Using plethystic identities from our previous work, we establish a decomposition of the regular representation as a sum of exterior powers of the modules Lien(2). By contrast, the classical result of Thrall decomposes the regular representation into a sum of symmetric powers of the representation Lien. We show that nearly every known property of Lien in the literature appears to have a counterpart for Lien(2), suggesting connections to the cohomology of configuration spaces and other areas.
The construction of Lien(2) can be generalised to a module
LienS indexed by subsets S of distinct primes.
This in turn yields new Schur-positivity results for multiplicity-free
sums of power sums, extending our previous results.
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