#### Green Polynomials for A6 #### W-rep key: # x[1] = [1, 1, 1, 1, 1, 1, 1] , orbit = [1, 1, 1, 1, 1, 1, 1] , A-rep = [] # x[2] = [2, 1, 1, 1, 1, 1] , orbit = [2, 1, 1, 1, 1, 1] , A-rep = [] # x[3] = [2, 2, 1, 1, 1] , orbit = [2, 2, 1, 1, 1] , A-rep = [] # x[4] = [2, 2, 2, 1] , orbit = [2, 2, 2, 1] , A-rep = [] # x[5] = [3, 1, 1, 1, 1] , orbit = [3, 1, 1, 1, 1] , A-rep = [] # x[6] = [3, 2, 1, 1] , orbit = [3, 2, 1, 1] , A-rep = [] # x[7] = [3, 2, 2] , orbit = [3, 2, 2] , A-rep = [] # x[8] = [3, 3, 1] , orbit = [3, 3, 1] , A-rep = [] # x[9] = [4, 1, 1, 1] , orbit = [4, 1, 1, 1] , A-rep = [] # x[10] = [4, 2, 1] , orbit = [4, 2, 1] , A-rep = [] # x[11] = [4, 3] , orbit = [4, 3] , A-rep = [] # x[12] = [5, 1, 1] , orbit = [5, 1, 1] , A-rep = [] # x[13] = [5, 2] , orbit = [5, 2] , A-rep = [] # x[14] = [6, 1] , orbit = [6, 1] , A-rep = [] # x[15] = [7] , orbit = [7] , A-rep = [] ### Green Polynomials by Orbit orbit #1 : [1, 1, 1, 1, 1, 1, 1] dim = 0 A(O) = 1 , |A(O)_0| = 1 g_s = 48*V[0] Z_G(x)_0 = A6 # Green Polys by orbit reps #1.1 : x[1] : [1, 1, 1, 1, 1, 1, 1],[] : [7] Qxi[A6,1,1] = (x[1])*q^21 + (x[2])*q^20 + (x[2]+x[3])*q^19 + (x[2]+x[3]+x[4]+x[5])*q^18 + (x[2]+2*x[3]+x[4]+x[5]+x[6])*q^17 + (x[2]+2*x[3]+x[4]+2*x[5]+2*x[6]+x[7])*q^16 + (x[2]+2*x[3]+2*x[4]+2*x[5]+3*x[6]+x[7]+x[8]+x[9])*q^15 + (2*x[3]+2*x[4]+3*x[5]+4*x[6]+2*x[7]+x[8]+x[9]+x[10])*q^14 + (2*x[3]+2*x[4]+2*x[5]+5*x[6]+2*x[7]+2*x[8]+2*x[9]+2*x[10])*q^13 + (x[3]+2*x[4]+2*x[5]+5*x[6]+3*x[7]+2*x[8]+3*x[9]+3*x[10]+x[11])*q^12 + (x[3]+x[4]+x[5]+5*x[6]+3*x[7]+3*x[8]+3*x[9]+4*x[10]+x[11]+x[12])*q^11 + (x[4]+x[5]+4*x[6]+3*x[7]+3*x[8]+3*x[9]+5*x[10]+x[11]+x[12]+x[13])*q^10 + (x[4]+3*x[6]+2*x[7]+3*x[8]+3*x[9]+5*x[10]+2*x[11]+2*x[12]+x[13])*q^9 + (2*x[6]+2*x[7]+2*x[8]+2*x[9]+5*x[10]+2*x[11]+2*x[12]+2*x[13])*q^8 + (x[6]+x[7]+2*x[8]+x[9]+4*x[10]+2*x[11]+3*x[12]+2*x[13])*q^7 + (x[7]+x[8]+x[9]+3*x[10]+2*x[11]+2*x[12]+2*x[13]+x[14])*q^6 + (x[8]+2*x[10]+x[11]+2*x[12]+2*x[13]+x[14])*q^5 + (x[10]+x[11]+x[12]+2*x[13]+x[14])*q^4 + (x[11]+x[12]+x[13]+x[14])*q^3 + (x[13]+x[14])*q^2 + (x[14])*q + x[15] # Green Polys by conj class in A(O) #1.1 : c = () |O_x_c^F| = 1 Qxc[A6,1,1] = (x[1])*q^21 + (x[2])*q^20 + (x[2]+x[3])*q^19 + (x[2]+x[3]+x[4]+x[5])*q^18 + (x[2]+2*x[3]+x[4]+x[5]+x[6])*q^17 + (x[2]+2*x[3]+x[4]+2*x[5]+2*x[6]+x[7])*q^16 + (x[2]+2*x[3]+2*x[4]+2*x[5]+3*x[6]+x[7]+x[8]+x[9])*q^15 + (2*x[3]+2*x[4]+3*x[5]+4*x[6]+2*x[7]+x[8]+x[9]+x[10])*q^14 + (2*x[3]+2*x[4]+2*x[5]+5*x[6]+2*x[7]+2*x[8]+2*x[9]+2*x[10])*q^13 + (x[3]+2*x[4]+2*x[5]+5*x[6]+3*x[7]+2*x[8]+3*x[9]+3*x[10]+x[11])*q^12 + (x[3]+x[4]+x[5]+5*x[6]+3*x[7]+3*x[8]+3*x[9]+4*x[10]+x[11]+x[12])*q^11 + (x[4]+x[5]+4*x[6]+3*x[7]+3*x[8]+3*x[9]+5*x[10]+x[11]+x[12]+x[13])*q^10 + (x[4]+3*x[6]+2*x[7]+3*x[8]+3*x[9]+5*x[10]+2*x[11]+2*x[12]+x[13])*q^9 + (2*x[6]+2*x[7]+2*x[8]+2*x[9]+5*x[10]+2*x[11]+2*x[12]+2*x[13])*q^8 + (x[6]+x[7]+2*x[8]+x[9]+4*x[10]+2*x[11]+3*x[12]+2*x[13])*q^7 + (x[7]+x[8]+x[9]+3*x[10]+2*x[11]+2*x[12]+2*x[13]+x[14])*q^6 + (x[8]+2*x[10]+x[11]+2*x[12]+2*x[13]+x[14])*q^5 + (x[10]+x[11]+x[12]+2*x[13]+x[14])*q^4 + (x[11]+x[12]+x[13]+x[14])*q^3 + (x[13]+x[14])*q^2 + (x[14])*q + x[15] orbit #2 : [2, 1, 1, 1, 1, 1] dim = 12 A(O) = 1 , |A(O)_0| = 1 g_s = 10*V[1]+V[2]+25*V[0] Z_G(x)_0 = GL5+GL1 # Green Polys by orbit reps #2.1 : x[2] : [2, 1, 1, 1, 1, 1],[] : [2, 1, 1, 1, 1, 1] Qxi[A6,2,1] = (x[2])*q^15 + (x[3]+x[5])*q^14 + (x[3]+x[4]+x[5]+x[6])*q^13 + (x[3]+x[4]+x[5]+2*x[6]+x[7]+x[9])*q^12 + (x[3]+x[4]+x[5]+3*x[6]+x[7]+x[8]+x[9]+x[10])*q^11 + (x[4]+x[5]+3*x[6]+2*x[7]+2*x[8]+2*x[9]+2*x[10])*q^10 + (x[4]+3*x[6]+2*x[7]+2*x[8]+2*x[9]+3*x[10]+x[11]+x[12])*q^9 + (2*x[6]+2*x[7]+2*x[8]+2*x[9]+4*x[10]+x[11]+x[12]+x[13])*q^8 + (x[6]+x[7]+2*x[8]+x[9]+4*x[10]+2*x[11]+2*x[12]+x[13])*q^7 + (x[7]+x[8]+x[9]+3*x[10]+2*x[11]+2*x[12]+2*x[13])*q^6 + (x[8]+2*x[10]+x[11]+2*x[12]+2*x[13]+x[14])*q^5 + (x[10]+x[11]+x[12]+2*x[13]+x[14])*q^4 + (x[11]+x[12]+x[13]+x[14])*q^3 + (x[13]+x[14])*q^2 + (x[14])*q + x[15] # Green Polys by conj class in A(O) #2.1 : c = () |O_x_c^F| = (q^2+q+1)*(q^3+1)*(q^7-1) Qxc[A6,2,1] = (x[2])*q^15 + (x[3]+x[5])*q^14 + (x[3]+x[4]+x[5]+x[6])*q^13 + (x[3]+x[4]+x[5]+2*x[6]+x[7]+x[9])*q^12 + (x[3]+x[4]+x[5]+3*x[6]+x[7]+x[8]+x[9]+x[10])*q^11 + (x[4]+x[5]+3*x[6]+2*x[7]+2*x[8]+2*x[9]+2*x[10])*q^10 + (x[4]+3*x[6]+2*x[7]+2*x[8]+2*x[9]+3*x[10]+x[11]+x[12])*q^9 + (2*x[6]+2*x[7]+2*x[8]+2*x[9]+4*x[10]+x[11]+x[12]+x[13])*q^8 + (x[6]+x[7]+2*x[8]+x[9]+4*x[10]+2*x[11]+2*x[12]+x[13])*q^7 + (x[7]+x[8]+x[9]+3*x[10]+2*x[11]+2*x[12]+2*x[13])*q^6 + (x[8]+2*x[10]+x[11]+2*x[12]+2*x[13]+x[14])*q^5 + (x[10]+x[11]+x[12]+2*x[13]+x[14])*q^4 + (x[11]+x[12]+x[13]+x[14])*q^3 + (x[13]+x[14])*q^2 + (x[14])*q + x[15] orbit #3 : [2, 2, 1, 1, 1] dim = 20 A(O) = 1 , |A(O)_0| = 1 g_s = 4*V[2]+12*V[1]+12*V[0] Z_G(x)_0 = GL3+GL2 # Green Polys by orbit reps #3.1 : x[3] : [2, 2, 1, 1, 1],[] : [2, 2, 1, 1, 1] Qxi[A6,3,1] = (x[3])*q^11 + (x[4]+x[5]+x[6])*q^10 + (x[4]+2*x[6]+x[7]+x[8]+x[9])*q^9 + (2*x[6]+2*x[7]+x[8]+x[9]+2*x[10])*q^8 + (x[6]+x[7]+2*x[8]+x[9]+3*x[10]+x[11]+x[12])*q^7 + (x[7]+x[8]+x[9]+3*x[10]+2*x[11]+x[12]+x[13])*q^6 + (x[8]+2*x[10]+x[11]+2*x[12]+2*x[13])*q^5 + (x[10]+x[11]+x[12]+2*x[13]+x[14])*q^4 + (x[11]+x[12]+x[13]+x[14])*q^3 + (x[13]+x[14])*q^2 + (x[14])*q + x[15] # Green Polys by conj class in A(O) #3.1 : c = () |O_x_c^F| = q*(q^4+q^3+q^2+q+1)*(q^2+1)*(q^6-1)*(q^7-1) Qxc[A6,3,1] = (x[3])*q^11 + (x[4]+x[5]+x[6])*q^10 + (x[4]+2*x[6]+x[7]+x[8]+x[9])*q^9 + (2*x[6]+2*x[7]+x[8]+x[9]+2*x[10])*q^8 + (x[6]+x[7]+2*x[8]+x[9]+3*x[10]+x[11]+x[12])*q^7 + (x[7]+x[8]+x[9]+3*x[10]+2*x[11]+x[12]+x[13])*q^6 + (x[8]+2*x[10]+x[11]+2*x[12]+2*x[13])*q^5 + (x[10]+x[11]+x[12]+2*x[13]+x[14])*q^4 + (x[11]+x[12]+x[13]+x[14])*q^3 + (x[13]+x[14])*q^2 + (x[14])*q + x[15] orbit #4 : [2, 2, 2, 1] dim = 24 A(O) = 1 , |A(O)_0| = 1 g_s = 9*V[2]+6*V[1]+9*V[0] Z_G(x)_0 = GL3+GL1 # Green Polys by orbit reps #4.1 : x[4] : [2, 2, 2, 1],[] : [2, 2, 2, 1] Qxi[A6,4,1] = (x[4])*q^9 + (x[6]+x[7])*q^8 + (x[6]+x[7]+x[8]+x[10])*q^7 + (x[7]+x[8]+x[9]+2*x[10]+x[11])*q^6 + (x[8]+2*x[10]+x[11]+x[12]+x[13])*q^5 + (x[10]+x[11]+x[12]+2*x[13])*q^4 + (x[11]+x[12]+x[13]+x[14])*q^3 + (x[13]+x[14])*q^2 + (x[14])*q + x[15] # Green Polys by conj class in A(O) #4.1 : c = () |O_x_c^F| = q^3*(q+1)*(q^2+1)*(q^5-1)*(q^6-1)*(q^7-1) Qxc[A6,4,1] = (x[4])*q^9 + (x[6]+x[7])*q^8 + (x[6]+x[7]+x[8]+x[10])*q^7 + (x[7]+x[8]+x[9]+2*x[10]+x[11])*q^6 + (x[8]+2*x[10]+x[11]+x[12]+x[13])*q^5 + (x[10]+x[11]+x[12]+2*x[13])*q^4 + (x[11]+x[12]+x[13]+x[14])*q^3 + (x[13]+x[14])*q^2 + (x[14])*q + x[15] orbit #5 : [3, 1, 1, 1, 1] dim = 22 A(O) = 1 , |A(O)_0| = 1 g_s = V[4]+9*V[2]+16*V[0] Z_G(x)_0 = GL4+GL1 # Green Polys by orbit reps #5.1 : x[5] : [3, 1, 1, 1, 1],[] : [3, 1, 1, 1, 1] Qxi[A6,5,1] = (x[5])*q^10 + (x[6]+x[9])*q^9 + (x[6]+x[7]+x[9]+x[10])*q^8 + (x[6]+x[8]+x[9]+2*x[10]+x[12])*q^7 + (x[7]+x[8]+x[9]+2*x[10]+x[11]+x[12]+x[13])*q^6 + (x[8]+2*x[10]+x[11]+2*x[12]+x[13])*q^5 + (x[10]+x[11]+x[12]+2*x[13]+x[14])*q^4 + (x[11]+x[12]+x[13]+x[14])*q^3 + (x[13]+x[14])*q^2 + (x[14])*q + x[15] # Green Polys by conj class in A(O) #5.1 : c = () |O_x_c^F| = q^5*(q^4+q^3+q^2+q+1)*(q^6-1)*(q^7-1) Qxc[A6,5,1] = (x[5])*q^10 + (x[6]+x[9])*q^9 + (x[6]+x[7]+x[9]+x[10])*q^8 + (x[6]+x[8]+x[9]+2*x[10]+x[12])*q^7 + (x[7]+x[8]+x[9]+2*x[10]+x[11]+x[12]+x[13])*q^6 + (x[8]+2*x[10]+x[11]+2*x[12]+x[13])*q^5 + (x[10]+x[11]+x[12]+2*x[13]+x[14])*q^4 + (x[11]+x[12]+x[13]+x[14])*q^3 + (x[13]+x[14])*q^2 + (x[14])*q + x[15] orbit #6 : [3, 2, 1, 1] dim = 28 A(O) = 1 , |A(O)_0| = 1 g_s = V[4]+2*V[3]+6*V[2]+6*V[1]+5*V[0] Z_G(x)_0 = GL2+2*GL1 # Green Polys by orbit reps #6.1 : x[6] : [3, 2, 1, 1],[] : [3, 2, 1, 1] Qxi[A6,6,1] = (x[6])*q^7 + (x[7]+x[8]+x[9]+x[10])*q^6 + (x[8]+2*x[10]+x[11]+x[12])*q^5 + (x[10]+x[11]+x[12]+2*x[13])*q^4 + (x[11]+x[12]+x[13]+x[14])*q^3 + (x[13]+x[14])*q^2 + (x[14])*q + x[15] # Green Polys by conj class in A(O) #6.1 : c = () |O_x_c^F| = q^5*(q^2+q+1)*(q+1)*(q^2+1)*(q^5-1)*(q^6-1)*(q^7-1) Qxc[A6,6,1] = (x[6])*q^7 + (x[7]+x[8]+x[9]+x[10])*q^6 + (x[8]+2*x[10]+x[11]+x[12])*q^5 + (x[10]+x[11]+x[12]+2*x[13])*q^4 + (x[11]+x[12]+x[13]+x[14])*q^3 + (x[13]+x[14])*q^2 + (x[14])*q + x[15] orbit #7 : [3, 2, 2] dim = 30 A(O) = 1 , |A(O)_0| = 1 g_s = V[4]+4*V[3]+5*V[2]+4*V[1]+4*V[0] Z_G(x)_0 = GL2+GL1 # Green Polys by orbit reps #7.1 : x[7] : [3, 2, 2],[] : [3, 2, 2] Qxi[A6,7,1] = (x[7])*q^6 + (x[8]+x[10])*q^5 + (x[10]+x[11]+x[13])*q^4 + (x[11]+x[12]+x[13])*q^3 + (x[13]+x[14])*q^2 + (x[14])*q + x[15] # Green Polys by conj class in A(O) #7.1 : c = () |O_x_c^F| = q^6*(q^2+q+1)*(q^4-1)*(q^5-1)*(q^6-1)*(q^7-1) Qxc[A6,7,1] = (x[7])*q^6 + (x[8]+x[10])*q^5 + (x[10]+x[11]+x[13])*q^4 + (x[11]+x[12]+x[13])*q^3 + (x[13]+x[14])*q^2 + (x[14])*q + x[15] orbit #8 : [3, 3, 1] dim = 32 A(O) = 1 , |A(O)_0| = 1 g_s = 4*V[4]+8*V[2]+4*V[0] Z_G(x)_0 = GL2+GL1 # Green Polys by orbit reps #8.1 : x[8] : [3, 3, 1],[] : [3, 3, 1] Qxi[A6,8,1] = (x[8])*q^5 + (x[10]+x[11])*q^4 + (x[11]+x[12]+x[13])*q^3 + (x[13]+x[14])*q^2 + (x[14])*q + x[15] # Green Polys by conj class in A(O) #8.1 : c = () |O_x_c^F| = q^8*(q^2+q+1)*(q^4-1)*(q^5-1)*(q^6-1)*(q^7-1) Qxc[A6,8,1] = (x[8])*q^5 + (x[10]+x[11])*q^4 + (x[11]+x[12]+x[13])*q^3 + (x[13]+x[14])*q^2 + (x[14])*q + x[15] orbit #9 : [4, 1, 1, 1] dim = 30 A(O) = 1 , |A(O)_0| = 1 g_s = V[6]+V[4]+6*V[3]+V[2]+9*V[0] Z_G(x)_0 = GL3+GL1 # Green Polys by orbit reps #9.1 : x[9] : [4, 1, 1, 1],[] : [4, 1, 1, 1] Qxi[A6,9,1] = (x[9])*q^6 + (x[10]+x[12])*q^5 + (x[10]+x[12]+x[13])*q^4 + (x[11]+x[12]+x[13]+x[14])*q^3 + (x[13]+x[14])*q^2 + (x[14])*q + x[15] # Green Polys by conj class in A(O) #9.1 : c = () |O_x_c^F| = q^9*(q+1)*(q^2+1)*(q^5-1)*(q^6-1)*(q^7-1) Qxc[A6,9,1] = (x[9])*q^6 + (x[10]+x[12])*q^5 + (x[10]+x[12]+x[13])*q^4 + (x[11]+x[12]+x[13]+x[14])*q^3 + (x[13]+x[14])*q^2 + (x[14])*q + x[15] orbit #10 : [4, 2, 1] dim = 34 A(O) = 1 , |A(O)_0| = 1 g_s = V[6]+3*V[4]+2*V[3]+4*V[2]+2*V[1]+2*V[0] Z_G(x)_0 = 3*GL1 # Green Polys by orbit reps #10.1 : x[10] : [4, 2, 1],[] : [4, 2, 1] Qxi[A6,10,1] = (x[10])*q^4 + (x[11]+x[12]+x[13])*q^3 + (x[13]+x[14])*q^2 + (x[14])*q + x[15] # Green Polys by conj class in A(O) #10.1 : c = () |O_x_c^F| = q^9*(q^2+q+1)*(q+1)*(q^4-1)*(q^5-1)*(q^6-1)*(q^7-1) Qxc[A6,10,1] = (x[10])*q^4 + (x[11]+x[12]+x[13])*q^3 + (x[13]+x[14])*q^2 + (x[14])*q + x[15] orbit #11 : [4, 3] dim = 36 A(O) = 1 , |A(O)_0| = 1 g_s = V[6]+2*V[5]+2*V[4]+2*V[3]+2*V[2]+2*V[1]+V[0] Z_G(x)_0 = 2*GL1 # Green Polys by orbit reps #11.1 : x[11] : [4, 3],[] : [4, 3] Qxi[A6,11,1] = (x[11])*q^3 + (x[13])*q^2 + (x[14])*q + x[15] # Green Polys by conj class in A(O) #11.1 : c = () |O_x_c^F| = q^10*(q+1)*(q^3-1)*(q^4-1)*(q^5-1)*(q^6-1)*(q^7-1) Qxc[A6,11,1] = (x[11])*q^3 + (x[13])*q^2 + (x[14])*q + x[15] orbit #12 : [5, 1, 1] dim = 36 A(O) = 1 , |A(O)_0| = 1 g_s = V[8]+V[6]+5*V[4]+V[2]+4*V[0] Z_G(x)_0 = GL2+GL1 # Green Polys by orbit reps #12.1 : x[12] : [5, 1, 1],[] : [5, 1, 1] Qxi[A6,12,1] = (x[12])*q^3 + (x[13]+x[14])*q^2 + (x[14])*q + x[15] # Green Polys by conj class in A(O) #12.1 : c = () |O_x_c^F| = q^12*(q^2+q+1)*(q^4-1)*(q^5-1)*(q^6-1)*(q^7-1) Qxc[A6,12,1] = (x[12])*q^3 + (x[13]+x[14])*q^2 + (x[14])*q + x[15] orbit #13 : [5, 2] dim = 38 A(O) = 1 , |A(O)_0| = 1 g_s = V[8]+V[6]+2*V[5]+V[4]+2*V[3]+2*V[2]+V[0] Z_G(x)_0 = 2*GL1 # Green Polys by orbit reps #13.1 : x[13] : [5, 2],[] : [5, 2] Qxi[A6,13,1] = (x[13])*q^2 + (x[14])*q + x[15] # Green Polys by conj class in A(O) #13.1 : c = () |O_x_c^F| = q^12*(q+1)*(q^3-1)*(q^4-1)*(q^5-1)*(q^6-1)*(q^7-1) Qxc[A6,13,1] = (x[13])*q^2 + (x[14])*q + x[15] orbit #14 : [6, 1] dim = 40 A(O) = 1 , |A(O)_0| = 1 g_s = V[10]+V[8]+V[6]+2*V[5]+V[4]+V[2]+V[0] Z_G(x)_0 = 2*GL1 # Green Polys by orbit reps #14.1 : x[14] : [6, 1],[] : [6, 1] Qxi[A6,14,1] = (x[14])*q + x[15] # Green Polys by conj class in A(O) #14.1 : c = () |O_x_c^F| = q^14*(q+1)*(q^3-1)*(q^4-1)*(q^5-1)*(q^6-1)*(q^7-1) Qxc[A6,14,1] = (x[14])*q + x[15] orbit #15 : [7] dim = 42 A(O) = 1 , |A(O)_0| = 1 g_s = V[12]+V[10]+V[8]+V[6]+V[4]+V[2] Z_G(x)_0 = GL1 # Green Polys by orbit reps #15.1 : x[15] : [7],[] : [7] Qxi[A6,15,1] = x[15] # Green Polys by conj class in A(O) #15.1 : c = () |O_x_c^F| = q^15*(q^2-1)*(q^3-1)*(q^4-1)*(q^5-1)*(q^6-1)*(q^7-1) Qxc[A6,15,1] = x[15]