Alexander Yong

Gröbner geometry of vertex decompositions and of flagged tableaux

Abstract We relate a classic algebro-geometric degeneration technique, dating at least to Hodge 1941 (J. London Math. Soc. 16: 245–255), 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. Our main example in this paper is the family of vexillary matrix Schubert varieties, whose ideals are also known as (one-sided) ladder determinantal ideals. Using a diagonal term order to specify the (Gröbner) degeneration, we show that these have geometric vertex decompositions into simpler varieties of the same type. From this, together with the combinatorics of the pipe dreams of Fomin-Kirillov 1996 (Discr. Math. 153: 123–143), we derive a new formula for the numerators of their multigraded Hilbert series, the double Grothendieck polynomials, in terms of flagged set-valued tableaux. This unifies work of Wachs 1985 (J. Combin. Th. (A) 40: 276–289) on flagged tableaux, and Buch 2002 (Acta. Math. 189: 37–78) on set-valued tableaux, giving geometric meaning to both. This work focuses on diagonal term orders, giving results complementary to those of Knutson-Miller 2005 (Ann. Math. 161: 1245–1318), where it was shown that the generating minors form a Gröbner basis for any antidiagonal term order and any matrix Schubert variety. We show here that under a diagonal term order, the only matrix Schubert varieties for which these minors form Gröbner bases are the vexillary ones, reaching an end toward which the ladder determinantal literature had been building.

The Direct Sum Map on Grassmannians and Jeu de Taquin for Increasing Tableaux

The direct sum map on Grassmannians induces a K-theory pullback that defines the splitting coefficients. We geometrically explain an identity from Buch [“Grothendieck classes of quiver varieties.” Duke Mathematical Journal 115, no. 1 (2002): 75–103] between the splitting coefficients and the Schubert structure constants for products of Schubert structure sheaves. This is related to the topic of product and splitting coefficients for Schubert boundary ideal sheaves. Our main results extend jeu de taquin for increasing tableaux [Thomas and Yong. “A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus.” Algebra and Number Theory Journal 3, no. 2 (2009): 121–48] by proving transparent analogues of Schutzenberger’s [“La Correspondance de Robinson.” In Combinatoire et Representation du Groupe Symetrique (Strasbourg, 1976), edited by D. Foata, 59–113. Lecture Notes in Mathematics 579. Berlin: Springer, 1977] fundamental theorems on well definedness of rectification. We then establish that jeu de taquin gives rules for each of these four kinds of coefficients.

Genomic tableaux

We explain how genomic tableaux [Pechenik–Yong ’15] are a semistandard complement to increasing tableaux [Thomas–Yong ’09]. From this perspective, one inherits genomic versions of jeu de taquin, Knuth equivalence, infusion and Bender–Knuth involutions, as well as Schur functions from (shifted) semistandard Young tableaux theory. These are applied to obtain new Littlewood–Richardson rules for K-theory Schubert calculus of Grassmannians (after [Buch ’02]) and maximal orthogonal Grassmannians (after [Clifford–Thomas–Yong ’14], [Buch–Ravikumar ’12]). For the unsolved case of Lagrangian Grassmannians, sharp upper and lower bounds using genomic tableaux are conjectured.

POLYNOMIALS FOR SYMMETRIC ORBIT CLOSURES IN THE FLAG VARIETY

In [WY] we introduced polynomial representatives of cohomology classes of orbit closures in the flag variety, for the symmetric pair (GLp+q, GLp × GLq). We present analogous results for the remaining symmetric pairs of the form (GLn, K), i.e., (GLn, On) and (GL2n, Sp2n). We establish “well-definedness” of certain representatives from [Wy1]. It is also shown that the representatives have the combinatorial properties of nonnegativity and stability. Moreover, we give some extensions to equivariant K-theory.

Poset edge densities, nearly reduced words, and barely set-valued tableaux

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 Youngs 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.

Governing singularities of symmetric orbit closures

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.

The Joseph Greenberg Problem: Combinatorics and Comparative Linguistics

We correct a 1957 combinatorial enumeration by the linguist J. Greenberg. The desired count, the Bell number B(25), supported using his Mass Comparison method for language classification. In 1987, he used this method to classify indigenous languages of the Americas into three families. Actually, the same combinatorics provides a back-of-the-envelope estimate for the number of families. This suggests that alternative classifications with over a hundred families possess the right order of magnitude.

Partition identities and quiver representations

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

A combinatorial rule for (co)minuscule Schubert calculus

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