46 research outputs found
A new subgroup lattice characterization of finite solvable groups
We show that if G is a finite group then no chain of modular elements in its
subgroup lattice L(G) is longer than a chief series. Also, we show that if G is
a nonsolvable finite group then every maximal chain in L(G) has length at least
two more than that of the chief length of G, thereby providing a converse of a
result of J. Kohler. Our results enable us to give a new characterization of
finite solvable groups involving only the combinatorics of subgroup lattices.
Namely, a finite group G is solvable if and only if L(G) contains a maximal
chain X and a chain M consisting entirely of modular elements, such that X and
M have the same length.Comment: 15 pages; v2 has minor changes for publication; v3 minor typos fixe
Chromatic quasisymmetric functions
We introduce a quasisymmetric refinement of Stanley's chromatic symmetric
function. We derive refinements of both Gasharov's Schur-basis expansion of the
chromatic symmetric function and Chow's expansion in Gessel's basis of
fundamental quasisymmetric functions. We present a conjectural refinement of
Stanley's power sum basis expansion, which we prove in special cases. We
describe connections between the chromatic quasisymmetric function and both the
-Eulerian polynomials introduced in our earlier work and, conjecturally,
representations of symmetric groups on cohomology of regular semisimple
Hessenberg varieties, which have been studied by Tymoczko and others. We
discuss an approach, using the results and conjectures herein, to the
-positivity conjecture of Stanley and Stembridge for incomparability graphs
of -free posets.Comment: 57 pages; final version, to appear in Advances in Mat
Eulerian quasisymmetric functions
We introduce a family of quasisymmetric functions called {\em Eulerian
quasisymmetric functions}, which specialize to enumerators for the joint
distribution of the permutation statistics, major index and excedance number on
permutations of fixed cycle type. This family is analogous to a family of
quasisymmetric functions that Gessel and Reutenauer used to study the joint
distribution of major index and descent number on permutations of fixed cycle
type. Our central result is a formula for the generating function for the
Eulerian quasisymmetric functions, which specializes to a new and surprising
-analog of a classical formula of Euler for the exponential generating
function of the Eulerian polynomials. This -analog computes the joint
distribution of excedance number and major index, the only of the four
important Euler-Mahonian distributions that had not yet been computed. Our
study of the Eulerian quasisymmetric functions also yields results that include
the descent statistic and refine results of Gessel and Reutenauer. We also
obtain -analogs, -analogs and quasisymmetric function analogs of
classical results on the symmetry and unimodality of the Eulerian polynomials.
Our Eulerian quasisymmetric functions refine symmetric functions that have
occurred in various representation theoretic and enumerative contexts including
MacMahon's study of multiset derangements, work of Procesi and Stanley on toric
varieties of Coxeter complexes, Stanley's work on chromatic symmetric
functions, and the work of the authors on the homology of a certain poset
introduced by Bj\"orner and Welker.Comment: Final version; to appear in Advances in Mathematics; 52 pages; this
paper was originally part of the longer paper arXiv:0805.2416v1, which has
been split into three paper