@article {le2024qbpd, AUTHOR = {Le, Tuong and Ouyang, Shuge and Tao, Leo and Restivo, Joseph and Zhang, Angelina}, TITLE = {Quantum bumpless pipe dreams}, JOURNAL = {Forum Math. Sigma}, FJOURNAL = {Forum of Mathematics. Sigma}, VOLUME = {13}, YEAR = {2025}, PAGES = {Paper No. e28, 21}, MRCLASS = {05E05 (14N15)}, MRNUMBER = {4860030}, DOI = {10.1017/fms.2024.112}, URL = {https://doi.org/10.1017/fms.2024.112}, } @incollection {vafa, AUTHOR = {Vafa, Cumrun}, TITLE = {Topological mirrors and quantum rings}, BOOKTITLE = {Essays on mirror manifolds}, PAGES = {96--119}, PUBLISHER = {Int. Press, Hong Kong}, YEAR = {1992}, MRCLASS = {81T40 (14J30 32G81 32J27 32J81 32S10)}, MRNUMBER = {1191421}, MRREVIEWER = {Bruce\ Hunt}, } @incollection {Givental97, AUTHOR = {Givental, Alexander}, TITLE = {A tutorial on quantum cohomology}, BOOKTITLE = {Symplectic geometry and topology ({P}ark {C}ity, {UT}, 1997)}, SERIES = {IAS/Park City Math. Ser.}, VOLUME = {7}, PAGES = {231--264}, PUBLISHER = {Amer. Math. Soc., Providence, RI}, YEAR = {1999}, MRCLASS = {14N35 (53D45)}, MRNUMBER = {1702945}, MRREVIEWER = {Jun\ S.\ Song}, DOI = {10.1090/pcms/007/06}, URL = {https://doi.org/10.1090/pcms/007/06}, } @incollection {witten, AUTHOR = {Witten, Edward}, TITLE = {Two-dimensional gravity and intersection theory on moduli space}, BOOKTITLE = {Surveys in differential geometry ({C}ambridge, {MA}, 1990)}, PAGES = {243--310}, PUBLISHER = {Lehigh Univ., Bethlehem, PA}, YEAR = {1991}, MRCLASS = {32G15 (14C17 14H15 32G81 58F07 81T40)}, MRNUMBER = {1144529}, MRREVIEWER = {Steven\ Rosenberg}, } @article {assaf, AUTHOR = {Assaf, Sami H.}, TITLE = {A bijective proof of {K}ohnert's rule for {S}chubert polynomials}, JOURNAL = {Comb. Theory}, FJOURNAL = {Combinatorial Theory}, VOLUME = {2}, YEAR = {2022}, NUMBER = {1}, PAGES = {Paper No. 5, 9 pp.}, MRCLASS = {05E05 (05A05 05A19)}, MRNUMBER = {4405994}, DOI = {10.5070/c62156877}, } @article {magyar, AUTHOR = {Magyar, Peter}, TITLE = {Schubert polynomials and {B}ott-{S}amelson varieties}, JOURNAL = {Comment. Math. Helv.}, FJOURNAL = {Commentarii Mathematici Helvetici}, VOLUME = {73}, YEAR = {1998}, NUMBER = {4}, PAGES = {603--636}, MRCLASS = {14M15 (05E15 16G20 20G05)}, MRNUMBER = {1639896}, MRREVIEWER = {Witold\ Kra\'skiewicz}, DOI = {10.1007/s000140050071}, URL = {https://doi.org/10.1007/s000140050071}, } @article{huang, author = "Huang, Daoji", title = {Bijective proofs of {M}onk's rule for {S}chubert and double {S}chubert polynomials with bumpless pipe dreams}, journal = "Electronic Journal of Combinatorics", doi = {https://doi.org/10.37236/11824}, year = 2023, volume = "30", number = "3", } @article{FOMIN1996123, title = {The {Y}ang-{B}axter equation, symmetric functions, and {S}chubert polynomials}, journal = {Discrete Math.}, volume = {153}, number = {1}, pages = {123-143}, year = {1996}, note = {Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics}, doi = {https://doi.org/10.1016/0012-365X(95)00132-G}, url = {https://www.sciencedirect.com/science/article/pii/0012365X9500132G}, author = {Sergey Fomin and Anatol N. Kirillov}, keywords = {Yang-Baxter equation, Schubert polynomials, Symmetric functions}, abstract = {We present an approach to the theory of Schubert polynomials, corresponding symmetric functions, and their generalizations that is based on exponential solutions of the Yang-Baxter equation. In the case of the solution related to the nilCoxeter algebra of the symmetric group, we recover the Schubert polynomials of Lascoux and Schützenberger, and provide simplified proofs of their basic properties, along with various generalizations thereof. Our techniques make use of an explicit combinatorial interpretation of these polynomials in terms of configurations of labelled pseudo-lines.