Second and fourth indices of plethysms

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Abstract
The direct product of several copies of a representation decomposes into a direct sum of components each with a definite permutation symmetry. The decomposition of any of the components into a direct sum of irreducible representations is the computation of a plethysm. The decomposition is often simply effected when the dimension and its analogs, the second and fourth indices of the plethysm, are known. The paper contains formulas for second and fourth indices of many specific plethysms as well as a prescription for the general plethysm. The same formula is valid for the plethysm based on any finite representation of any semisimple Lie algebra. Applications are illustrated by decomposition of all plethysms of degree 3 based on the E8 representation of dimension 3875; all fourth‐degree E8‐scalars are enumerated.

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