On a variant of Lie_n

This talk will discuss a curious variant of the celebrated representation Lie(n) of the symmetric group  on the multilinear component of the free Lie algebra on n generators. Introduced by the speaker a few years ago, the variant Lie(n,2) satisfies the analogue of almost every known property of Lie(n).

As one example, the exterior powers of the variant Lie(n,2) decompose the regular representation of the symmetric group. The classical free Lie algebra counterpart of this result is the 1942 theorem of Robert Thrall which also follows from the earlier Poincar\’e-Birkhoff-Witt theorem: it is the well-known decomposition of the regular representation given by the symmetrised powers of the representations Lie(n), that is, the higher Lie modules.

The talk will survey this and other properties of the variant, including some recent developments.

For information about the Richard P. Stanley Seminar in Combinatorics, visit… https://math.mit.edu/combin/