Subword complexes in Coxeter groups

ABSTRACT


Let W be a Coxeter group generated by a set S, and suppose we're given a word Q (an ordered list of elements of S) as well as an element w in W. The subword complex _{}(Q,w) is a simplicial complex whose faces are subwords of Q, with facets defined by the following:
Taking W = S_n and Q = a certain reduced expression for the long element yields the main examples, whose facets correspond to the rcgraphs of FominKirillov and BergeronBilley; rcgraphs reflect the combinatorics and geometry of Schubert varieties in flag manifolds. Subword complexes are vertex decomposable (hence shellable) topological manifolds, homoemorphic to balls or spheres. The Hilbert series of their StanleyReisner rings can be computed in terms of Demazure products of elements in Coxeter groups (to be defined). The main examples, together with a Gröbner basis theorem for certain determinantal ideals, provide (i) a new proof that Schubert varieties are CohenMacaulay in type A; (ii) a combinatorial interpretation for the coefficients of Grothendieck polynomials; and (iii) an algorithm by induction on the weak Bruhat order for producing the monomials in Schubert polynomials. Both (iii) and the sum of lowest degree terms in (ii) recover the BilleyJockush StanleyFomin formula for Schubert polynomials, while (iii) is a geometrically motivated alternative to Kohnert's conjecture. This talk represents joint work with Allen Knutson. 
Combinatorics Seminar, Mathematics Department, MIT, sara(atsign)math.mit.edu 

