Papers etc. by Alexander Yong

84. RSK as a linear operator (with Ada Stelzer), preprint version of October 30, 2024.

We study RSK as a linear operator on the coordinate ring of matrices.

We study representations coming from Levi groups actions on ``bicrystalline'' varieties in matrix space.

An algorithm is presented that generates sets of size equal to the degree of a given variety defined by a homogeneous ideal.

We prove that the Hilbert function and Hilbert polynomials associated to a Schubert determinantal ideal agree for all nonnegative integers.

We give a root-system uniform classification of Levi-spherical Schubert varieties, with an application to the multiplicity-free decomposition of Demazure modules into irreducibles for a Levi (sub)group.

This chapter combines an introduction and research survey about Schubert varieties. The theme is to combinatorially classify their singularities using a family of polynomial ideals generated by determinants.

Establishes an exponential worst case lower bound on the number of generators needed to present the cohomology ring of a Schubert variety in type A.

Discusses earlier work on the classification problem of the title together with some new results, conjecture, and evidence for that conjecture.

Survey of work on the Newell-Littlewood numbers.

Explicit minimal generators for Fulton's Schubert determinantal ideals are determined along with some implications.

We continue our study of Newell-Littlewood numbers.

Extended notes for my talk at ``Recent developments in Grobner Geometry'', June 2021.

We classify Levi spherical Schubert varieties in the complete flag variety GL_n/B.

We formulate a combinatorial conjecture about the expansion of Grothendieck polynomials into Lascoux polynomials.

We study the Hilbert basis of the Kostka semigroup.

We introduce and study proper permutations, in connection to spherical Schubert geometry.

We use the theory of generalized permutahedra to give a vanishing criterion for Schubert intersection numbers.

We continue our study of Newell-Littlewood numbers.

We introduce spherical elements of Coxeter groups, in connection to Levi spherical Schubert varieties.

We classify multiplicity-free key polynomials. This is the companion paper to ``Coxeter combinatorics and spherical Schubert geometry''.

We study Newell-Littlewood numbers by analogy with modern research on their celebrated cousins, the Littlewood-Richardson coefficients.

We discuss results concerning the objects of the title. We introduce shifted edge labeled tableau in connection to a ring of Anderson-Fulton.

We collect "Atiyah-Bott Combinatorial Dreams" in Schubert calculus.

61. An estimation method for game complexity (with David Yong), preprint, version of January 29, 2019.

We study a method for estimating the game tree size of strategic games.

We study algorithmic and complexity aspects of reduced word enumeration.

This is the main technical part of this preprint whose material was split into this text and the one below.

We discuss an algebraic combinatorics paradigm for computation complexity. Our main results concern Schubert polynomials.

This is an extended abstract of this preprint whose material was split into this text and the one above.

We study the circuit complexity of some tropicalized symmetric polynomials.

We construct and study Barbasch-Evens-Magyar varieties.

We give a strongly polynomial time algorithm for deciding the vanishing of Littlewood-Richardson polynomials.

A remark on the representation theory of symmetric groups.

We prove symmetric Grothendieck polynomials have saturated Newton polytopes.

We study Newton polytopes of various families of polynomials in algebraic combinatorics.

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)xGL(q) acting on the classical flag variety.

We establish a specific connection between classical partition combinatorics and the theory of quiver representations.

We observe a connection between Elnitsky's rhombic tilings and Magyar's description of Bott-Samelson varieties.

We study a certain coincidence of expectations of down-degree of posets.

We describe a theory of genomic tableaux.

We propose a new model for Schubert polynomials.

We resolve a conjecture of A. Knutson-R. Vakil.

We give a combinatorial rule for multiplication of the basis of Schubert structure sheaves in the equivariant K-theory ring of the Grassmannian. (See also the announcement version below.)

Announcement of results on (equivariant) K-theory of Grassmannians (and maximal orthogonal, Lagrangian Grassmannians).

We use the Euler-Gauss partition identity to offer a critique of the bibliometric h-index.

We continue our study of polynomial representatives of orbit closures in the flag variety for symmetric pairs (G,K) in type A.

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 .

We introduce polynomial representatives of GL_p x GL_q orbit closures in the flag variety.

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.)

We present combinatorial rules (one theorem and two conjectures) concerning three bases of Z[x1,x2,...]. Errata note added March 10, 2017: the published version has a misstatement of Conjectures 1.4 and 1.6. The version above has the correction.

We establish a connection between the eigenvalue problem of S. Friedland and equivariant Schubert calculus.

We prove essentially all singularity questions about Richardson varieties reduce to the case of Schubert varieties.

We develop equivariant analogues of jeu de taquin, in relation to Schubert calculus of Grassmannians.

We study patch ideals of Peterson varieties and some other subvarieties of GL_n/B.

We suggest a parallel between Kazhdan-Lusztig polynomials and H-polynomials of local rings of Schubert varieties.

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.

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.

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 splitting coefficients defining the direct sum map, as well as products of Schubert boundary ideal sheaves, in K-theory 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), (revised July 2013). To appear in MSJ-SI 2012 Proceedings. 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.

Correction (October, 2019): Example 6.18 has a typo; the multiplicity is 3 (as confirmed by the code, and consistent with Proposition 6.21). We thank Jordan Lambert for bringing this to our attention.

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.)