) #3.1 : c = () |O_x_c^F| = q^2*(q^4-1)*(q^6-1) Qxc[D4,3,1] = (x[3])*q^6 + (x[7])*q^5 + (x[6]+x[8]+x[9])*q^4 + (x[7]+x[12])*q^3 + (x[9]+x[10]+x[11])*q^2 + (x[12])*q + x[13] orbit #4 : [2, 2, 2, 2] dim = 12 A(O) = 1 , |A(O)_0| = 1 g_s = 6*V[2]+10*V[0] Z_G(x)_0 = Sp4 # Green Polys by orbit reps #4.1 : x[4] : [2, 2, 2, 2],[] : [[1, 1], [1, 1], 1] Qxi[D4,4,1] = (x[4])*q^6 + (x[7])*q^5 + (x[6]+x[8]+x[10])*q^4 + (x[7]+x[12])*q^3 + (x[9]+x[10]+x[11])*q^2 + (x[12])*q + x[13] # Green Polys by conj class in A(O) #4.1 : c = () |O_x_c^F| = q^2*(q^4-1)*(q^6-1) Qxc[D4,4,1] = (x[4])*q^6 + (x[7])*q^5 + (x[6]+x[8]+x[10])*q^4 + (x[7]+x[12])*q^3 + (x[9]+x[10]+x[11])*q^2 + (x[12])*q + x[13] orbit #5 : [3, 1, 1, 1, 1, 1] dim = 12 A(O) = 1 , |A(O)_0| = 1 g_s = 6*V[2]+10*V[0] Z_G(x)_0 = O5+O1 # Green Polys by orbit reps #5.1 : x[5] : [3, 1, 1, 1, 1, 1],[] : [[], [2, 1, 1]] Qxi[D4,5,1] = (x[5])*q^6 + (x[7])*q^5 + (x[6]+x[8]+x[11])*q^4 + (x[7]+x[12])*q^3 + (x[9]+x[10]+x[11])*q^2 + (x[12])*q + x[13] # Green Polys by conj class in A(O) #5.1 : c = () |O_x_c^F| = q^2*(q^4-1)*(q^6-1) Qxc[D4,5,1] = (x[5])*q^6 + (x[7])*q^5 + (x[6]+x[8]+x[11])*q^4 + (x[7]+x[12])*q^3 + (x[9]+x[10]+x[11])*q^2 + (x[12])*q + x[13] orbit #6 : [3, 2, 2, 1] dim = 16 A(O) = 1 , |A(O)_0| = 1 g_s = 2*V[3]+3*V[2]+4*V[1]+3*V[0] Z_G(x)_0 = Sp2+2*O1 # Green Polys by orbit reps #6.1 : x[6] : [3, 2, 2, 1],[] : [[], [2, 2]] Qxi[D4,6,1] = (x[6])*q^4 + (x[7])*q^3 + (x[9]+x[10]+x[11])*q^2 + (x[12])*q + x[13] # Green Polys by conj class in A(O) #6.1 : c = () |O_x_c^F| = q^2*(q^4-1)^2*(q^6-1) Qxc[D4,6,1] = (x[6])*q^4 + (x[7])*q^3 + (x[9]+x[10]+x[11])*q^2 + (x[12])*q + x[13] orbit #7 : [3, 3, 1, 1] dim = 18 A(O) = Z2 , |A(O)_0| = 2 g_s = V[4]+7*V[2]+2*V[0] Z_G(x)_0 = 2*O2 # Green Polys by orbit reps #7.1 : x[7] : [3, 3, 1, 1],[1] : [[1], [2, 1]] Qxi[D4,7,1] = (x[7])*q^3 + (x[9]+x[10]+x[11])*q^2 + (x[12])*q + x[13] #7.2 : x[8] : [3, 3, 1, 1],[-1] : [[2], [1, 1]] Qxi[D4,7,2] = (x[8])*q^3 + (x[12])*q^2 # Green Polys by conj class in A(O) #7.1 : c = () |O_x_c^F| = 1/2*q^4*(q+1)^2*(q^2+1)*(q^4-1)*(q^6-1) Qxc[D4,7,1] = (x[7]+x[8])*q^3 + (x[9]+x[10]+x[11]+x[12])*q^2 + (x[12])*q + x[13] #7.2 : c = (1) |O_x_c^F| = 1/2*q^4*(q-1)^2*(q^2+1)*(q^4-1)*(q^6-1) Qxc[D4,7,2] = (x[7]-x[8])*q^3 + (x[9]+x[10]+x[11]-x[12])*q^2 + (x[12])*q + x[13] orbit #8 : [4, 4, 0] dim = 20 A(O) = 1 , |A(O)_0| = 1 g_s = V[6]+3*V[4]+V[2]+3*V[0] Z_G(x)_0 = Sp2 # Green Polys by orbit reps #8.1 : x[9] : [4, 4, 0],[] : [[2], [2], 2] Qxi[D4,8,1] = (x[9])*q^2 + (x[12])*q + x[13] # Green Polys by conj class in A(O) #8.1 : c = () |O_x_c^F| = q^6*(q^4-1)^2*(q^6-1) Qxc[D4,8,1] = (x[9])*q^2 + (x[12])*q + x[13] orbit #9 : [4, 4] dim = 20 A(O) = 1 , |A(O)_0| = 1 g_s = V[6]+3*V[4]+V[2]+3*V[0] Z_G(x)_0 = Sp2 # Green Polys by orbit reps #9.1 : x[10] : [4, 4],[] : [[2], [2], 1] Qxi[D4,9,1] = (x[10])*q^2 + (x[12])*q + x[13] # Green Polys by conj class in A(O) #9.1 : c = () |O_x_c^F| = q^6*(q^4-1)^2*(q^6-1) Qxc[D4,9,1] = (x[10])*q^2 + (x[12])*q + x[13] orbit #10 : [5, 1, 1, 1] dim = 20 A(O) = 1 , |A(O)_0| = 1 g_s = V[6]+3*V[4]+V[2]+3*V[0] Z_G(x)_0 = O3+O1 # Green Polys by orbit reps #10.1 : x[11] : [5, 1, 1, 1],[] : [[], [3, 1]] Qxi[D4,10,1] = (x[11])*q^2 + (x[12])*q + x[13] # Green Polys by conj class in A(O) #10.1 : c = () |O_x_c^F| = q^6*(q^4-1)^2*(q^6-1) Qxc[D4,10,1] = (x[11])*q^2 + (x[12])*q + x[13] orbit #11 : [5, 3] dim = 22 A(O) = 1 , |A(O)_0| = 1 g_s = 2*V[6]+V[4]+3*V[2] Z_G(x)_0 = 2*O1 # Green Polys by orbit reps #11.1 : x[12] : [5, 3],[] : [[1], [3]] Qxi[D4,11,1] = (x[12])*q + x[13] # Green Polys by conj class in A(O) #11.1 : c = () |O_x_c^F| = q^6*(q^2-1)*(q^4-1)^2*(q^6-1) Qxc[D4,11,1] = (x[12])*q + x[13] orbit #12 : [7, 1] dim = 24 A(O) = 1 , |A(O)_0| = 1 g_s = V[10]+2*V[6]+V[2] Z_G(x)_0 = 2*O1 # Green Polys by orbit reps #12.1 : x[13] : [7, 1],[] : [[], [4]] Qxi[D4,12,1] = x[13] # Green Polys by conj class in A(O) #12.1 : c = () |O_x_c^F| = q^8*(q^2-1)*(q^4-1)^2*(q^6-1) Qxc[D4,12,1] = x[13]