Some noteworthy spin plethysms

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Abstract
The spin plethysms lambda G(X) Delta that arise in the reduction of Delta under SO(N) to G when (1) to lambda G are considered. It is shown that, for the simple Lie algebras of rank k, if lambda G= phi G, the adjoint representation of G, then phi G(X) Delta =2(k2)/ delta G where delta G is the representation of G whose highest weight is half the sum of the positive roots. Certain results for other representations are described. A remarkable series of S-functions is introduced leading to a new dimensional equality between certain representations of O(2k) and Sp(2k).

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