Gröbner geometry of vertex decompositions and of flagged tableaux

Abstract

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.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], 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] on flagged tableaux, and [Buch 2002] 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], 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.

Tableau complexes

“…Flagged Young tableaux are used to compute the Schubert polynomials for vexillary permutations [Wa85], and one choice of vertex decomposition parallels the "transition formula" for vexillary double Grothendieck polynomials [La01,La03]. Hence we are able to give set-valued-tableaux-based formulae for double Grothendieck polynomials of vexillary permutations, via Theorem C. The second formula in the following corollary, which appeared already in [KMY05], is a common generalization of Buch's and Wachs's formulae, each of which specializes to the usual tableau formula for Schur polynomials. The other two parts give new formulae for these polynomials.…”

“…To show the difference between subword and tableau complexes, consider the Young tableau complex for the 2 × 2 square shape with entries at most 3; it has dimension 3 and eight vertices, none of which is a cone vertex. On the other hand, deleting all cone points from the subword complex in [KMY05] having the same tableaux for facets yields a simplicial complex of dimension 2 with seven vertices.…”

“…This gives formulae for the Hilbert series and K-polynomials of vexillary matrix Schubert varieties (also known as (one-sided) ladder determinantal varieties). See [KMY05] for a treatment of the related algebraic geometry. 5.1.…”

“…The second of these formulae for vexillary double Grothendieck polynomials was based on the algebraic geometry of matrix Schubert varieties [KMY05]. It was that geometry that first motivated us to fit Young tableaux into a simplicial complex.…”

A jeu de taquin theory for increasing tableaux, with applications to K-theoretic Schubert calculus

“…Indeed, there has been significant interest in the Grothendieck ring of X and of related varieties; see work on, for example, quiver loci [Buch 2002a;2005a;Buch et al 2008], Hilbert series of determinantal ideals [Knutson and Miller 2005;Knutson et al 2008;2009], applications to invariants of matroids [Speyer 2006], and in relation to representation theory [Griffeth and Ram 2004;Lenart and Postnikov 2007;Willems 2006]. See also work of concerning combinatorial Hopf algebras.…”