Related papers: Duality of antidiagonals and pipe dreams
In their study of infinite flag varieties, Lam, Lee, and Shimozono (2021) introduced bumpless pipe dreams in a new combinatorial formula for double Schubert polynomials. These polynomials are the TxT-equivariant cohomology classes of matrix…
Knutson and Miller (2005) established a connection between the anti-diagonal Gr\"obner degenerations of matrix Schubert varieties and the pre-existing combinatorics of pipe dreams. They used this correspondence to give a…
Schubert polynomials for the classical groups were defined by S.Billey and M.Haiman in 1995; they are polynomial representatives of Schubert classes in a full flag variety of a classical group. We provide a combinatorial description for…
Schubert polynomials represent a basis for the cohomology of the complete flag variety and thus play a central role in geometry and combinatorics. In this context, Schubert polynomials are generating functions over various combinatorial…
Schubert polynomials are polynomial representatives of Schubert classes in the cohomology of the complete flag variety and have a combinatorial formulation in terms of bumpless pipe dreams. Quantum double Schubert polynomials are polynomial…
The geometric naturality of Schubert polynomials and their combinatorial pipe dream representations was established by Knutson and Miller (2005) via antidiagonal Gr\"obner degeneration of matrix Schubert varieties. We consider instead…
We study the factorization of Schubert polynomials into elementary symmetric polynomials. We conjecture that this occurs when the permutation corresponding to the Schubert polynomial does not contain the patterns $1432$, $1423$, $4132$, and…
In recent work, Hamaker, Pechenik, Speyer, and Weigandt showed that a certain differential operator $\nabla$ expands positively in the basis of Schubert polynomials. For $\pi$ a dominant permutation, Gaetz and Tung showed an analogous…
In [KU23] were introduced hybrid pipe dreams interpolating between classic and bumpless pipe dreams, each hybridization giving a different formula for double Schubert polynomials. A bijective proof was given (following [GH23]) of the…
We develop a family of new combinatorial models for key polynomials. It is similar to the hybrid pipe dream model for Schubert polynomials defined recently by Knutson and Udell.
Lascoux and Sch\"utzenberger introduced Schubert and Grothendieck polynomials to study the cohomology and K-theory of the complete flag variety. We present explicit combinatorial rules for expressing Grothendieck polynomials in the basis of…
Schubert polynomials are distinguished representatives of Schubert cycles in the cohomology of the flag variety. In the spirit of Bergeron and Sottile, we use the Bruhat order to give $(n-1)!$ different combinatorial formulas for the…
We give bijective proofs of Monk's rule for Schubert and double Schubert polynomials computed with bumpless pipe dreams. In particular, they specialize to bijective proofs of transition and cotransition formulas of Schubert and double…
Involution Schubert polynomials represent cohomology classes of $K$-orbit closures in the complete flag variety, where $K$ is the orthogonal or symplectic group. We show they also represent $T$-equivariant cohomology classes of subvarieties…
We construct a bijection between marked bumpless pipedreams with reverse compatible pairs, which are in bijection with not-necessarily-reduced pipedreams. This directly unifies various formulas for Grothendieck polynomials in the…
Convex algebraic geometry concerns the interplay between optimization theory and real algebraic geometry. Its objects of study include convex semialgebraic sets that arise in semidefinite programming and from sums of squares. This article…
Schubert polynomials are a basis for the polynomial ring that represent Schubert classes for the flag manifold. In this paper, we introduce and develop several new combinatorial models for Schubert polynomials that relate them to other…
We recall the lower-upper varieties from [Knutson '05] and give a formula for their equivariant cohomology classes, as a sum over generic pipe dreams. We recover as limits the classic and bumpless pipe dream formulae for double Schubert…
We consider quadrangles of perimeter $2$ in the plane with marked directed edge. To such quadrangle $Q$ a two-dimensional plane $\Pi\in\mathbb{R}^4$ with orthonormal base is corresponded. Orthogonal plane $\Pi^\bot$ defines a plane…
We give a new proof that three families of polynomials coincide: the double Schubert polynomials of Lascoux and Sch\"utzenberger defined by divided difference operators, the pipe dream polynomials of Bergeron and Billey, and the equivariant…