Related papers: Bumpless Pipe Dream Fragments -- Equivariant Geome…
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…
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…
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…
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…
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…
Extending results of Wyser, we determine formulas for the equivariant cohomology classes of closed orbits of certain families of spherical subgroups of $GL_n$ on the flag variety $GL_n/B$. Putting this together with a slight extension of…
Pipedreams and bumpless pipedreams are two combinatorial models that compute double Grothendieck polynomials. While studying matrix Schubert varieties, Pechenik, Speyer, and Weigandt defined a permutation statistic$\mathsf{rajcode}(\cdot)$…
In this note, a new puzzle is introduced where the pipe dream and bumpless pipe dream can be played simultaneously. Using these, a combinatorial proof of the (ordinary) Schubert polynomials in terms of bumpless pipe dream is given. The main…
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…
In their work on the infinite flag variety, Lam, Lee, and Shimozono (2018) introduced objects called bumpless pipe dreams and used them to give a formula for double Schubert polynomials. We extend this formula to the setting of K-theory,…
We define skew Schubert polynomials to be normal form (polynomial) representatives of certain classes in the cohomology of a flag manifold. We show that this definition extends a recent construction of Schubert polynomials due to Bergeron…
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 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…
Knutson and Zinn-Justin recently found a puzzle rule for the expansion of the product $\mathfrak{G}_{u}(x,t)\cdot \mathfrak{G}_{v}(x,t)$ of two double Grothendieck polynomials indexed by permutations with separated descents. We establish…
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…
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…
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…
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…
In this paper, we first discuss the topological properties of projective Stiefel manifolds, we compute their cohomology rings and classify their cohomology endomorphisms; Then by embedding the flag manifold of a classical Lie group into its…
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…