# The conjugacy action of S_n and variations on the module  Lie_n

Sheila Sundaram, Pierrepont School
Wednesday, May 16, 2018 - 3:30pm
PDL C-401

We introduce a new \$S_n\$-module $Lie_n^{(2)}$ which interpolates between the representation \$Lie_n\$ of the symmetric group \$S_n\$ afforded by the free Lie algebra, and the module \$Conj_n\$ of the conjugacy action of \$S_n\$ on \$n\$-cycles. Using plethystic identities from previous work, we establish a decomposition of the regular representation as a sum of exterior powers of the modules \$Lie_n^{(2)}\$. By contrast, the classical result of Thrall decomposes the regular representation into a sum of symmetric powers of the representation \$Lie_n\$. We show that nearly every known property of \$Lie_n\$ in the literature appears to have a counterpart for \$Lie_n^{(2)}\$, suggesting connections to the cohomology of configuration spaces and other areas. The construction of \$Lie_n^{(2)}\$ can be generalised to a module \$Lie_n^{S}\$ indexed by subsets \$S\$ of distinct primes. This in turn yields new Schur-positivity results for multiplicity-free sums of power sums, extending previous results.

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