Subword complexes in Coxeter groups
and applications to Schubert varieties

Ezra Miller

MIT

February 16,
4:15pm
refreshments at 3:45pm
2-338

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 Delta(Q,w) is a simplicial complex whose faces are subwords of Q, with facets defined by the following:

    a subword P of Q is a reduced expression for w   iff   Q\P is a facet of Delta(Q,w).

Taking W = S_n and Q = a certain reduced expression for the long element yields the main examples, whose facets correspond to the rc-graphs of Fomin-Kirillov and Bergeron-Billey; rc-graphs 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 Stanley-Reisner 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 Cohen-Macaulay 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 Billey-Jockush- Stanley-Fomin formula for Schubert polynomials, while (iii) is a geometrically motivated alternative to Kohnert's conjecture. This talk represents joint work with Allen Knutson.


Speaker's Contact Info: ezra(at-sign)math.mit.edu


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