Bruhat order, smooth Schubert varieties, and hyperplane arrangements

The aim of this article is to link Schubert varieties in the flag manifold with hyperplane arrangements. For a permutation, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincare polynomial of the corresponding Schubert variety if and only if the Schubert variety is smooth. We give an explicit combinatorial formula for the Poincare polynomial. Our main technical tools are chordal graphs and perfect elimination orderings.Comment: 14 pages, 2 figure

From Bruhat intervals to intersection lattices and a conjecture of Postnikov

We prove the conjecture of A. Postnikov that (A) the number of regions in the inversion hyperplane arrangement associated with a permutation w\in \Sn is at most the number of elements below $w$ in the Bruhat order, and (B) that equality holds if and only if $w$ avoids the patterns 4231, 35142, 42513 and 351624. Furthermore, assertion (A) is extended to all finite reflection groups. A byproduct of this result and its proof is a set of inequalities relating Betti numbers of complexified inversion arrangements to Betti numbers of closed Schubert cells. Another consequence is a simple combinatorial interpretation of the chromatic polynomial of the inversion graph of a permutation which avoids the above patterns.Comment: 24 page

A pattern avoidance criterion for free inversion arrangements

We show that the hyperplane arrangement of a coconvex set in a finite root system is free if and only if it is free in corank 4. As a consequence, we show that the inversion arrangement of a Weyl group element w is free if and only if w avoids a finite list of root system patterns. As a key part of the proof, we use a recent theorem of Abe and Yoshinaga to show that if the root system does not contain any factors of type C or F, then Peterson translation of coconvex sets preserves freeness. This also allows us to give a Kostant-Shapiro-Steinberg rule for the coexponents of a free inversion arrangement in any type.Comment: 20 pages. Corrects some errors from a preliminary version that was privately circulate

Combinatorics in Schubert varieties and Specht modules

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, June 2011."June 2011." Cataloged from PDF version of thesis.Includes bibliographical references (p. 57-59).This thesis consists of two parts. Both parts are devoted to finding links between geometric/algebraic objects and combinatorial objects. In the first part of the thesis, we link Schubert varieties in the full flag variety with hyperplane arrangements. Schubert varieties are parameterized by elements of the Weyl group. For each element of the Weyl group, we construct certain hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincaré polynomial if and only if the Schubert variety is rationally smooth. For classical types the arrangements are (signed) graphical arrangements coning from (signed) graphs. Using this description, we also find an explicit combinatorial formula for the Poincaré polynomial in type A. The second part is about Specht modules of general diagram. For each diagram, we define a new class of polytopes and conjecture that the normalized volume of the polytope coincides with the dimension of the corresponding Specht module in many cases. We give evidences to this conjecture including the proofs for skew partition shapes and forests, as well as the normalized volume of the polytope for the toric staircase diagrams. We also define new class of toric tableaux of certain shapes, and conjecture the generating function of the tableaux is the Frobenius character of the corresponding Specht module. For a toric ribbon diagram, this is consistent with the previous conjecture. We also show that our conjecture is intimately related to Postnikov's conjecture on toric Specht modules and McNamara's conjecture of cylindric Schur positivity.by Hwanchul Yoo.Ph.D