We study the depth statistic defined by Petersen and Tenner in the case of the type B and D Coxeter groups and give explicit formulas for the depth of an element.
Slides from a talk at the Special Session on Patterns in Permutations and Words at the AMS Eastern Sectional Meeting in Washington, DC, March 2015.
We use facts about Richardson varieties (from the paper below) to characterize various singularity properties for these orbit closures.
Slides from a talk at
We characterize the permutations whose Bruhat graphs can be drawn on the plane or on a torus in terms of the properties of the permutation. One of our characterizations is in terms of pattern avoidance
This was an undergraduate summer research project for Chris in the summer of 2011.
The Waring rank of a polynomial is the least number of linear forms such that the polyomial is the sum of powers of the linear forms. We use classical facts about complex reflection groups and apolarity to determine the Waring rank of the Vandermonde determinant and some related products of linear forms.
Slides from a talk at the SIAM Conference on Applied Algebraic Geometry, August 2013.
We show that all local questions about Richardson varieties can be reduced to corresponding questions about Schubert varieties.
We characterize by pattern avoidance the Schubert varieties (for GL_n) that are local complete intersections. These turn out to include the Schubert varieties defined by inclusions as an important special case. Our proof is partly through obtaining an explicit minimal set of generators for the ideal cutting out a neighborhood of the identity, and we derive some consequences thereof.
Slides from a talk at the Algebraic Geometry Seminar at the University of British Columbia, February 2012.
Deodhar gave a formula for Kazhdan-Lusztig polynomials requiring an unknown (but known to exist) set of subwords of a reduced expression. We give two explicit constructions of such sets for cograssmannian permutations.
Slides from a talk at the Special Session on Combinatorial Representation Theory, AMS Central Sectional Meeting, St. Paul, MN, April 2010.
We show that certain obvious equations for local neighborhoods of Schubert varieties are a Groebner basis. We use this to explain a Grothendieck polynomial formula and give another multiplicity rule for Grassmannians.
Slides from a talk at the Combinatorics Seminar at the University of Minnesota, October 2009.
We give relatively short presentations for the cohomology rings of Schubert varieties.
Slides from a talk at the Special Session on Algebra, Geometry, and Combinatorics at the AMS Central Sectional Meeting in Urbana, IL, March 2009.
We prove a conjecture of Billey and Braden characterizing the permutations mentioned in the title using resolutions of singularities due to Cortez.
Slides from a talk Formal Power Series and Algebraic Combinatorics, Hagenburg, Austria, July 2009.
We extend the idea of interval pattern avoidance to arbitrary Weyl groups using the definition of pattern avoidance due to Billey and Braden, and Billey and Postnikov. We show that it characterizes Schubert varieties satisfying almost any singularity property for arbitrary semisimple Lie groups.
We define the combinatorial notion of interval pattern avoidance and show that it can be used to characterize Schubert varieties satisfying almost any singularity property. We analyze this notion for various singularity properties and measures and provide Macaulay 2 code for further exploration.
We determine which Schubert varieties are Gorenstein (a condition on singularities) in terms of a combinatorial characterization using generalized pattern avoidance conditions. We also give an explicit description as a line bundle of the canonical sheaf of a Gorenstein Schubert variety.
Notes from Mark Haiman's lectures at the Park City Mathematics Institute, July 2004, on the connection between classical q-analogs in combinatorics and q,t-analogs coming from the theory of Macdonald polynomials.
We show that the Schubert polynomial S_w specializes to the Catalan number C_n when w=1(n+1)...2. Several proofs of this result as well as a q-analog are given. An application to the singularities of Schubert varieties is given.
The geometric results of this and the following paper have been subsumed by a paper of Li Li and Alexander Yong, Some degenerations of Kazhdan-Lusztig ideals and multiplicities of Schubert varieties. They disproved the conjecture that Catalan numbers give the maximum multiplicity. The combinatorial results are generalized and subsumed by a paper of Luis Serrano and Christian Stump, Maximal fillings of moon polyominoes, simplicial complexes, and Schubert polynomials
We calculate using Macaulay 2 the multiplicities of the most singular point on Schubert varieties on Gl(n)/B for n=5,6. The method of computation is described and tables of the results as well as code are included.
This contains the main results from an REU project in 1995. I must confess I have forgotten most of the contents.