} } @article{qbsm, author = {Thomas Lam and Mark Shimozono}, JOURNAL = {Proc. Amer. Math. Soc.}, number = {3}, pages = {835--850}, publisher = {American Mathematical Society}, title = {Quantum double {S}chubert polynomials represent {S}chubert classes}, volume = {142}, year = {2014}, DOI = {10.1090/S0002-9939-2013-11831-9}, } @article{KNUTSON2004161, title = {Subword complexes in {C}oxeter groups}, journal = {Adv. Math.}, volume = {184}, number = {1}, pages = {161-176}, year = {2004}, doi = {https://doi.org/10.1016/S0001-8708(03)00142-7}, url = {https://www.sciencedirect.com/science/article/pii/S0001870803001427}, author = {Allen Knutson and Ezra Miller}, keywords = {Coxeter group, Reduced composition, Reduced word, Reduced expression, Subword, Simplicial complex, Shellable, Vertex-decomposable, Hilbert series, Grothendieck polynomial}, abstract = {Let (Π,Σ) be a Coxeter system. An ordered list of elements in Σ and an element in Π determine a subword complex, as introduced in Knutson and Miller (Ann. of Math. (2) (2003), to appear). Subword complexes are demonstrated here to be homeomorphic to balls or spheres, and their Hilbert series are shown to reflect combinatorial properties of reduced expressions in Coxeter groups. Two formulae for double Grothendieck polynomials, one of which appeared in Fomin and Kirillov (Proceedings of the Sixth Conference in Formal Power Series and Algebraic Combinatorics, DIMACS, 1994, pp. 183–190), are recovered in the context of simplicial topology for subword complexes. Some open questions related to subword complexes are presented.} } @article{KIRILLOV2000191, title = {Quantum double {S}chubert polynomials, quantum {S}chubert polynomials and {V}afa-{I}ntriligator formula}, journal = {Discrete Math.}, volume = {217}, number = {1}, pages = {191-223}, year = {2000}, doi = {https://doi.org/10.1016/S0012-365X(99)00263-0}, url = {https://www.sciencedirect.com/science/article/pii/S0012365X99002630}, author = {Anatol N. Kirillov and Toshiaki Maeno}, keywords = {Double quantum Schubert polynomials, Ehresman-Brunhat graph, Quantum Pieri's rule}, abstract = {We study algebraic aspects of equivariant quantum cohomology algebra of the flag manifold. We introduce and study the quantum double Schubert polynomials S̃w(x,y), which are the Lascoux–Schützenberger type representatives of the equivariant quantum cohomology classes. Our approach is based on the quantum Cauchy identity. We define also quantum Schubert polynomials S̃w(x) as the Gram–Schmidt orthogonalization of some set of monomials with respect to the scalar product, defined by the Grothendieck residue. Using quantum Cauchy identity, we prove that S̃w(x)=S̃w(x,y)|y=0 and as a corollary obtain a simple formula for the quantum Schubert polynomials S̃w(x)=∂ww0(y)S̃w0(x,y)|y=0. We also prove the higher genus analog of Vafa–Intriligator's formula for the flag manifolds and study the quantum residues generating function. We introduce the Ehresmann–Bruhat graph on the symmetric group and formulate the equivariant quantum Pieri rule. Résumé Nous étudions les aspects algébriques de la cohomologie quantique de la varieété de drapeaux. Nous introduisons et étudions les polynômes de Schubert doubles quantiques S̃w(x,y), qui sont les