arXiv subjects
Publications and source records attributed to Alexander Yong.
This report formulates a conjectural combinatorial rule that positively expands Grothendieck polynomials into Lascoux polynomials. It generalizes one such formula expanding Schubert polynomials into key polynomials, and refines another one expanding stable Grothendieck polynomials.
We define and study proper permutations. Properness is a geometrically natural necessary criterion for a Schubert variety to be Levi-spherical. We prove the probability that a random permutation is proper goes to zero in the limit.
We connect generalized permutahedra with Schubert calculus. Thereby, we give sufficient vanishing criteria for Schubert intersection numbers of the flag variety. Our argument utilizes recent developments in the study of Schubitopes, which are Newton polytopes of Schubert polynomials. The resulting tableau test executes in polynomial time.
The key polynomials, defined by A. Lascoux-M.-P. Schützenberger, are characters for the Demazure modules of type A. We classify multiplicity-free key polynomials. The proof uses two combinatorial models for key polynomials. The first is due to A. Kohnert. The second is by S. Assaf-D. Searles, in terms of quasi-key polynomials. Our argument proves a sufficient condition for a quasi-key polynomial to be multiplicity-free.
The Newell-Littlewood numbers are defined in terms of their celebrated cousins, the Littlewood-Richardson coefficients. Both arise as tensor product multiplicities for a classical Lie group. They are the structure coefficients of the K. Koike-I. Terada basis of the ring of symmetric functions. Recent work of H. Hahn studies them, motivated by R. Langlands' beyond endoscopy proposal; we address her work with a simple characterization of detection of Weyl modules. This motivates further study of the combinatorics of the numbers. We consider analogues of ideas of J. De Loera-T. McAllister, H. Derksen-J. Weyman, S. Fomin-W. Fulton-C.-K. Li-Y.-T. Poon, W. Fulton, R. King-C. Tollu-F. Toumazet, M. Kleber, A. Klyachko, A. Knutson-T. Tao, T. Lam-A. Postnikov-P. Pylyavskyy, K. Mulmuley-H. Narayanan-M. Sohoni, H. Narayanan, A. Okounkov, J. Stembridge, and H. Weyl.
This chapter concerns edge labeled Young tableaux, introduced by H. Thomas and the third author. It is used to model equivariant Schubert calculus of Grassmannians. We survey results, problems, conjectures, together with their influences from combinatorics, algebraic and symplectic geometry, linear algebra, and computational complexity. We report on a new shifted analogue of edge labeled tableaux. Conjecturally, this gives a Littlewood-Richardson rule for the structure constants of the D. Anderson-W. Fulton ring, which is related to the equivariant cohomology of isotropic Grassmannians.
We collect Atiyah-Bott Combinatorial Dreams (A.B.C.Ds) in Schubert calculus. One result relates equivariant structure coefficients for two isotropic flag manifolds, with consequences to the thesis of C. Monical. We contextualize using work of N. Bergeron-F. Sottile, S. Billey-M. Haiman, P. Pragacz, and T. Ikeda-L. Mihalcea-I. Naruse. The relation complements a theorem of A. Kresch-H. Tamvakis in quantum cohomology. Results of A. Buch-V. Ravikumar rule out a similar correspondence in K-theory.
A reduced word of a permutation $w$ is a minimal length expression of $w$ as a product of simple transpositions. We examine the computational complexity, formulas and (randomized) algorithms for their enumeration. In particular, we prove that the Edelman-Greene statistic, defined by S. Billey-B. Pawlowski, is typically exponentially large. This implies a result of B. Pawlowski, that it has exponentially growing expectation. Our result is established by a formal run-time analysis of A. Lascoux-M.-P. Schützenberger's transition algorithm. The more general problem of Hecke word enumeration, and its closely related question of counting set-valued standard Young tableaux, is also investigated. The latter enumeration problem is further motivated by work on Brill-Noether varieties due to M. Chan-N. Pflueger and D. Anderson-L. Chen-N. Tarasca.
We looked at a method for estimating the complexity measure of game tree size (the number of legal games). It seems effective for a number of children's games such as Tic-Tac-Toe, Connect Four and Othello.
