#### Green Polynomials for C4 #### 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, 1, 1, 1, 1, 1, 1] , A-rep = [] # x[3] = [[1], [1, 1, 1]] , orbit = [2, 2, 1, 1, 1, 1] , A-rep = [1] # x[4] = [[], [2, 1, 1]] , orbit = [2, 2, 1, 1, 1, 1] , A-rep = [-1] # x[5] = [[1, 1, 1], [1]] , orbit = [2, 2, 2, 1, 1] , A-rep = [] # x[6] = [[1, 1], [1, 1]] , orbit = [2, 2, 2, 2] , A-rep = [1] # x[7] = [[], [2, 2]] , orbit = [2, 2, 2, 2] , A-rep = [-1] # x[8] = [[1], [2, 1]] , orbit = [3, 3, 1, 1] , A-rep = [] # x[9] = [[1, 1], [2]] , orbit = [3, 3, 2] , A-rep = [] # x[10] = [[2, 1, 1], []] , orbit = [4, 1, 1, 1, 1] , A-rep = [] # x[11] = [[2], [1, 1]] , orbit = [4, 2, 1, 1] , A-rep = [1] # x[12] = [[], [3, 1]] , orbit = [4, 2, 1, 1] , A-rep = [-1] # x[13] = [[2, 1], [1]] , orbit = [4, 2, 2] , A-rep = [1] # x[14] = [[2, 2], []] , orbit = [4, 2, 2] , A-rep = [-1] # x[15] = [[2], [2]] , orbit = [4, 4] , A-rep = [1] # x[16] = [[1], [3]] , orbit = [4, 4] , A-rep = [-1] # x[17] = [[3, 1], []] , orbit = [6, 1, 1] , A-rep = [] # x[18] = [[3], [1]] , orbit = [6, 2] , A-rep = [1] # x[19] = [[], [4]] , orbit = [6, 2] , A-rep = [-1] # x[20] = [[4], []] , orbit = [8] , 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 = 36*V[0] Z_G(x)_0 = C4 # Green Polys by orbit reps #1.1 : x[1] : [1, 1, 1, 1, 1, 1, 1, 1],[] : [[4], []] Qxi[C4,1,1] = (x[1])*q^16 + (x[3])*q^15 + (x[4]+x[6])*q^14 + (x[3]+x[5]+x[8])*q^13 + (x[2]+x[4]+x[6]+x[7]+x[9]+x[11])*q^12 + (x[3]+x[5]+2*x[8]+x[13])*q^11 + (x[4]+2*x[6]+x[9]+x[10]+x[11]+x[12]+x[15])*q^10 + (x[3]+x[5]+2*x[8]+2*x[13]+x[16])*q^9 + (x[6]+x[7]+2*x[9]+x[10]+2*x[11]+x[12]+x[14]+x[15])*q^8 + (x[5]+2*x[8]+2*x[13]+x[16]+x[18])*q^7 + (x[6]+x[9]+x[10]+x[11]+x[12]+2*x[15]+x[17])*q^6 + (x[8]+2*x[13]+x[16]+x[18])*q^5 + (x[9]+x[11]+x[14]+x[15]+x[17]+x[19])*q^4 + (x[13]+x[16]+x[18])*q^3 + (x[15]+x[17])*q^2 + (x[18])*q + x[20] # Green Polys by conj class in A(O) #1.1 : c = () |O_x_c^F| = 1 Qxc[C4,1,1] = (x[1])*q^16 + (x[3])*q^15 + (x[4]+x[6])*q^14 + (x[3]+x[5]+x[8])*q^13 + (x[2]+x[4]+x[6]+x[7]+x[9]+x[11])*q^12 + (x[3]+x[5]+2*x[8]+x[13])*q^11 + (x[4]+2*x[6]+x[9]+x[10]+x[11]+x[12]+x[15])*q^10 + (x[3]+x[5]+2*x[8]+2*x[13]+x[16])*q^9 + (x[6]+x[7]+2*x[9]+x[10]+2*x[11]+x[12]+x[14]+x[15])*q^8 + (x[5]+2*x[8]+2*x[13]+x[16]+x[18])*q^7 + (x[6]+x[9]+x[10]+x[11]+x[12]+2*x[15]+x[17])*q^6 + (x[8]+2*x[13]+x[16]+x[18])*q^5 + (x[9]+x[11]+x[14]+x[15]+x[17]+x[19])*q^4 + (x[13]+x[16]+x[18])*q^3 + (x[15]+x[17])*q^2 + (x[18])*q + x[20] orbit #2 : [2, 1, 1, 1, 1, 1, 1] dim = 8 A(O) = 1 , |A(O)_0| = 1 g_s = 6*V[1]+V[2]+21*V[0] Z_G(x)_0 = O1+Sp6 # Green Polys by orbit reps #2.1 : x[2] : [2, 1, 1, 1, 1, 1, 1],[] : [[1, 1, 1, 1], []] Qxi[C4,2,1] = (x[2])*q^12 + (x[5])*q^11 + (x[6]+x[10])*q^10 + (x[3]+x[5]+x[13])*q^9 + (x[6]+x[9]+x[10]+x[11]+x[14])*q^8 + (x[5]+x[8]+2*x[13])*q^7 + (x[6]+x[9]+x[10]+x[11]+x[15]+x[17])*q^6 + (x[8]+2*x[13]+x[18])*q^5 + (x[9]+x[11]+x[14]+x[15]+x[17])*q^4 + (x[13]+x[16]+x[18])*q^3 + (x[15]+x[17])*q^2 + (x[18])*q + x[20] # Green Polys by conj class in A(O) #2.1 : c = () |O_x_c^F| = q^8-1 Qxc[C4,2,1] = (x[2])*q^12 + (x[5])*q^11 + (x[6]+x[10])*q^10 + (x[3]+x[5]+x[13])*q^9 + (x[6]+x[9]+x[10]+x[11]+x[14])*q^8 + (x[5]+x[8]+2*x[13])*q^7 + (x[6]+x[9]+x[10]+x[11]+x[15]+x[17])*q^6 + (x[8]+2*x[13]+x[18])*q^5 + (x[9]+x[11]+x[14]+x[15]+x[17])*q^4 + (x[13]+x[16]+x[18])*q^3 + (x[15]+x[17])*q^2 + (x[18])*q + x[20] orbit #3 : [2, 2, 1, 1, 1, 1] dim = 14 A(O) = Z2 , |A(O)_0| = 2 g_s = 8*V[1]+3*V[2]+11*V[0] Z_G(x)_0 = Sp4+O2 # Green Polys by orbit reps #3.1 : x[3] : [2, 2, 1, 1, 1, 1],[1] : [[1], [1, 1, 1]] Qxi[C4,3,1] = (x[3])*q^9 + (x[6]+x[11])*q^8 + (x[5]+x[8]+x[13])*q^7 + (x[6]+x[9]+x[10]+x[11]+x[15])*q^6 + (x[8]+2*x[13]+x[18])*q^5 + (x[9]+x[11]+x[14]+x[15]+x[17])*q^4 + (x[13]+x[16]+x[18])*q^3 + (x[15]+x[17])*q^2 + (x[18])*q + x[20] #3.2 : x[4] : [2, 2, 1, 1, 1, 1],[-1] : [[], [2, 1, 1]] Qxi[C4,3,2] = (x[4])*q^9 + (x[8])*q^8 + (x[7]+x[9]+x[12])*q^7 + (x[8]+x[16])*q^6 + (x[12]+x[15])*q^5 + (x[16])*q^4 + (x[19])*q^3 # Green Polys by conj class in A(O) #3.1 : c = () |O_x_c^F| = 1/2*q*(q^2+q+1)*(q^3+1)*(q^8-1) Qxc[C4,3,1] = (x[3]+x[4])*q^9 + (x[6]+x[8]+x[11])*q^8 + (x[5]+x[7]+x[8]+x[9]+x[12]+x[13])*q^7 + (x[6]+x[8]+x[9]+x[10]+x[11]+x[15]+x[16])*q^6 + (x[8]+x[12]+2*x[13]+x[15]+x[18])*q^5 + (x[9]+x[11]+x[14]+x[15]+x[16]+x[17])*q^4 + (x[13]+x[16]+x[18]+x[19])*q^3 + (x[15]+x[17])*q^2 + (x[18])*q + x[20] #3.2 : c = (1) |O_x_c^F| = 1/2*q*(q^2-q+1)*(q^3-1)*(q^8-1) Qxc[C4,3,2] = (x[3]-x[4])*q^9 + (x[6]-x[8]+x[11])*q^8 + (x[5]-x[7]+x[8]-x[9]-x[12]+x[13])*q^7 + (x[6]-x[8]+x[9]+x[10]+x[11]+x[15]-x[16])*q^6 + (x[8]-x[12]+2*x[13]-x[15]+x[18])*q^5 + (x[9]+x[11]+x[14]+x[15]-x[16]+x[17])*q^4 + (x[13]+x[16]+x[18]-x[19])*q^3 + (x[15]+x[17])*q^2 + (x[18])*q + x[20] orbit #4 : [2, 2, 2, 1, 1] dim = 18 A(O) = 1 , |A(O)_0| = 1 g_s = 6*V[2]+6*V[1]+6*V[0] Z_G(x)_0 = O3+Sp2 # Green Polys by orbit reps #4.1 : x[5] : [2, 2, 2, 1, 1],[] : [[1, 1, 1], [1]] Qxi[C4,4,1] = (x[5])*q^7 + (x[6]+x[9]+x[10])*q^6 + (x[8]+2*x[13])*q^5 + (x[9]+x[11]+x[14]+x[15]+x[17])*q^4 + (x[13]+x[16]+x[18])*q^3 + (x[15]+x[17])*q^2 + (x[18])*q + x[20] # Green Polys by conj class in A(O) #4.1 : c = () |O_x_c^F| = q^2*(q^2+1)*(q^6-1)*(q^8-1) Qxc[C4,4,1] = (x[5])*q^7 + (x[6]+x[9]+x[10])*q^6 + (x[8]+2*x[13])*q^5 + (x[9]+x[11]+x[14]+x[15]+x[17])*q^4 + (x[13]+x[16]+x[18])*q^3 + (x[15]+x[17])*q^2 + (x[18])*q + x[20] orbit #5 : [2, 2, 2, 2] dim = 20 A(O) = Z2 , |A(O)_0| = 2 g_s = 10*V[2]+6*V[0] Z_G(x)_0 = O4 # Green Polys by orbit reps #5.1 : x[6] : [2, 2, 2, 2],[1] : [[1, 1], [1, 1]] Qxi[C4,5,1] = (x[6])*q^6 + (x[8]+x[13])*q^5 + (x[9]+x[11]+x[14]+x[15])*q^4 + (x[13]+x[16]+x[18])*q^3 + (x[15]+x[17])*q^2 + (x[18])*q + x[20] #5.2 : x[7] : [2, 2, 2, 2],[-1] : [[], [2, 2]] Qxi[C4,5,2] = (x[7])*q^6 + (x[8])*q^5 + (x[12]+x[15])*q^4 + (x[16])*q^3 + (x[19])*q^2 # Green Polys by conj class in A(O) #5.1 : c = () |O_x_c^F| = 1/2*q^4*(q^2+1)*(q^6-1)*(q^8-1) Qxc[C4,5,1] = (x[6]+x[7])*q^6 + (2*x[8]+x[13])*q^5 + (x[9]+x[11]+x[12]+x[14]+2*x[15])*q^4 + (x[13]+2*x[16]+x[18])*q^3 + (x[15]+x[17]+x[19])*q^2 + (x[18])*q + x[20] #5.2 : c = (1) |O_x_c^F| = 1/2*q^4*(q^2-1)*(q^6-1)*(q^8-1) Qxc[C4,5,2] = (x[6]-x[7])*q^6 + (x[13])*q^5 + (x[9]+x[11]-x[12]+x[14])*q^4 + (x[13]+x[18])*q^3 + (x[15]+x[17]-x[19])*q^2 + (x[18])*q + x[20] orbit #6 : [3, 3, 1, 1] dim = 22 A(O) = 1 , |A(O)_0| = 1 g_s = 3*V[4]+5*V[2]+6*V[0] Z_G(x)_0 = 2*Sp2 # Green Polys by orbit reps #6.1 : x[8] : [3, 3, 1, 1],[] : [[1], [2, 