> G := AlternatingGroup(5); > load Chern; Loading "Chern" Character Table of Group G -------------------------- --------------------------- Class | 1 2 3 4 5 Size | 1 15 20 12 12 Order | 1 2 3 5 5 --------------------------- p = 2 1 1 3 5 4 p = 3 1 2 1 5 4 p = 5 1 2 3 1 1 --------------------------- X.1 + 1 1 1 1 1 X.2 + 3 -1 0 Z1 Z1#2 X.3 + 3 -1 0 Z1#2 Z1 X.4 + 4 0 1 -1 -1 X.5 + 5 1 -1 0 0 Explanation of Symbols: ----------------------- # denotes algebraic conjugation, that is, #k indicates replacing the root of unity w by w^k Z1 = (CyclotomicField(5)) ! [ RationalField() | 1, 0, 1, 1 ] Chern classes of repn 2 i.e. ( 3, -1, 0, zeta_5^3 + zeta_5^2 + 1, -zeta_5^3 - zeta_5^2 ) c_ 1 = [ 3, -1, 0, 0, 0 ] = ( 0, 4, 3, -zeta_5^3 - zeta_5^2 + 2, zeta_5^3 + zeta_5^2 + 3 ) c_ 2 = [ 3, -1, 0, 0, 0 ] = ( 0, 4, 3, -zeta_5^3 - zeta_5^2 + 2, zeta_5^3 + zeta_5^2 + 3 ) c_ 3 = [ 0, 0, 0, 0, 0 ] = ( 0, 0, 0, 0, 0 ) Chern classes of repn 3 i.e. ( 3, -1, 0, -zeta_5^3 - zeta_5^2, zeta_5^3 + zeta_5^2 + 1 ) c_ 1 = [ 3, 0, -1, 0, 0 ] = ( 0, 4, 3, zeta_5^3 + zeta_5^2 + 3, -zeta_5^3 - zeta_5^2 + 2 ) c_ 2 = [ 3, 0, -1, 0, 0 ] = ( 0, 4, 3, zeta_5^3 + zeta_5^2 + 3, -zeta_5^3 - zeta_5^2 + 2 ) c_ 3 = [ 0, 0, 0, 0, 0 ] = ( 0, 0, 0, 0, 0 ) Chern classes of repn 4 i.e. ( 4, 0, 1, -1, -1 ) c_ 1 = [ 4, 0, 0, -1, 0 ] = ( 0, 4, 3, 5, 5 ) c_ 2 = [ 6, 1, 1, -3, 0 ] = ( 0, 4, 3, 10, 10 ) c_ 3 = [ 4, 2, 2, -4, 0 ] = ( 0, 0, 0, 10, 10 ) c_ 4 = [ 2, 1, 1, -2, 0 ] = ( 0, 0, 0, 5, 5 ) Chern classes of repn 5 i.e. ( 5, 1, -1, 0, 0 ) c_ 1 = [ 5, 0, 0, 0, -1 ] = ( 0, 4, 6, 5, 5 ) c_ 2 = [ 10, 1, 1, 1, -4 ] = ( 0, 4, 15, 10, 10 ) c_ 3 = [ 10, 2, 2, 2, -6 ] = ( 0, 0, 18, 10, 10 ) c_ 4 = [ 5, 1, 1, 1, -3 ] = ( 0, 0, 9, 5, 5 ) c_ 5 = [ 0, 0, 0, 0, 0 ] = ( 0, 0, 0, 0, 0 ) So, omitting 1st Chern classes, since these ar obvious, we find c2(X.2) = c1(X.2) = 3 - X.2 c3(X.2) = 0 c2(X.3) = c1(X.3) = 3 - X.3 c3(X.3) = 0 c2(X.4) = 6 + X.2 + X.3 - 3X.4 c3(X.4) = 4 + 2X.2 + 2X.3 - 4X.4 = 2c4(X.4) c4(X.4) = 2 + X.2 + X.3 - 2X.4 c2(X.5) = 10 + X.2 + X.3 + X.4 - 4 X.5 c3(X.5) = 10 + 2X.2 + 2X.3 + 2X.4 - 6X.5 c4(X.5) = (c3(X.5))/2 c5(X.5) = 0