## ON THE (CO)HOMOLOGY OF THE PARTITION LATTICE AND THE FREE LIE
ALGEBRA

**
Michelle Wachs**
**Abstract:**
There is a well-known relationship between the homology of the
partition lattice and the free Lie algebra, namely that the representation of
the symmetric group on the homology of the partition lattice is isomorphic to the
representation of the symmetric group on the $1^n$ component of the free Lie
algebra tensored with the sign representation. Barcelo combinatorially explained
this result by comparing the representation matrices for a basis for
homology due to Bj\"orner with the
representation matrices for the Lyndon basis for the free Lie algebra.
In this paper we further explain the result by giving an elegant $\Cal
S_n$-module isomorphism from the cohomology of the partition lattice to the
$1^n$ component of the free Lie algebra tensored with the sign representation.
This isomorphism is described as a map from the natural defining generating set
for one module to the natural defining generating set for the other module,
rather than as a bijection between bases. The natural generating set for the
cohomology module contains the basis dual to the Bj\"orner basis and the natural
generating set for the free Lie algebra module contains the Lyndon basis.
Moreover, our isomorphism is shown to map the dual Bj\"orner basis to the Lyndon
basis thereby recovering Barcelo's result. The natural generating sets contain
other known bases for cohomology and the free Lie algebra; and we show that our
isomorphism maps the splitting basis for cohomology to the comb basis for the
free Lie algebra. A new basis for homology is introduced and studied in this
context.

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