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1. Bijective Proofs of Monk's rule for Schubert and Double Schubert Polynomials with Bumpless Pipe Dreams

   https://doi.org/10.37236/11824  

   Huang, Daoji (July 2023, The Electronic Journal of Combinatorics)

   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 Schubert polynomials, which can be used to establish bijections with ordinary pipe dreams. 

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2. The Canonical Bijection between Pipe Dreams and Bumpless Pipe Dreams

   https://doi.org/10.1093/imrn/rnad083  

   Gao, Yibo; Huang, Daoji (April 2023, International Mathematics Research Notices)

   Abstract We present a direct bijection between reduced pipe dreams and reduced bumpless pipe dreams (BPDs) by interpreting reduced compatible sequences on BPDs and show that this is the unique bijection preserving bijective realizations of Monk’s formula, establishing its canonical nature. 

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3. Grothendieck Shenanigans: Permutons from pipe dreams via integrable probability

   Morales, Alejandro H; Panova, Greta; Petrov, Leonid; Yeliussizov, Damir (July 2024, arXiv)

   We study random permutations arising from reduced pipe dreams. Our main model is motivated by Grothendieck polynomials with parameter $$\beta=1$$ arising in K-theory of the flag variety. The probability weight of a permutation is proportional to the principal specialization (setting all variables to 1) of the corresponding Grothendieck polynomial. By mapping this random permutation to a version of TASEP (Totally Asymmetric Simple Exclusion Process), we describe the limiting permuton and fluctuations around it as the order $$n$$ of the permutation grows to infinity. The fluctuations are of order $$n^{\frac13}$$ and have the Tracy-Widom GUE distribution, which places this algebraic (K-theoretic) model into the Kardar-Parisi-Zhang universality class. We also investigate non-reduced pipe dreams and make progress on a recent open problem on the asymptotic number of inversions of the resulting permutation. Inspired by Stanley's question for the maximal value of principal specializations of Schubert polynomials, we resolve the analogous question for $$\beta=1$$ Grothendieck polynomials, and provide bounds for general $$\beta$$. 

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4. Back Stable K -Theory Schubert Calculus

   https://doi.org/10.1093/imrn/rnac315  

   Lam, Thomas; Lee, Seung Jin; Shimozono, Mark (November 2022, International Mathematics Research Notices)

   Abstract We study the back stable $$K$$-theory Schubert calculus of the infinite flag variety. We define back stable (double) Grothendieck polynomials and double $$K$$-Stanley functions and establish coproduct expansion formulae. Applying work of Weigandt, we extend our previous results on bumpless pipedreams from cohomology to $$K$$-theory. We study finiteness and positivity properties of the ring of back stable Grothendieck polynomials and divided difference operators in $$K$$-homology. 

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5. Double Schubert polynomials do have saturated Newton polytopes

   https://doi.org/10.1017/fms.2023.101  

   Castillo, Federico; Cid-Ruiz, Yairon; Mohammadi, Fatemeh; Montaño, Jonathan (January 2023, Forum of Mathematics, Sigma)

   Abstract We prove that double Schubert polynomials have the saturated Newton polytope property. This settles a conjecture by Monical, Tokcan and Yong. Our ideas are motivated by the theory of multidegrees. We introduce a notion of standardization of ideals that enables us to study nonstandard multigradings. This allows us to show that the support of the multidegree polynomial of each Cohen–Macaulay prime ideal in a nonstandard multigrading, and in particular, that of each Schubert determinantal ideal is a discrete polymatroid. 

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