Eulerian actions on posets and the combinatorics of peaks
Berkeley Combinatorics Seminar
13 March 2000

Frank Sottile
University of Massachusetts-Amherst

   The algebraic combinatorics of a polytope is encoded by its flag 
f-vector, which satisfies the generalized Dehn-Sommerville relations 
of Bayer and Billera.  The flag f-vector is seen to arise from a 
combinatorial action of a free associative algebra on the face poset 
of the polytope, and the common kernel of these actions is the 
generalized Dehn-Sommerville relations.  Billera and Liu elegantly 
reformulated these subtle relations, showing that this ideal is generated 
by the obvious Euler relations, and the resulting quotient has Hilbert 
function the Fibonacci sequence.

    While studying `the combinatorics of peaks', Stembridge introduced the
Peak subalgebra of the algebra of quasi-symmetric functions, as a target
for a generating function for peak enumeration.   The Hilbert function 
of this peak algebra is also the Fibonacci sequence.

   This talk, which describes joint work with Bergeron, Mykytiuk, and 
van Willigenberg, will connect these two stories via our theory of 
combinatorial actions on graded posets and the resulting quasi-symmetric
function.  In particular, we show the peak algebra and the algebra of
Billera and Liu are dual Hopf algebras.  This leads to the notion of an 
Eulerian combinatorial action - one satisfying the generalized Dehn-
Sommerville relations.   Such Eulerian actions abound in the isotropic
Schubert calculus, polytope theory, and in the `combinatorics of peaks',
and this theory should allow us to apply techniques developed for study
flag f-vectors to these other combinatorial domains.