Let G be a real compact Lie group, such that G = G 0 ⋊ C 2 , with G 0 simple. Here G 0 is the connected component of G containing the identity and C 2 is the cyclic group of order 2. We give a criterion whether an orthogonal representation π : G → O(V ) lifts to Pin(V ) in terms of the highest weights of π. We also calculate the first and second Stiefel-Whitney classes of the representations of the Orthogonal groups.
We compute the total Stiefel Whitney class for a real representation [Formula: see text] of [Formula: see text], where [Formula: see text] is odd. The obstruction class of [Formula: see text] is defined to be the Stiefel Whitney class of lowest positive degree that does not vanish. We provide an expression for the obstruction class of [Formula: see text] in terms of its character values if [Formula: see text].
Fix a partition µ = (µ 1 , . . . , µ m ) of an integer k and positive integer d. For each n > k, let χ λ µ denote the value of the irreducible character of S n at a permutation with cycle type (µ 1 , . . . , µ m , 1 n−k ). We show that the proportion of partitions λ of n such that χ λ µ is divisible by d approaches 1 as n approaches infinity.2010 Mathematics Subject Classification. 20C30,05A16,05A17.
We compute the total Stiefel Whitney class for a real representation π of GLn(Fq), where q is odd in terms of character values of π on order 2 diagonal elements. We also compute the total Stiefel Whitney class of GLn(R) and GLn(C).
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