The cd-index of Bruhat intervals.

Nathan Reading

The electronic journal of combinatorics, Tome 11 (2004) no. 1

Résumé

We study flag enumeration in intervals in the Bruhat order on a Coxeter group by means of a structural recursion on intervals in the Bruhat order. The recursion gives the isomorphism type of a Bruhat interval in terms of smaller intervals, using basic geometric operations which preserve PL sphericity and have a simple effect on the cd-index. This leads to a new proof that Bruhat intervals are PL spheres as well a recursive formula for the cd-index of a Bruhat interval. This recursive formula is used to prove that the cd-indices of Bruhat intervals span the space of cd-polynomials. The structural recursion leads to a conjecture that Bruhat spheres are "smaller" than polytopes. More precisely, we conjecture that if one fixes the lengths of $x$ and $y$, then the cd-index of a certain dual stacked polytope is a coefficientwise upper bound on the cd-indices of Bruhat intervals $[x,y]$. We show that this upper bound would be tight by constructing Bruhat intervals which are the face lattices of these dual stacked polytopes. As a weakening of a special case of the conjecture, we show that the flag h-vectors of lower Bruhat intervals are bounded above by the flag h-vectors of Boolean algebras (i. e. simplices).

Détail

DOI : 10.37236/1827

Classification : 20F55, 06A07, 52B05, 52C35
Mots-clés : flag enumeration, Bruhat order, Coxeter groups, PL sphericity, cd-indices, Bruhat intervals, face lattices