# Papers etc. by Alexander Yong

My papers on the Mathematics ArXiV.

42. Critique of Hirsch's citation index: a combinatorial Fermi problem preprint (10 pages+appendix), accepted to Notices of the AMS , version of October 13, 2014.

We use the Euler-Gauss partition identity to offer a critique of the bibliometric h-index.
41. Polynomials for symmetric orbit closures in the flag variety (with Benjamin Wyser), preprint (22 pages), version of September 30, 2014.
We continue our study of polynomial representatives of orbit closures in the flag variety for symmetric pairs (G,K) in type A.
40. The Joseph Greenberg problem: combinatorics and comparative linguistics , preprint (6 pages), version of October 10, 2013.

We discuss some combinatorics related to work of linguist J. Greenberg. We give an interpretation of the consensus range of the number of indigenious language families of the Americas. Best read along with the video .
39. Polynomials for GLp x GLq orbit closures in the flag variety (with Benjamin Wyser), to appear in Selecta Math (25 pages), version of March 4, 2014.

We introduce polynomial representatives of GLp x GLq orbit closures in the flag variety.
38. Root-theoretic Young diagrams and Schubert calculus: planarity and the adjoint varieties (with Dominic Searles), preprint (45 pages), version of August 29, 2013.

We study the existence of a root-system uniform and nonnegative combinatorial rule for Schubert calculus. Our main cases are the adjoint varieties of classical type. (An preliminary version appeared as an extended abstract in FPSAC 2013.)
37. Combinatorial rules for three bases of polynomials (with Colleen Ross), preprint (9 pages), version of February 1, 2013.

We present combinatorial rules (one theorem and two conjectures) concerning three bases of Z[x1,x2,...].
36. Eigenvalues of Hermitian matrices and equivariant cohomology of Grassmannians (with David Anderson and Edward Richmond), accepted to Compositio Math (14 pages), version of April 2, 2013.

We establish a connection between the eigenvalue problem of S. Friedland and equivariant Schubert calculus.
35. Singularities of Richardson varieties (with Allen Knutson and Alexander Woo), Math. Res. Letters (10 pages), version of September 14, 2012.

We prove essentially all singularity questions about Richardson varieties reduce to the case of Schubert varieties.
34. Equivariant Schubert calculus and jeu de taquin (with Hugh Thomas), Annales de l'Institut Fourier (31 pages), accepted pending minor revisions, version of July 11, 2012.

We develop equivariant analogues of jeu de taquin, in relation to Schubert calculus of Grassmannians.
33. Patch ideals and Peterson varieties (with Erik Insko), Transformation Groups (23 pages), version of February 3, 2012.

We study patch ideals of Peterson varieties and some other subvarieties of GLn/B.
32. Kazhdan-Lusztig polynomials and drift configurations (with Li Li), Algebra and Number Theory , (25 pages), June 19, 2010.

We suggest a parallel between Kazhdan-Lusztig polynomials and H-polynomials of local rings of Schubert varieties.
31. Some degenerations of Kazhdan-Lusztig ideals and multiplicities of Schubert varieties (with Li Li), Advances in Math , (31 pages), version of July 1, 2011.

We prove a combinatorial rule for the Hilbert-Samuel multiplicities of covexillary Schubert varieties at singular points. Generalizations of our method to general Schubert varieties in the complete flag variety are suggested.
30. K-theoretic Schubert calculus for OG(n,2n+1) and jeu de taquin for shifted increasing tableaux (with Edward Clifford and Hugh Thomas), to appear in J. Reine Angew Math (Crelle) (13 pages), Feb 15, 2012.

Using recent work of Buch--Ravikumar and Feigenbaum-Sergel we prove the conjectural rule from [Thomas-Yong '09] for $K$-theory Schubert calculus for maximal odd orthogonal Grassmannians.
29. A Grobner basis for Kazhdan-Lusztig ideals (with Alexander Woo), preprint (40 pages), American J. Math , 2012.

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.
28. The direct sum map on Grassmannians and Jeu de taquin for increasing tableaux (with Hugh Thomas), International Math. Res. Notices, (20 pages), 2010.

We establish analogues of Schutzenberger's fundamental theorems of jeu de taquin for splitting coefficients defining the direct sum map, as well as products of Schubert boundary ideal sheaves, in K-theory of Grassmannians.
27. Presenting the cohomology of a Schubert variety (with Victor Reiner and Alexander Woo), Transactions of the AMS, (22 pages), 2009.

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.
26. An approximation algorithm for contingency tables (with Alexander Barvinok, Zur Luria and Alex Samorodnitsky), accepted to Random Structures and Algorithms , 2009 (45 pages).

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.

25. Longest strictly increasing subsequences, Plancherel measure and the Hecke insertion algorithm (with Hugh Thomas), Advances in Applied Math (Dennis Stanton special issue) (34 pages), 10/02/02. (Contains an appendix with Ofer Zeitouni and myself.)
Companion Maple code HeckeLIS.v0.1.txt.

We define and study the Plancherel-Hecke probability measure on Young diagrams.
24. A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus (with Hugh Thomas), Algebra and Number Theory , 2008 (22 pages).

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 (revised July 2013). 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), 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), Israel Journal of Math , 163 (2008), 317-343.

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), 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.
12. Grobner geometry of Schubert transition formulae and Littlewood-Richardson rules (with A. Knutson), preprint.

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.

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

(Code for a particular paper is found with the .ps file above.)

• Maple 7 code to compute Schubert calculus in G/B , website exclusive, 2006.
• 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.)

• C++ code for estimating permanents, hafnians and the number of forests in a graph, 2003.