Mahir Bilen Can

Chains in weak order posets associated to involutions

The W -set of an element of a weak order poset is useful in the cohomological study of the closures of spherical subgroups in generalized flag varieties. We explicitly describe in a purely combinatorial manner the W -sets of three different weak order posets: the set of all involutions in the symmetric group, the set of fixed point free involutions of the symmetric group, and the set of charged involutions. These distinguished sets of involutions parametrize Borel orbits in the classical symmetric spaces associated to the general linear group. In particular, we characterize the maximal chains of an arbitrary lower order ideal in any of these three posets.

A proof of the q, t -square conjecture

We prove a combinatorial formula conjectured by Loehr and Warrington for the coefficient of the sign character in ∇ (pn). Here ∇ denotes the Bergeron-Garsia nabla operator, and pn is a power-sum symmetric function. The combinatorial formula enumerates lattice paths in an n × n square according to two suitable statistics.

Calculating Heegaard–Floer homology by counting lattice points in tetrahedra

We introduce a notion of complexity for Seifert homology spheres by establishing a correspondence between lattice point counting in tetrahedra and the Heegaard-Floer homology. This complexity turns out to be equivalent to a version of Casson invariant and it is monotone under a natural partial order on the set of Seifert homology spheres. Using this interpretation we prove that there are finitely many Seifert homology spheres with a prescribed Heegaard–Floer homology. As an application, we characterize L-spaces and weakly elliptic manifolds among Seifert homology spheres. Also, we list all the Seifert homology spheres up to complexity two.

Weak order on complete quadrics

Using an action of the Richardson-Springer monoid on involutions, we study the weak order on the variety of complete quadrics. Maximal chains in the poset are explicitly determined. Applying results of Brion, our calculations describe certain cohomology classes in the complete flag variety.

Lexicographic shellability of the Bruhat-Chevalley order on fixed-point-free involutions

The main purpose of this paper is to prove that the Bruhat-Chevalley ordering of the symmetric group when restricted to the fixed-point-free involutions forms an EL-shellable poset whose order complex triangulates a ball. Another purpose of this article is to prove that the Deodhar-Srinivasan poset is a proper, graded subposet of the Bruhat-Chevalley poset on fixed-point-free involutions.

Wonderful Symmetric Varieties and Schubert Polynomials

Extending results of Wyser, we determine formulas for the equivariant cohomology classes of closed orbits of certain families of spherical subgroups of

Lexicographic shellability of partial involutions

GL_n

Ordered Bell numbers, Hermite polynomials, skew Young tableaux, and Borel orbits

on the flag variety

H-POLYNOMIALS AND ROOK POLYNOMIALS

GL_n/B

Algebraic monoids, group embeddings, and algebraic combinatorics

. Putting this together with a slight extension of the results of Can-Joyce-Wyser, we arrive at a family of polynomial identities which show that certain explicit sums of Schubert polynomials factor as products of linear forms.