Two New Characterizations of Lattice Supersolvability

Supersolvable lattices were introduced in 1972 by Stanley. Examples include finite distributive lattices, the lattice of partitions of $[n]$ and the lattice subgroups of a supersolvable group (hence the terminology.) Stanley showed that the edges of the Hasse diagram of a supersolvable lattice can be labeled to give an EL-labeling with the additional property that the labels along any maximal chain form a permutation. We call such a labeling an $S_n$ EL-labeling and we show that the converse result is true: if a finite lattice has an $S_n$ EL-labeling then it must be supersolvable.

In the second part of the talk we investigate a natural action on the maximal chains of an $S_n$ EL-labeled lattice. We show that this action gives a representation of the Hecke algebra of type $A$ at $q=0$. As a further desirable property, the character of this representation has Frobenius characteristic that is closely related to Ehrenborg's flag quasisymmetric function. We ask what other classes of lattices have an action with these properties and we show that finite graded lattices have such an action if and only if they have an $S_n$ EL-labeling.