Alexander Woo

Stick Numbers and Composition of Knots and Links

We address the concept of stick number for knots and links under various restrictions concerning the length of the sticks, the angles between sticks, and placements of the vertices. In particular, we focus on the effect of composition on the various stick numbers. Ultimately, we determine the traditional stick number for an infinite class of knots, which are the (n,n-1)-torus knots together with all of the possible compositions of such knots. The exact stick number was previously known for only seven knots.

Governing Singularities of Schubert Varieties

We present a combinatorial and computational commutative algebra methodology for studying singularities of Schubert varieties of flag manifolds. We define the combinatorial notion of interval pattern avoidance. For “reasonable” invariants P of singularities, we geometrically prove that this governs (1) the P-locus of a Schubert variety, and (2) which Schubert varieties are globally not P. The prototypical case is P=“singular”; classical pattern avoidance applies admirably for this choice [V. Lakshmibai, B. Sandhya, Criterion for smoothness of Schubert varieties in SL(n)/B, Proc. Indian Acad. Sci. Math. Sci. 100 (1) (1990) 45–52, MR 91c:14061], but is insufficient in general. Our approach is analyzed for some common invariants, including Kazhdan–Lusztig polynomials, multiplicity, factoriality, and Gorensteinness, extending [A. Woo, A. Yong, When is a Schubert variety Gorenstein?, Adv. Math. 207 (1) (2006) 205–220, MR 2264071]; the description of the singular locus (which was independently proved by [S. Billey, G. Warrington, Maximal singular loci of Schubert varieties in SL(n)/B, Trans. Amer. Math. Soc. 335 (2003) 3915–3945, MR 2004f:14071; A. Cortez, Singularites generiques et quasi-resolutions des varietes de Schubert pour le groupe lineaire, Adv. Math. 178 (2003) 396–445, MR 2004i:14056; C. Kassel, A. Lascoux, C. Reutenauer, The singular locus of a Schubert variety, J. Algebra 269 (2003) 74–108, MR 2005f:14096; L. Manivel, Le lieu singulier des varietes de Schubert, Int. Math. Res. Not. 16 (2001) 849–871, MR 2002i:14045]) is also thus reinterpreted. Our methods are amenable to computer experimentation, based on computing with Kazhdan–Lusztig ideals (a class of generalized determinantal ideals) using Macaulay 2. This feature is supplemented by a collection of open problems and conjectures.

Interval pattern avoidance for arbitrary root systems

Author(s): Woo, Alexander | Abstract: We extend the idea of interval pattern avoidance defined by Yong and the author for

Mask Formulas for Cograssmannian Kazhdan-Lusztig Polynomials

S_n

Governing singularities of symmetric orbit closures

to arbitrary Weyl groups using the definition of pattern avoidance due to Billey and Braden, and Billey and Postnikov. We show that, as previously shown by Yong and the author for

Power sum decompositions of defining equations of reflection arrangements

GL_n

Specht Polytopes and Specht Matroids

, interval pattern avoidance is a universal tool for characterizing which Schubert varieties have certain local properties, and where these local properties hold.

WHEN IS A SCHUBERT VARIETY GORENSTEIN

We give two constructions of sets of masks on cograssmannian permutations that can be used in Deodhar’s formula for Kazhdan–Lusztig basis elements of the Iwahori–Hecke algebra. The constructions are respectively based on a formula of Lascoux–Schützenberger and its geometric interpretation by Zelevinsky. The first construction relies on a basis of the Hecke algebra constructed from principal lower order ideals in Bruhat order and a translation of this basis into sets of masks. The second construction relies on an interpretation of masks as cells of the Bott–Samelson resolution. These constructions give distinct answers to a question of Deodhar.

A Gröbner basis for Kazhdan-Lusztig ideals

We develop interval pattern avoidance and Mars-Springer ideals to study singularities of symmetric orbit closures in a flag variety. This paper focuses on the case of the Levi subgroup GL_p x GL_q acting on the classical flag variety. We prove that all reasonable singularity properties can be classified in terms of interval patterns of clans.

Presenting the cohomology of a Schubert variety

We determine the Waring rank and a Waring decomposition of the fundamental skew invariant of any complex reflection group whose highest degree is a regular number. This includes all irreducible real reflection groups.