1]] Qxi[C4,6,1] = (x[8])*q^5 + (x[9]+x[11]+x[12]+x[15])*q^4 + (x[13]+2*x[16]+x[18])*q^3 + (x[15]+x[17]+x[19])*q^2 + (x[18])*q + x[20] # Green Polys by conj class in A(O) #6.1 : c = () |O_x_c^F| = q^6*(q^2+1)*(q^6-1)*(q^8-1) Qxc[C4,6,1] = (x[8])*q^5 + (x[9]+x[11]+x[12]+x[15])*q^4 + (x[13]+2*x[16]+x[18])*q^3 + (x[15]+x[17]+x[19])*q^2 + (x[18])*q + x[20] orbit #7 : [3, 3, 2] dim = 24 A(O) = 1 , |A(O)_0| = 1 g_s = 3*V[4]+2*V[3]+2*V[2]+2*V[1]+3*V[0] Z_G(x)_0 = O1+Sp2 # Green Polys by orbit reps #7.1 : x[9] : [3, 3, 2],[] : [[1, 1], [2]] Qxi[C4,7,1] = (x[9])*q^4 + (x[13]+x[16])*q^3 + (x[15]+x[17])*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[C4,7,1] = (x[9])*q^4 + (x[13]+x[16])*q^3 + (x[15]+x[17])*q^2 + (x[18])*q + x[20] orbit #8 : [4, 1, 1, 1, 1] dim = 20 A(O) = 1 , |A(O)_0| = 1 g_s = V[6]+4*V[3]+V[2]+10*V[0] Z_G(x)_0 = O1+Sp4 # Green Polys by orbit reps #8.1 : x[10] : [4, 1, 1, 1, 1],[] : [[2, 1, 1], []] Qxi[C4,8,1] = (x[10])*q^6 + (x[13])*q^5 + (x[11]+x[14]+x[17])*q^4 + (x[13]+x[18])*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^6*(q^6-1)*(q^8-1) Qxc[C4,8,1] = (x[10])*q^6 + (x[13])*q^5 + (x[11]+x[14]+x[17])*q^4 + (x[13]+x[18])*q^3 + (x[15]+x[17])*q^2 + (x[18])*q + x[20] orbit #9 : [4, 2, 1, 1] dim = 24 A(O) = Z2 , |A(O)_0| = 2 g_s = V[6]+V[4]+2*V[3]+3*V[2]+2*V[1]+3*V[0] Z_G(x)_0 = Sp2+2*O1 # Green Polys by orbit reps #9.1 : x[11] : [4, 2, 1, 1],[1] : [[2], [1, 1]] Qxi[C4,9,1] = (x[11])*q^4 + (x[13]+x[18])*q^3 + (x[15]+x[17])*q^2 + (x[18])*q + x[20] #9.2 : x[12] : [4, 2, 1, 1],[-1] : [[], [3, 1]] Qxi[C4,9,2] = (x[12])*q^4 + (x[16])*q^3 + (x[19])*q^2 # Green Polys by conj class in A(O) #9.1 : c = () |O_x_c^F| = 1/2*q^6*(q^4-1)*(q^6-1)*(q^8-1) Qxc[C4,9,1] = (x[11]+x[12])*q^4 + (x[13]+x[16]+x[18])*q^3 + (x[15]+x[17]+x[19])*q^2 + (x[18])*q + x[20] #9.2 : c = (1) |O_x_c^F| = 1/2*q^6*(q^4-1)*(q^6-1)*(q^8-1) Qxc[C4,9,2] = (x[11]-x[12])*q^4 + (x[13]-x[16]+x[18])*q^3 + (x[15]+x[17]-x[19])*q^2 + (x[18])*q + x[20] orbit #10 : [4, 2, 2] dim = 26 A(O) = Z2 , |A(O)_0| = 2 g_s = V[6]+2*V[4]+6*V[2]+V[0] Z_G(x)_0 = O2+O1 # Green Polys by orbit reps #10.1 : x[13] : [4, 2, 2],[1] : [[2, 1], [1]] Qxi[C4,10,1] = (x[13])*q^3 + (x[15]+x[17])*q^2 + (x[18])*q + x[20] #10.2 : x[14] : [4, 2, 2],[-1] : [[2, 2], []] Qxi[C4,10,2] = (x[14])*q^3 # Green Polys by conj class in A(O) #10.1 : c = () |O_x_c^F| = 1/2*q^7*(q+1)*(q^4-1)*(q^6-1)*(q^8-1) Qxc[C4,10,1] = (x[13]+x[14])*q^3 + (x[15]+x[17])*q^2 + (x[18])*q + x[20] #10.2 : c = (1) |O_x_c^F| = 1/2*q^7*(q-1)*(q^4-1)*(q^6-1)*(q^8-1) Qxc[C4,10,2] = (x[13]-x[14])*q^3 + (x[15]+x[17])*q^2 + (x[18])*q + x[20] orbit #11 : [4, 4] dim = 28 A(O) = Z2 , |A(O)_0| = 2 g_s = 3*V[6]+V[4]+3*V[2]+V[0] Z_G(x)_0 = O2 # Green Polys by orbit reps #11.1 : x[15] : [4, 4],[1] : [[2], [2]] Qxi[C4,11,1] = (x[15])*q^2 + (x[18])*q + x[20] #11.2 : x[16] : [4, 4],[-1] : [[1], [3]] Qxi[C4,11,2] = (x[16])*q^2 + (x[19])*q # Green Polys by conj class in A(O) #11.1 : c = () |O_x_c^F| = 1/2*q^9*(q+1)*(q^4-1)*(q^6-1)*(q^8-1) Qxc[C4,11,1] = (x[15]+x[16])*q^2 + (x[18]+x[19])*q + x[20] #11.2 : c = (1) |O_x_c^F| = 1/2*q^9*(q-1)*(q^4-1)*(q^6-1)*(q^8-1) Qxc[C4,11,2] = (x[15]-x[16])*q^2 + (x[18]-x[19])*q + x[20] orbit #12 : [6, 1, 1] dim = 28 A(O) = 1 , |A(O)_0| = 1 g_s = V[10]+V[6]+2*V[5]+V[2]+3*V[0] Z_G(x)_0 = O1+Sp2 # Green Polys by orbit reps #12.1 : x[17] : [6, 1, 1],[] : [[3, 1], []] Qxi[C4,12,1] = (x[17])*q^2 + (x[18])*q + x[20] # Green Polys by conj class in A(O) #12.1 : c = () |O_x_c^F| = q^10*(q^4-1)*(q^6-1)*(q^8-1) Qxc[C4,12,1] = (x[17])*q^2 + (x[18])*q + x[20] orbit #13 : [6, 2] dim = 30 A(O) = Z2 , |A(O)_0| = 2 g_s = V[10]+2*V[6]+V[4]+2*V[2] Z_G(x)_0 = 2*O1 # Green Polys by orbit reps #13.1 : x[18] : [6, 2],[1] : [[3], [1]] Qxi[C4,13,1] = (x[18])*q + x[20] #13.2 : x[19] : [6, 2],[-1] : [[], [4]] Qxi[C4,13,2] = (x[19])*q # Green Polys by conj class in A(O) #13.1 : c = () |O_x_c^F| = 1/2*q^10*(q^2-1)*(q^4-1)*(q^6-1)*(q^8-1) Qxc[C4,13,1] = (x[18]+x[19])*q + x[20] #13.2 : c = (1) |O_x_c^F| = 1/2*q^10*(q^2-1)*(q^4-1)*(q^6-1)*(q^8-1) Qxc[C4,13,2] = (x[18]-x[19])*q + x[20] orbit #14 : [8] 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 #14.1 : x[20] : [8],[] : [[4], []] Qxi[C4,14,1] = x[20] # Green Polys by conj class in A(O) #14.1 : c = () |O_x_c^F| = q^12*(q^2-1)*(q^4-1)*(q^6-1)*(q^8-1) Qxc[C4,14,1] = x[20]