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Alexander Woo

Publications and source records attributed to Alexander Woo.

27 records · Page 2Linked to original sources
arXiv2011

A Gröbner basis for Kazhdan-Lusztig ideals

Kazhdan-Lusztig ideals, a family of generalized determinantal ideals investigated in [Woo-Yong '08], provide an explicit choice of coordinates and equations encoding a neighbourhood of a torus-fixed point of a Schubert variety on a type A flag variety. Our main result is a Grobner basis for these ideals. This provides a single geometric setting to transparently explain the naturality of pipe dreams on the Rothe diagram of a permutation, and their appearance in: * combinatorial formulas [Fomin-Kirillov '94] for Schubert and Grothendieck polynomials of [Lascoux-Schutzenberger '82]; * the equivariant K-theory specialization formula of [Buch-Rimanyi '04]; and * a positive combinatorial formula for multiplicities of Schubert varieties in good cases, including those for which the associated Kazhdan-Lusztig ideal is homogeneous under the standard grading. Our results generalize (with alternate proofs) [Knutson-Miller '05]'s Grobner basis theorem for Schubert determinantal ideals and their geometric interpretation of the monomial positivity of Schubert polynomials. We also complement recent work of [Knutson '08,'09] on degenerations of Kazhdan-Lusztig varieties in general Lie type, as well as work of [Goldin '01] on equivariant localization and of [Lakshmibai-Weyman '90], [Rosenthal-Zelevinsky '01], and [Krattenthaler '01] on Grassmannian multiplicity formulas.

math.CO↗
arXiv2010

Mask formulas for cograssmannian Kazhdan-Lusztig polynomials

We give two contructions 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-Schutzenberger 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.

math.CO↗
arXiv2009

Presenting the cohomology of a Schubert variety

We extend the short presentation due to [Borel '53] of the cohomology ring of a generalized flag manifold to a relatively short presentation of the cohomology of any of its Schubert varieties. Our result is stated in a root-system uniform manner by introducing the essential set of a Coxeter group element, generalizing and giving a new characterization of [Fulton '92]'s definition for permutations. Further refinements are obtained in type A.

math.CO↗
arXiv2008

Permutations with Kazhdan-Lusztig polynomial P_{id,w}(q) = 1 + q^h

Using resolutions of singularities introduced by Cortez and a method for calculating Kazhdan-Lusztig polynomials due to Polo, we prove the conjecture of Billey and Braden characterizing permutations w with Kazhdan-Lusztig polynomial P_{id,w}(q)=1+q^h for some h.

math.AG↗
arXiv2006

Interval pattern avoidance for arbitrary root systems

We extend the idea of interval pattern avoidance defined by Yong and the author for $S_n$ 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 $GL_n$, interval pattern avoidance is a universal tool for characterizing which Schubert varieties have certain local properties, and where these local properties hold.

math.CO↗
arXiv2006

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 [Lakshmibai-Sandhya'90], but is insufficient in general. Our approach is analyzed for some common invariants, including Kazhdan-Lusztig polynomials, multiplicity, factoriality, and Gorensteinness, extending [Woo-Yong'05]; the description of the singular locus (which was independently proved by [Billey-Warrington '03], [Cortez '03], [Kassel-Lascoux-Reutenauer'03], [Manivel'01]) 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.

math.AG↗
arXiv2005

When is a Schubert variety Gorenstein?

A (normal) variety is Gorenstein if it is Cohen-Macualay and its canonical sheaf is a line bundle. This property, which measures the ``pathology'' of the singularities of a variety, is thus stronger than Cohen-Macualayness, but is also weaker than smoothness. We determine which Schubert varieties are Gorenstein 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.

math.AG↗