> G := DihedralGroup(8); > load Chern; Loading "Chern" Character Table of Group G -------------------------- ------------------------------- Class | 1 2 3 4 5 6 7 Size | 1 1 4 4 2 2 2 Order | 1 2 2 2 4 8 8 ------------------------------- p = 2 1 1 1 1 2 5 5 ------------------------------- X.1 + 1 1 1 1 1 1 1 X.2 + 1 1 1 -1 1 -1 -1 X.3 + 1 1 -1 1 1 -1 -1 X.4 + 1 1 -1 -1 1 1 1 X.5 + 2 2 0 0 -2 0 0 X.6 + 2 -2 0 0 0 Z1 -Z1 X.7 + 2 -2 0 0 0 -Z1 Z1 Explanation of Symbols: ----------------------- # denotes algebraic conjugation, that is, #k indicates replacing the root of unity w by w^k Z1 = (CyclotomicField(8)) ! [ RationalField() | 0, 1, 0, -1 ] Chern classes of repn 2 i.e. ( 1, 1, 1, -1, 1, -1, -1 ) c_ 1 = [ 1, -1, 0, 0, 0, 0, 0 ] = ( 0, 0, 0, 2, 0, 2, 2 ) Chern classes of repn 3 i.e. ( 1, 1, -1, 1, 1, -1, -1 ) c_ 1 = [ 1, 0, -1, 0, 0, 0, 0 ] = ( 0, 0, 2, 0, 0, 2, 2 ) Chern classes of repn 4 i.e. ( 1, 1, -1, -1, 1, 1, 1 ) c_ 1 = [ 1, 0, 0, -1, 0, 0, 0 ] = ( 0, 0, 2, 2, 0, 0, 0 ) Chern classes of repn 5 i.e. ( 2, 2, 0, 0, -2, 0, 0 ) c_ 1 = [ 2, 0, 0, 0, -1, 0, 0 ] = ( 0, 0, 2, 2, 4, 2, 2 ) c_ 2 = [ 1, 0, 0, 1, -1, 0, 0 ] = ( 0, 0, 0, 0, 4, 2, 2 ) Chern classes of repn 6 i.e. ( 2, -2, 0, 0, 0, -zeta_8^3 + zeta_8, zeta_8^3 - zeta_8 ) c_ 1 = [ 2, 0, 0, 0, 0, -1, 0 ] = ( 0, 4, 2, 2, 2, zeta_8^3 - zeta_8 + 2, -zeta_8^3 + zeta_8 + 2 ) c_ 2 = [ 1, 0, 0, 1, 0, -1, 0 ] = ( 0, 4, 0, 0, 2, zeta_8^3 - zeta_8 + 2, -zeta_8^3 + zeta_8 + 2 ) Chern classes of repn 7 i.e. ( 2, -2, 0, 0, 0, zeta_8^3 - zeta_8, -zeta_8^3 + zeta_8 ) c_ 1 = [ 2, 0, 0, 0, 0, 0, -1 ] = ( 0, 4, 2, 2, 2, -zeta_8^3 + zeta_8 + 2, zeta_8^3 - zeta_8 + 2 ) c_ 2 = [ 1, 0, 0, 1, 0, 0, -1 ] = ( 0, 4, 0, 0, 2, -zeta_8^3 + zeta_8 + 2, zeta_8^3 - zeta_8 + 2 ) > The expression in square brackets gives the linear combination of irreducibles, while the expression in parentheses gives the character. In other words, c1(X.2) = 1 - X.2 c1(X.3) = 1 - X.3 c1(X.4) = 1 - X.4 c1(X.5) = 2 - X.5 c1(X.6) = 2 - X.6 c1(X.7) = 2 - X.7 (as they must: c1(V) = dim(V) - V for any V) and c2(X.5) = X.1 + X.4 - X.5 c2(X.6) = X.1 + X.4 - X.6 c2(X.7) = X.1 + X.4 - X.7