In certain finite posets, the expected down-degree of their elements is the same whether computed with respect to either the uniform distribution or the distribution weighting an element by the number of maximal chains passing through it. We show that this coincidence of expectations holds for Cartesian products of chains, connected minuscule posets, weak Bruhat orders on finite Coxeter groups, certain lower intervals in Young's lattice, and certain lower intervals in the weak Bruhat order below dominant permutations. Our tools involve formulas for counting nearly reduced factorizations in 0-Hecke algebras; that is, factorizations that are one letter longer than the Coxeter group length.
D. Grigoriev-G. Koshevoy recently proved that tropical Schur polynomials have (at worst) polynomial tropical semiring complexity. They also conjectured tropical skew Schur polynomials have at least exponential complexity; we establish a polynomial complexity upper bound. Our proof uses results about (stable) Schubert polynomials, due to R. P. Stanley and S. Billey-W. Jockusch-R. P. Stanley, together with a sufficient condition for polynomial complexity that is connected to the saturated Newton polytope property.
J. DeLoera-T. McAllister and K. D. Mulmuley-H. Narayanan-M. Sohoni independently proved that determining the vanishing of Littlewood-Richardson coefficients has strongly polynomial time computational complexity. Viewing these as Schubert calculus numbers, we prove the generalization to the Littlewood-Richardson polynomials that control equivariant cohomology of Grassmannians. We construct a polytope using the edge-labeled tableau rule of H. Thomas-A. Yong. Our proof then combines a saturation theorem of D. Anderson-E. Richmond-A. Yong, a reading order independence property, and E. Tardos' algorithm for combinatorial linear programming.
We discuss implications of the following statement about the representation theory of symmetric groups: every integer appears infinitely often as an irreducible character evaluation, and every nonnegative integer appears infinitely often as a Littlewood-Richardson coefficient and as a Kronecker coefficient.
Symmetric Grothendieck polynomials are inhomogeneous versions of Schur polynomials that arise in combinatorial $K$-theory. A polynomial has saturated Newton polytope (SNP) if every lattice point in the polytope is an exponent vector. We show Newton polytopes of these Grothendieck polynomials and their homogeneous components have SNP. Moreover, the Newton polytope of each homogeneous component is a permutahedron. This addresses recent conjectures of C. Monical-N. Tokcan-A. Yong and of A. Fink-K. Mészáros-A. St. Dizier in this special case.
A polynomial has saturated Newton polytope (SNP) if every lattice point of the convex hull of its exponent vectors corresponds to a monomial. We compile instances of SNP in algebraic combinatorics (some with proofs, others conjecturally): skew Schur polynomials; symmetric polynomials associated to reduced words, Redfield--Polya theory, Witt vectors, and totally nonnegative matrices; resultants; discriminants (up to quartics); Macdonald polynomials; key polynomials; Demazure atoms; Schubert polynomials; and Grothendieck polynomials, among others. Our principal construction is the Schubitope. For any subset of [n] x [n], we describe it by linear inequalities. This generalized permutahedron conjecturally has positive Ehrhart polynomial. We conjecture it describes the Newton polytope of Schubert and key polynomials. We also define dominance order on permutations and study its poset-theoretic properties.
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.
We present a particular connection between classical partition combinatorics and the theory of quiver representations. Specifically, we give a bijective proof of an analogue of A. L. Cauchy's Durfee square identity to multipartitions. We then use this result to give a new proof of M. Reineke's identity in the case of quivers $Q$ of Dynkin type $A$ of arbitrary orientation. Our identity is stated in terms of the lacing diagrams of S. Abeasis - A. Del Fra, which parameterize orbits of the representation space of $Q$ for a fixed dimension vector.
S.~Elnitsky (1997) gave an elegant bijection between rhombic tilings of $2n$-gons and commutation classes of reduced words in the symmetric group on $n$ letters. P.~Magyar (1998) found an important construction of the Bott-Samelson varieties introduced by H.C.~Hansen (1973) and M.~Demazure (1974). We explain a natural connection between S.~Elnitsky's and P.~Magyar's results. This suggests using tilings to encapsulate Bott-Samelson data (in type $A$). It also indicates a geometric perspective on S.~Elnitsky's combinatorics. We also extend this construction by assigning desingularizations to the zonotopal tilings considered by B.~Tenner (2006).