Papers etc. by Alexander Yong

30. A Grobner basis for Kazhdan-Lusztig ideals (with Alexander Woo), preprint (41 pages), 2009.

We prove a Grobner basis theorem for Kazhdan-Lusztig ideals, and apply this to the study of specializations of Schubert/Grothendieck polynomials, and multiplicities of Schubert varieties.

We establish analogues of Schutzenberger's fundamental theorems of jeu de taquin for coproducts in K-theory of Grassmannians.

We develop equivariant and equivariant $K$-theory analogues of jeu de taquin, in relation to Schubert calculus of Grassmannians.

We extend the short presentation of the cohomology ring of a generalized flag manifold, due to [Borel '53], to a relatively short presentation of the cohomology of any of its Schubert varieties.

We present a randomized approximation algorithm for counting contingency tables, m x n non-negative integer matrices with given row sums R=(r_1,...,r_m) and column sums C=(c_1, ..., c_n).

Update (July, 2008): My coauthor A. Barvinok has further studied the typical table in his paper.

We define and study the Plancherel-Hecke probability measure on Young diagrams.

We introduce a theory of jeu de taquin for increasing tableaux, extending fundamental work of [Sch\"{u}tzenberger '77] for standard Young tableaux.

23. An S_3 symmetric Littlewood-Richardson rule (with Hugh Thomas), Mathematical Research Letters , volume 15, no.5-6, 2008; (9 pages).

The classical Littlewood-Richardson coefficients carry a natural $S_3$ symmetry via permutation of indices. Our ``carton rule'' for computing these numbers transparently and uniformly explains these six symmetries; previous rules manifest at most three of the six.

22. Counting magic squares in quasi-polynomial time (with Alexander Barvinok and Alexander Samorodnitsky), preprint (30 pages), 2007.
Companion Maple code contingency.txt. Here is a supplemental webpage comparing various ways of enumerating contingency tables and magic squares.

We present a randomized algorithm which approximates the number of n x n non-negative integer matrices (magic squares) with the row and column sums equal to t.

21. Cominuscule tableau combinatorics (with Hugh Thomas), preprint, 2007. Here is the Maple code to check Lemma 2.4 for Lie types E6 and E7.

We continue our study of "cominuscule tableau combinatorics" by generalizing constructions of M. Haiman, S. Fomin and M.-P. Schützenberger.

20. What is a Young tableau?
Notices of the AMS , Volume 54, Number 2, February 2007.

A short expository piece on Young tableaux and their basic theorems.

19. A combinatorial rule for (co)minuscule Schubert calculus (with Hugh Thomas), to appear in Advances in Math , 2009.
Companion Maple code cominrule.v1.0.txt. Here's an extra note about comparing cominrule.v1.0.txt with Schubert.v0.2.txt, given below.

We prove a root system uniform, concise combinatorial rule for Schubert calculus of _minuscule_ and _cominuscule_ flag manifolds G/P.

18. Stable Grothendieck polynomials and K-theoretic factor sequences (full version) (with A. Buch, A. Kresch, M. Shimozono and H. Tamvakis), Math. Annalen, volume 340, number 2, 2008.

We formulate a nonrecursive combinatorial rule for the expansion of the stable Grothendieck polynomials of [Fomin-Kirillov '94] in the basis of stable Grothendieck polynomials for partitions.

This is the complete version of the announcement (presented at FPSAC 2005).

17. Tableau complexes (with A. Knutson and E. Miller), to appear in Israel Journal of Math , 163 (2008), 317-343. A picture (due to A. Knutson).

Let X, Y be finite sets and T a set of functions from X->Y, which we will call ``tableaux''. We define a simplicial complex whose facets, all the same dimension, correspond to these tableaux, and call it a ``tableau complex''.

16. Multiplicity-free Schubert calculus (with H. Thomas), to appear in Canadian Math. Bulletin, 2007.

We give a short combinatorial classification of which products of Schubert classes in the Grassmannian are multiplicity-free. This answers a question of W. Fulton, and extends work of J. Stembridge.

15. Governing singularities of Schubert varieties (with A. Woo), accepted Journal of Algebra (section on computational algebra), 2007.
Companion Macaulay 2 code Schubsingular.v0.2.m2

We present a combinatorial and computational commutative algebra methodology for studying singularities of Schubert varieties of flag manifolds.

Update (Nov 11, 2006): 1. My coauthor A. Woo has extended the interval pattern avoidance ideas of the above paper here.

Update (April 6, 2007): N. Perrin has proved our Gorenstein locus conjecture for the case of minuscule G/P flag varieties. See this paper.

14. Grobner geometry of vertex decompositions and of flagged tableaux (with A. Knutson and E. Miller), Journal fur die reine und angewandte Mathematik (Crelle's Journal) , accepted, 2007.

We relate a classic algebro-geometric degeneration technique, dating at least to [Hodge 1941], to the notion of vertex decompositions of simplicial complexes. The good case is when the degeneration is reduced, and we call this a geometric vertex decomposition .

13. When is a Schubert variety Gorenstein? (with A. Woo),
Advances in Math , Vol 207 (2006), Issue 1 205--220. Companion Maple code available.

