Ph.D. Thesis - On the combinatorics of the free Lie algebra with multiple brackets

R. S. González D'León

University of Miami, June 2014.
Advisor Michelle Wachs

Abstract:

This thesis is concerned with the connection between Lie algebras with multiple brackets and the topology of partially ordered sets. From a partially ordered set (poset) one obtains a simplicial complex, called the order complex, whose faces are the chains of the poset. There is a long tradition of using topological properties of the order complex to study various geometric and algebraic structures.

It is a classical result that the multilinear component of the free Lie algebra is isomorphic (as a representation of the symmetric group) to the top (co)homology of the order complex of the proper part of the poset of partitions \(\Pi_n\) tensored with the sign representation. We generalize this result in order to study the multilinear component of the free Lie algebra on \(n\) generators with multiple compatible Lie brackets. We consider the poset of weighted partitions \(\Pi_n^w\), introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads and we introduce a new poset of weighted partitions \(\Pi_n^k\) that allows us to generalize the result. The maximal intervals of \(\Pi_n^w\) provide a generalization of the lattice \(\Pi_n\) of partitions, which we show possesses many of the well-known properties of \(\Pi_n\); the new poset \(\Pi_n^k\) is a generalization of both \(\Pi_n\) and \(\Pi_n^w\). Indeed, \(\Pi_n^1\simeq \Pi_n\) and \(\Pi_n^2\simeq \Pi_n^w\).

An important combinatorial tool for studying the topology of the order complex is provided by the theory of shellability. We prove that the poset \(\Pi_n^k\) with a top element added is EL-shellable and hence Cohen-Macaulay. This enables us in the case \(k=2\) to use the poset theoretic M"obius function to recover results of Dotsenko-Khoroshkin and Liu giving the dimension of the multilinear component of the free doubly bracketed Lie algebra \(\mathcal{L}ie_2(n)\) as \(n^{n-1}\). We show that the M"obius invariant of each maximal interval of \(\Pi_n^w\) is given up to sign by the number of rooted trees on node set \({1,2,\dots,n}\) having a fixed number of descents. Moreover, we construct a nice combinatorial basis for the homology of these intervals consisting of fundamental cycles indexed by such rooted trees, generalizing Bj"orner’s NBC basis for the homology of \(\Pi_n\). We also show that the characteristic polynomial of \(\Pi_n^w\) has a nice factorization analogous to that of \(\Pi_n\).

EL-shellability and other properties of the more general poset \(\Pi_n^k\) enable us to answer questions posed by Liu on free multibracketed Lie algebras. In particular, we obtain various dimension formulas and multicolored generalizations of the classical Lyndon and comb bases for the multilinear component of the free Lie algebra. We obtain and rely on an interesting bijection between the colored Lyndon trees and the colored combs. This bijection is a generalization of the classical bijection between the classical Lyndon trees and combs.

The multilinear component of the free multibracketed Lie algebra decomposes in a natural way into more refined components according to the number of brackets of each type used in its generators. Indeed, for a weak composition \(\mu=(\mu_1,\mu_2,\dots)\) we consider the component \(\mathcal{L}ie(\mu)\) whose generators contain \(\mu_j\) brackets of type \(j\) for each \(j\). We prove that the generating function of \(\dim \mathcal{L}ie(\mu)\) is an e-positive symmetric function, that is, it has positive coefficients in the basis of elementary symmetric functions. We give various combinatorial descriptions of the e-coefficients in terms of leaf-labeled binary trees and in terms of the Stirling permutations introduced by Gessel and Stanley.

We also use poset theoretic techniques to obtain a plethystic formula for the Frobenius characteristic of the representation of the symmetric group on the multilinear component of the free multibracketed Lie algebra.