#### Green Polynomials for A5 #### W-rep key: # x[1] = [1, 1, 1, 1, 1, 1] , orbit = [1, 1, 1, 1, 1, 1] , A-rep = [] # x[2] = [2, 1, 1, 1, 1] , orbit = [2, 1, 1, 1, 1] , A-rep = [] # x[3] = [2, 2, 1, 1] , orbit = [2, 2, 1, 1] , A-rep = [] # x[4] = [2, 2, 2] , orbit = [2, 2, 2] , A-rep = [] # x[5] = [3, 1, 1, 1] , orbit = [3, 1, 1, 1] , A-rep = [] # x[6] = [3, 2, 1] , orbit = [3, 2, 1] , A-rep = [] # x[7] = [3, 3] , orbit = [3, 3] , A-rep = [] # x[8] = [4, 1, 1] , orbit = [4, 1, 1] , A-rep = [] # x[9] = [4, 2] , orbit = [4, 2] , A-rep = [] # x[10] = [5, 1] , orbit = [5, 1] , A-rep = [] # x[11] = [6] , orbit = [6] , 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 = 35*V[0] Z_G(x)_0 = A5 # Green Polys by orbit reps #1.1 : x[1] : [1, 1, 1, 1, 1, 1],[] : [6] Qxi[A5,1,1] = (x[1])*q^15 + (x[2])*q^14 + (x[2]+x[3])*q^13 + (x[2]+x[3]+x[4]+x[5])*q^12 + (x[2]+2*x[3]+x[5]+x[6])*q^11 + (x[2]+x[3]+x[4]+2*x[5]+2*x[6])*q^10 + (2*x[3]+x[4]+2*x[5]+2*x[6]+x[7]+x[8])*q^9 + (x[3]+x[4]+2*x[5]+3*x[6]+x[8]+x[9])*q^8 + (x[3]+x[5]+3*x[6]+x[7]+2*x[8]+x[9])*q^7 + (x[4]+x[5]+2*x[6]+x[7]+2*x[8]+2*x[9])*q^6 + (2*x[6]+x[7]+2*x[8]+x[9]+x[10])*q^5 + (x[6]+x[8]+2*x[9]+x[10])*q^4 + (x[7]+x[8]+x[9]+x[10])*q^3 + (x[9]+x[10])*q^2 + (x[10])*q + x[11] # Green Polys by conj class in A(O) #1.1 : c = () |O_x_c^F| = 1 Qxc[A5,1,1] = (x[1])*q^15 + (x[2])*q^14 + (x[2]+x[3])*q^13 + (x[2]+x[3]+x[4]+x[5])*q^12 + (x[2]+2*x[3]+x[5]+x[6])*q^11 + (x[2]+x[3]+x[4]+2*x[5]+2*x[6])*q^10 + (2*x[3]+x[4]+2*x[5]+2*x[6]+x[7]+x[8])*q^9 + (x[3]+x[4]+2*x[5]+3*x[6]+x[8]+x[9])*q^8 + (x[3]+x[5]+3*x[6]+x[7]+2*x[8]+x[9])*q^7 + (x[4]+x[5]+2*x[6]+x[7]+2*x[8]+2*x[9])*q^6 + (2*x[6]+x[7]+2*x[8]+x[9]+x[10])*q^5 + (x[6]+x[8]+2*x[9]+x[10])*q^4 + (x[7]+x[8]+x[9]+x[10])*q^3 + (x[9]+x[10])*q^2 + (x[10])*q + x[11] orbit #2 : [2, 1, 1, 1, 1] dim = 10 A(O) = 1 , |A(O)_0| = 1 g_s = 8*V[1]+V[2]+16*V[0] Z_G(x)_0 = GL4+GL1 # Green Polys by orbit reps #2.1 : x[2] : [2, 1, 1, 1, 1],[] : [2, 1, 1, 1, 1] Qxi[A5,2,1] = (x[2])*q^10 + (x[3]+x[5])*q^9 + (x[3]+x[4]+x[5]+x[6])*q^8 + (x[3]+x[5]+2*x[6]+x[8])*q^7 + (x[4]+x[5]+2*x[6]+x[7]+x[8]+x[9])*q^6 + (2*x[6]+x[7]+2*x[8]+x[9])*q^5 + (x[6]+x[8]+2*x[9]+x[10])*q^4 + (x[7]+x[8]+x[9]+x[10])*q^3 + (x[9]+x[10])*q^2 + (x[10])*q + x[11] # Green Polys by conj class in A(O) #2.1 : c = () |O_x_c^F| = (q^4+q^3+q^2+q+1)*(q^6-1) Qxc[A5,2,1] = (x[2])*q^10 + (x[3]+x[5])*q^9 + (x[3]+x[4]+x[5]+x[6])*q^8 + (x[3]+x[5]+2*x[6]+x[8])*q^7 + (x[4]+x[5]+2*x[6]+x[7]+x[8]+x[9])*q^6 + (2*x[6]+x[7]+2*x[8]+x[9])*q^5 + (x[6]+x[8]+2*x[9]+x[10])*q^4 + (x[7]+x[8]+x[9]+x[10])*q^3 + (x[9]+x[10])*q^2 + (x[10])*q + x[11] orbit #3 : [2, 2, 1, 1] dim = 16 A(O) = 1 , |A(O)_0| = 1 g_s = 4*V[2]+8*V[1]+7*V[0] Z_G(x)_0 = 2*GL2 # Green Polys by orbit reps #3.1 : x[3] : [2, 2, 1, 1],[] : [2, 2, 1, 1] Qxi[A5,3,1] = (x[3])*q^7 + (x[4]+x[5]+x[6])*q^6 + (2*x[6]+x[7]+x[8])*q^5 + (x[6]+x[8]+2*x[9])*q^4 + (x[7]+x[8]+x[9]+x[10])*q^3 + (x[9]+x[10])*q^2 + (x[10])*q + x[11] # Green Polys by conj class in A(O) #3.1 : c = () |O_x_c^F| = q*(q^2+q+1)*(q^2+1)*(q^5-1)*(q^6-1) Qxc[A5,3,1] = (x[3])*q^7 + (x[4]+x[5]+x[6])*q^6 + (2*x[6]+x[7]+x[8])*q^5 + (x[6]+x[8]+2*x[9])*q^4 + (x[7]+x[8]+x[9]+x[10])*q^3 + (x[9]+x[10])*q^2 + (x[10])*q + x[11] orbit #4 : [2, 2, 2] dim = 18 A(O) = 1 , |A(O)_0| = 1 g_s = 9*V[2]+8*V[0] Z_G(x)_0 = GL3 # Green Polys by orbit reps #4.1 : x[4] : [2, 2, 2],[] : [2, 2, 2] Qxi[A5,4,1] = (x[4])*q^6 + (x[6])*q^5 + (x[6]+x[9])*q^4 + (x[7]+x[8]+x[9])*q^3 + (x[9]+x[10])*q^2 + (x[10])*q + x[11] # Green Polys by conj class in A(O) #4.1 : c = () |O_x_c^F| = q^3*(q^4-1)*(q^5-1)*(q^6-1) Qxc[A5,4,1] = (x[4])*q^6 + (x[6])*q^5 + (x[6]+x[9])*q^4 + (x[7]+x[8]+x[9])*q^3 + (x[9]+x[10])*q^2 + (x[10])*q + x[11] orbit #5 : [3, 1, 1, 1] dim = 18 A(O) = 1 , |A(O)_0| = 1 g_s = V[4]+7*V[2]+9*V[0] Z_G(x)_0 = GL3+GL1 # Green Polys by orbit reps #5.1 : x[5] : [3, 1, 1, 1],[] : [3, 1, 1, 1] Qxi[A5,5,1] = (x[5])*q^6 + (x[6]+x[8])*q^5 + (x[6]+x[8]+x[9])*q^4 + (x[7]+x[8]+x[9]+x[10])*q^3 + (x[9]+x[10])*q^2 + (x[10])*q + x[11] # Green Polys by conj class in A(O) #5.1 : c = () |O_x_c^F| = q^4*(q+1)*(q^2+1)*(q^5-1)*(q^6-1) Qxc[A5,5,1] = (x[5])*q^6 + (x[6]+x[8])*q^5 + (x[6]+x[8]+x[9])*q^4 + (x[7]+x[8]+x[9]+x[10])*q^3 + (x[9]+x[10])*q^2 + (x[10])*q + x[11] orbit #6 : [3, 2, 1] dim = 22 A(O) = 1 , |A(O)_0| = 1 g_s = V[4]+2*V[3]+4*V[2]+4*V[1]+2*V[0] Z_G(x)_0 = 3*GL1 # Green Polys by orbit reps #6.1 : x[6] : [3, 2, 1],[] : [3, 2, 1] Qxi[A5,6,1] = (x[6])*q^4 + (x[7]+x[8]+x[9])*q^3 + (x[9]+x[10])*q^2 + (x[10])*q + x[11] # Green Polys by conj class in A(O) #6.1 : c = () |O_x_c^F| = q^4*(q^2+q+1)*(q+1)*(q^4-1)*(q^5-1)*(q^6-1) Qxc[A5,6,1] = (x[6])*q^4 + (x[7]+x[8]+x[9])*q^3 + (x[9]+x[10])*q^2 + (x[10])*q + x[11] orbit #7 : [3, 3] dim = 24 A(O) = 1 , |A(O)_0| = 1 g_s = 4*V[4]+4*V[2]+3*V[0] Z_G(x)_0 = GL2 # Green Polys by orbit reps #7.1 : x[7] : [3, 3],[] : [3, 3] Qxi[A5,7,1] = (x[7])*q^3 + (x[9])*q^2 + (x[10])*q + x[11] # Green Polys by conj class in A(O) #7.1 : c = () |O_x_c^F| = q^6*(q^3-1)*(q^4-1)*(q^5-1)*(q^6-1) Qxc[A5,7,1] = (x[7])*q^3 + (x[9])*q^2 + (x[10])*q + x[11] orbit #8 : [4, 1, 1] dim = 24 A(O) = 1 , |A(O)_0| = 1 g_s = V[6]+V[4]+4*V[3]+V[2]+4*V[0] Z_G(x)_0 = GL2+GL1 # Green Polys by orbit reps #8.1 : x[8] : [4, 1, 1],[] : [4, 1, 1] Qxi[A5,8,1] = (x[8])*q^3 + (x[9]+x[10])*q^2 + (x[10])*q + x[11] # Green Polys by conj class in A(O) #8.1 : c = () |O_x_c^F| = q^7*(q^2+q+1)*(q^4-1)*(q^5-1)*(q^6-1) Qxc[A5,8,1] = (x[8])*q^3 + (x[9]+x[10])*q^2 + (x[10])*q + x[11] orbit #9 : [4, 2] dim = 26 A(O) = 1 , |A(O)_0| = 1 g_s = V[6]+3*V[4]+4*V[2]+V[0] Z_G(x)_0 = 2*GL1 # Green Polys by orbit reps #9.1 : x[9] : [4, 2],[] : [4, 2] Qxi[A5,9,1] = (x[9])*q^2 + (x[10])*q + x[11] # Green Polys by conj class in A(O) #9.1 : c = () |O_x_c^F| = q^7*(q+1)*(q^3-1)*(q^4-1)*(q^5-1)*(q^6-1) Qxc[A5,9,1] = (x[9])*q^2 + (x[10])*q + x[11] orbit #10 : [5, 1] dim = 28 A(O) = 1 , |A(O)_0| = 1 g_s = V[8]+V[6]+3*V[4]+V[2]+V[0] Z_G(x)_0 = 2*GL1 # Green Polys by orbit reps #10.1 : x[10] : [5, 1],[] : [5, 1] Qxi[A5,10,1] = (x[10])*q + x[11] # Green Polys by conj class in A(O) #10.1 : c = () |O_x_c^F| = q^9*(q+1)*(q^3-1)*(q^4-1)*(q^5-1)*(q^6-1) Qxc[A5,10,1] = (x[10])*q + x[11] orbit #11 : [6] dim = 30 A(O) = 1 , |A(O)_0| = 1 g_s = V[10]+V[8]+V[6]+V[4]+V[2] Z_G(x)_0 = GL1 # Green Polys by orbit reps #11.1 : x[11] : [6],[] : [6] Qxi[A5,11,1] = x[11] # Green Polys by conj class in A(O) #11.1 : c = () |O_x_c^F| = q^10*(q^2-1)*(q^3-1)*(q^4-1)*(q^5-1)*(q^6-1) Qxc[A5,11,1] = x[11]