We determine which Schubert varieties are Gorenstein in terms of a combinatorial characterization using generalized pattern avoidance conditions.

See slides for a talk which discusses this (as well as related work below).

11. A formula for K-theory truncation Schubert calculus (with A. Knutson),
International Mathematics Research Notices , 70 (2004), 3741-3756.

In certain good cases, the truncation of a Schubert or Grothendieck polynomial may again be a Schubert or Grothendieck polynomial. We use this phenomenon to give subtraction-free formulae for certain Schubert structure constants in K(Flags(C^n)).

10. Lecture notes on the K-theory of the flag variety and the Fomin-Kirillov quadratic algebra (with C. Lenart), 2004.

We construct a commutative subalgebra of the Fomin-Kirillov quadratic algebra and explain the (conjectural) connection to the K-theory ring of the flag variety and the associated Schubert structure constants.

Update (Feb 12, 2006): 1. My coauthor C. Lenart has reported on our work in his paper "K-theory of the flag variety and the Fomin-Kirillov quadratic algebra" (published in the Journal of Algebra). 2. Conjecture 3.5 of our text above (as also reported in Lenart's paper) has been proved by A.N. Kirillov and T. Maeno (published in IMRN).

9. Quiver coefficients are Schubert structure constants (with A. Buch and F. Sottile),
Mathematical Research Letters , Volume 12, Issue 4, 567-574 (2005).

We give a new explanation of the alternation in sign of the K-theory quiver coefficients (via a theorem of Brion) as well as new combinatorial formulas for what they count.

8. Grothendieck polynomials and Quiver formulas (with A. Buch, A. Kresch and H. Tamvakis),
American Journal of Math , 127 (2005), 551-567.

We give a formula that proves that ``splitting coefficients'' for Grothendieck polynomials alternate in sign according to degree.

7. On Combinatorics of Degeneracy Loci and H*(G/B), a dissertation submitted to the Rackham graduate school, University of Michigan, 2003. Companion Maple code for part 2 available. (Includes an implementation of the classical Monk-Chevalley formula for arbitrary Lie type.)

Contains: 1. the content of the paper 6 below; 2. a Grobner based construction of an alternative (combinatorial) linear basis for H*(G/B) related to the Schubert calculus of a generalized flag manifold.

6. On Combinatorics of Quiver Component Formulas,
Journal of Algebraic Combinatorics , 21, 351-371, 2005.

Buch and Fulton conjectured the nonnegativity of the quiver coefficients appearing in their formula for a quiver variety. Knutson, Miller and Shimozono proved this conjecture as an immediate consequence of their ``component formula''. We present an alternative proof of the component formula by substituting combinatorics for Grobner degeneration.

5. Schubert polynomials and Quiver formulas (with A. Buch, A. Kresch and H. Tamvakis),
Duke Math Journal , Volume 122, Issue 1, 125-143 (2004).

Fulton's Universal Schubert polynomials represent general degeneracy loci for maps of vector bundles with rank conditions coming from a permutation. The Buch-Fulton Quiver formula expresses this polynomial as an integer linear combination of products of Schur polynomials in the differences of the bundles. We present a positive combinatorial formula for the coefficients.

4. Degree bounds in quantum Schubert calculus
Proceedings of the AMS , Volume 131, Number 9, 2649-2655 (2003).

We give a combinatorial proof of a conjecture of Fulton (after Fulton-Woodward) for the smallest degree of q that appears in the expansion of the product of two Schubert classes in the (small) quantum cohomology ring of a Grassmannian. We present a combinatorial proof of this result.

3. Tree-like properties of cycle factorizations (with I.P. Goulden)
Journal of Combinatorial Theory Series A, 98 , 106-117 (2002).

We provide a bijection between the set of factorizations, that is, ordered (n-1)-tuples of transpositions in ${\mathcal S}_{n}$ whose product is (12...n), and labelled trees on $n$ vertices. We prove a refinement of a 1959 theorem and question of Dénes that establishes new tree-like properties of factorizations.

2. Dyck paths and a bijection for multisets of hook numbers (with I.P. Goulden),
Discrete Math, 254 , no.1-3, 153-164 (2002).

We give a bijective proof of a conjecture of Regev and Vershik on the equality of two multisets of hook numbers of certain skew-Young diagrams. The bijection proves a result that is stronger and more symmetric than the original conjecture, by means of a construction involving Dyck paths, a particular type of lattice path.

1. Seeing the factorizations for the trees, M.Math. thesis (1999), University of Waterloo. Available upon request.

Contains 1. the results of paper ``2'' above; and 2. a Prufer type code for factorizations providing the first direct bijective proof that n^{n-2} counts the number of factorizations of the long cycle.

More software

At the end of the 2002 CRM Workshop on "Computational Lie Theory", a "wish list" of software as compiled. One of these wishes was for code to compute Schubert calculus for arbitrary G/B. This code implements Allen Knutson's recursion for Schubert calculus (in the classical setting). (Warning: this was written for Maple 7. I do not know if it works for later editions of Maple.) (Code updated June 4, 2006.)