Cohomology theory by S. T Hu

By S. T Hu

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Let xK = xxl - > -x^" be a monomial. , kn are equal, then the isotropy group (En)xK of xK contains the involution switching those two indices. Let Z/2 < En be the subgroup generated by this involution. Then T r ^ x * ) = T r ^ T r ^ V ) ) = T r f / 2 ( ^ +xK) = 0, since F has characteristic 2. The only monomials of degree less than or equal to (n) with all indices distinct are the n\ monomials in the En-orbit of the monomial x\ -x\ • • -x^z\. Therefore, the image of the transfer is the principal ideal generated by ir [XX -X2 ' • 'Xn-l) 2Ls X°W ^(2) ' ' 'Xd(n-l)' With the aid of the Leibniz rule for expanding determinants applied to the Vandermonde determinant, and the fact that +1 = -1 e F , one sees that x +x n ( > >y-= det \

2) and the radical of Im(Tr ) is also invariant under Steenrod operations, there are only a small number, among all possibilities, for y I m ( T r ). The interaction between the Dickson algebra and the Steenrod operations also shows that some power of the top-degree Dickson polynomial d^o always belongs to the image of the transfer, whereas if the characteristic of ¥q divides the order of G, no power of the bottom Dickson class An>n-\ does. We will examine these results about invariant ideals in Chapter 9.

368], [130], as well as the references already cited. EXAMPLE 3 : Consider the dihedral group Dlk of order 2k represented in GL(2, R) as the group of symmetries of a regular £-gon centered at the origin. This representation is orthogonal, so we may identify it with its own dual. In this representation the group D2k is generated by the matrices D cos -y -sin-T^ _ sin ^ cos ^ andA Q S = [ Q1 0_ 1 1L where D is a rotation through 2u/k radians and S is a reflection in an axis. Thus the elements of D2k are the identity, the k - 1 rotations D^ i = 1 , .

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