représentants des classes de cohomologie équivariantes du type des polynômes de Lascoux-Schützenberger. Notre approche est fondée sur une identité de cauchy quantique. Nous définissons aussi les polynômes de Schubert quantiques par un procédé d'orthogonalisation de Gram–Schmidt, par rapport à un produit scalaire défini à l'aide d'un résidu de Grothendieck. Utilisant la formule de Cauchy quantique, nous montrons que S̃w(x)=S̃w(x,y)|y=0, et comme corollaire, nous obtenons une formule simple pour les polynômes de Schubert quantiqes: S̃w(x)=∂ww0(y)S̃w0(x,y)|y=0. Nous prouvons aussi l'analogue, en genre plus élevé, de la formule de Vafa–Intriligator pour les variétés de drapeaux, et étudions la fonction génératrice des résidus quantiques. Nous introduisons enfin le graphe d'Ehresmann–Bruhat sur le groupe symétrique et énoncons la régle de Pieri quantique équivariante.} } @misc{knutsonschubert, title={SCHUBERT POLYNOMIALS AND SYMMETRIC FUNCTIONS NOTES FOR THE LISBON COMBINATORICS SUMMER SCHOOL 2012}, author={Knutson, Allen} } @article{billey1993some, title={Some combinatorial properties of Schubert polynomials}, author={Billey, Sara C and Jockusch, William and Stanley, Richard P}, journal={Journal of Algebraic Combinatorics}, volume={2}, number={4}, pages={345--374}, year={1993}, publisher={Springer} } @article{quantum, ISSN = {08940347, 10886834}, author = {Sergey Fomin and Sergei Gelfand and Alexander Postnikov}, JOURNAL = {J. Amer. Math. Soc.}, number = {3}, pages = {565--596}, publisher = {American Mathematical Society}, title = {Quantum {S}chubert polynomials}, volume = {10}, year = {1997}, DOI = {10.1090/S0894-0347-97-00237-3}, } @Book{ Book, title={ Notes on Schubert polynomials }, authors={ Ian G. MacDonald }, year={ 1990 }, publisher={ LACIM, Université du Québec à Montréal }, } @article{ bergeron, AUTHOR = {Bergeron, Nantel and Billey, Sara}, TITLE = {R{C}-graphs and {S}chubert polynomials}, JOURNAL = {Experiment. Math.}, FJOURNAL = {Experimental Mathematics}, VOLUME = {2}, YEAR = {1993}, NUMBER = {4}, PAGES = {257--269}, URL = {http://projecteuclid.org/euclid.em/1048516036}, } @article {erdosrenyi, AUTHOR = {Erd\H{o}s, P. and R\'{e}nyi, A.}, TITLE = {On the evolution of random graphs}, JOURNAL = {Magyar Tud. Akad. Mat. Kutat\'{o} Int. K\"{o}zl.}, FJOURNAL = {A Magyar Tudom\'{a}nyos Akad\'{e}mia. Matematikai Kutat\'{o} Int\'{e}zet\'{e}nek K\"{o}zlem\'{e}nyei}, VOLUME = {5}, YEAR = {1960}, PAGES = {17--61}, } @article{ckw, Author = {Calegari, Danny and Koch, Sarah and Walker, Alden}, Doi = {10.1017/etds.2016.17}, Fjournal = {Ergodic Theory and Dynamical Systems}, Journal = {Ergodic Theory Dynam. Systems}, Mrclass = {37F45 (28A80 37F30)}, Mrnumber = {3719268}, Number = {8}, Pages = {2487--2555}, Title = {Roots, {S}chottky semigroups, and a proof of {B}andt's conjecture}, Url = {https://doi-org.proxy.lib.umich.edu/10.1017/etds.2016.17}, Volume = {37}, Year = {2017}, Bdsk-Url-1 = {https://doi-org.proxy.lib.umich.edu/10.1017/etds.2016.17}, Bdsk-Url-2 = {https://doi.org/10.1017/etds.2016.17}} @unpublished{bousch88, Author = {Bousch, Thierry}, Date-Added = {2019-02-05 11:29:00 -0600}, Date-Modified = {2019-02-05 11:29:49 -0600}, Note = {preprint, available from author's webpage, \url{https://www.imo.universite-paris-saclay.fr/~bousch/preprints/}}, Title = {Paires de similtudes}, Year = {1988}} @inbook{cf, title = {Quantum double Schubert polynomials}, booktitle = { Appendix J in Schubert Varieties and Degeneracy Loci}, author = {Fulton, William}, year = {1998}, pages = {134--138}, series = {Lecture Notes in Mathematics}, publisher = {Springer} } @article{WEIGANDT2021105470, title = {Bumpless pipe dreams and alternating sign matrices}, JOURNAL = {J. Combin. Theory Ser. A}, volume = {182}, PAGES = {Paper No. 105470, 52 pp.}, year = {2021}, doi = {https://doi.org/10.1016/j.jcta.2021.105470}, url = {https://www.sciencedirect.com/science/article/pii/S0097316521000698}, author = {Anna Weigandt}, keywords = {Grothendieck polynomials, Bumpless pipe dreams, Alternating sign matrices}, abstract = {In their work on the infinite flag variety, Lam, Lee, and Shimozono [30] 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, giving an expression for double Grothendieck polynomials as a sum over a larger class of bumpless pipe dreams. Our proof relies on techniques found in an unpublished manuscript of Lascoux [27]. Lascoux showed how to write double Grothendieck polynomials as a sum over alternating sign matrices. We explain how to view the Lam-Lee-Shimozono formula as a disguised special case of Lascoux's alternating sign matrix formula. Knutson, Miller, and Yong [21] gave a tableau formula for vexillary Grothendieck polynomials. We recover this formula by showing vexillary marked bumpless pipe dreams and flagged set-valued tableaux are in weight preserving bijection. Finally, we give a bijection between Hecke bumpless pipe dreams and decreasing tableaux. The restriction of this bijection to Edelman-Greene bumpless pipe dreams solves a problem of Lam, Lee, and Shimozono.} } @inproceedings{fomin1994grothendieck, title={Grothendieck polynomials and the Yang-Baxter equation}, author={Fomin, Sergey and Kirillov, Anatol N}, booktitle={Proc. formal power series and alg. comb}, pages={183--190}, year={1994} } @article{thompson17, Author = {Thompson, Daniel J.}, Doi = {10.4171/CMH/424}, Fjournal = {Commentarii Mathematici Helvetici. A Journal of the Swiss Mathematical Society}, Journal = {Comment. Math. Helv.}, Mrclass = {37E05 (11R06 30C15 37B40)}, Mrnumber = {3718487}, Mrreviewer = {Joseph Andrew Vandehey}, Number = {4}, Pages = {777--800}, Title = {Generalized {$\beta$}-transformations and the entropy of unimodal maps}, Url = {https://doi-org.proxy.lib.umich.edu/10.4171/CMH/424}, Volume = {92}, Year = {2017}, Bdsk-Url-1 = {https://doi-org.proxy.lib.umich.edu/10.4171/CMH/424}, Bdsk-Url-2 = {https://doi.org/10.4171/CMH/424}} @article{Lam_2021, title={Back stable {S}chubert calculus}, volume={157}, ISSN={1570-5846}, url={http://dx.doi.org/10.1112/S0010437X21007028}, DOI={10.1112/s0010437x21007028}, number={5}, journal={Compos. Math.}, publisher={Wiley}, author={Lam, Thomas and Lee, Seung Jin and Shimozono, Mark}, year={2021}, pages={883–962} } @article {Lascoux82, AUTHOR = {Lascoux, Alain and Sch\"{u}tzenberger, Marcel-Paul}, TITLE = {Polyn\^{o}mes de {S}chubert}, JOURNAL = {C. R. Acad. Sci. Paris S\'{e}r. I Math.}, FJOURNAL = {Comptes Rendus des S\'{e}ances de l'Acad\'{e}mie des Sciences. S\'{e}rie I. Math\'{e}matique}, VOLUME = {294}, YEAR = {1982}, NUMBER = {13}, PAGES = {447--450}, MRCLASS = {14M17 (05A10 14N10)}, MRNUMBER = {660739}, } @article{Gao_2023, title={The Canonical Bijection between Pipe Dreams and Bumpless Pipe Dreams}, volume={2023}, ISSN={1687-0247}, url={http://dx.doi.org/10.1093/imrn/rnad083}, DOI={10.1093/imrn/rnad083}, number={21}, journal={International Mathematics Research Notices}, publisher={Oxford University Press (OUP)}, author={Gao, Yibo and Huang, Daoji}, year={2023}, month=apr, pages={18629–18663} }