@book{DLRS-triang-book, title={Triangulations: Structures for algorithms and applications}, author={De Loera, J. and Rambau, J. and Santos, F.}, isbn={9783642129711}, lccn={2010933308}, series={Algorithms and Computation in Mathematics}, year={2010}, publisher={Springer Berlin Heidelberg} } @article{PRW08, AUTHOR = {Postnikov, Alex and Reiner, Victor and Williams, Lauren}, TITLE = {Faces of generalized permutohedra}, JOURNAL = {Doc. Math.}, FJOURNAL = {Documenta Mathematica}, VOLUME = {13}, YEAR = {2008}, PAGES = {207--273}, DOI = {10.4171/DM/248} } @article{AT10, AUTHOR = {Anderson, Dave and Tymoczko, Julianna}, TITLE = {Schubert polynomials and classes of {H}essenberg varieties}, JOURNAL = {J. Algebra}, FJOURNAL = {Journal of Algebra}, VOLUME = {323}, YEAR = {2010}, NUMBER = {10}, PAGES = {2605--2623}, DOI = {10.1016/j.jalgebra.2010.03.001}, URL = {https://doi.org/10.1016/j.jalgebra.2010.03.001}, } @incollection {AT20, AUTHOR = {Abe, Hiraku and Horiguchi, Tatsuya}, TITLE = {A survey of recent developments on {H}essenberg varieties}, BOOKTITLE = {Schubert calculus and its applications in combinatorics and representation theory}, SERIES = {Springer Proc. Math. Stat.}, VOLUME = {332}, PAGES = {251--279}, PUBLISHER = {Springer, Singapore}, YEAR = {2020}, ISBN = {978-981-15-7451-1; 978-981-15-7450-4}, MRCLASS = {14M15}, MRNUMBER = {4167519}, DOI = {10.1007/978-981-15-7451-1\_10}, URL = {https://doi.org/10.1007/978-981-15-7451-1_10}, } @article{Atk90, AUTHOR = {Atkinson, M. D.}, TITLE = {On computing the number of linear extensions of a tree}, JOURNAL = {Order}, FJOURNAL = {Order. A Journal on the Theory of Ordered Sets and its Applications}, VOLUME = {7}, YEAR = {1990}, NUMBER = {1}, PAGES = {23--25}, DOI = {10.1007/BF00383170}, URL = {https://doi.org/10.1007/BF00383170}, } @article {LMP21, AUTHOR = {Lee, Eunjeong and Masuda, Mikiya and Park, Seonjeong}, TITLE = {Toric {B}ruhat interval polytopes}, JOURNAL = {J. Combin. Theory Ser. A}, FJOURNAL = {Journal of Combinatorial Theory. Series A}, VOLUME = {179}, YEAR = {2021}, PAGES = {Paper No. 105387, 41}, MRCLASS = {52B05 (14M25)}, MRNUMBER = {4190574}, MRREVIEWER = {Margherita\ Barile}, DOI = {10.1016/j.jcta.2020.105387}, URL = {https://doi.org/10.1016/j.jcta.2020.105387}, } @article{GH24, title={Poset topology, moves, and Bruhat interval polytope lattices}, author={Gaetz, Christian and Hersh, Patricia}, eprinttype = {arXiv}, eprint = {2410.08076}, year={2024} } @misc{And07, title={Double Schubert polynomials and double Schubert varieties}, author={Anderson, Dave}, note={Notes}, url = {https://people.math.osu.edu/anderson.2804/papers/geomschpolyn.pdf}, year={2007} } @article {ADHM18, AUTHOR = {Abe, Hiraku and DeDieu, Lauren and Galetto, Federico and Harada, Megumi}, TITLE = {Geometry of {H}essenberg varieties with applications to {N}ewton-{O}kounkov bodies}, JOURNAL = {Selecta Math. (N.S.)}, FJOURNAL = {Selecta Mathematica. New Series}, VOLUME = {24}, YEAR = {2018}, NUMBER = {3}, PAGES = {2129--2163}, MRCLASS = {14M17 (14M10 14M25)}, MRNUMBER = {3816501}, MRREVIEWER = {Lucas\ Fresse}, DOI = {10.1007/s00029-018-0405-3}, URL = {https://doi.org/10.1007/s00029-018-0405-3}, } @article{CR11, abstract = {Let P(M) be the matroid base polytope of a matroid M. A matroid base polytope decomposition of P(M) is a decomposition of the form P(M)=Ui=1tP(M i) where each P(Mi) is also a matroid base polytope for some matroid Mi, and for each 1≤i≠'j≤t, the intersection P(Mi)∩P(Mj) is a face of both P(Mi) and P(Mj). In this paper, we investigate hyperplane splits, that is, polytope decompositions when t=2. We give sufficient conditions for M so P(M) has a hyperplane split and characterize when P(M1⊕M 2) has a hyperplane split where M1⊕M2 denote the direct sum of matroids M1 and M2. We also prove that P(M) has not a hyperplane split if M is binary. Finally, we show that P(M) has not a decomposition if its 1-skeleton is the hypercube. © 2010 Elsevier Inc. All rights reserved.}, author = {Vanessa Chatelain and Jorge Luis Ramírez Alfonsín}, doi = {10.1016/j.aam.2010.04.005}, issue = {1}, journal = {Advances in Applied Mathematics}, title = {Matroid base polytope decomposition}, volume = {47}, year = {2011}, } @article {BEW24, AUTHOR = {Boretsky, Jonathan and Eur, Christopher and Williams, Lauren}, TITLE = {Polyhedral and tropical geometry of flag positroids}, JOURNAL = {Algebra Number Theory}, FJOURNAL = {Algebra \& Number Theory}, VOLUME = {18}, YEAR = {2024}, NUMBER = {7}, PAGES = {1333--1374}, MRCLASS = {05E14 (14T15)}, MRNUMBER = {4757308}, DOI = {10.2140/ant.2024.18.1333}, URL = {https://doi.org/10.2140/ant.2024.18.1333}, } @article {TW15, AUTHOR = {Tsukerman, E. and Williams, L.}, TITLE = {Bruhat interval polytopes}, JOURNAL = {Adv. Math.}, FJOURNAL = {Advances in Mathematics}, VOLUME = {285}, YEAR = {2015}, PAGES = {766--810}, MRCLASS = {52B40 (05A05 06A07 20F55 52B05)}, MRNUMBER = {3406515}, MRREVIEWER = {Joseph\ Kung}, DOI = {10.1016/j.aim.2015.07.030}, URL = {https://doi.org/10.1016/j.aim.2015.07.030}, } @article{HHMP19, author = {Megumi Harada and Tatsuya Horiguchi and Mikiya Masuda and Seonjeong Park}, doi = {10.1134/S0081543819030192}, journal = {Proc. Steklov Inst. Math.}, title = {The volume polynomial of regular semisimple Hessenberg varieties and the {G}elfand--{Z}etlin polytope}, volume = {305}, pages = {318--344}, year = {2019}, } @article{NT23, abstract = {We compute the expansion of the cohomology class of the permutahedral variety in the basis of Schubert classes. The resulting structure constants aw are expressed as a sum of normalized mixed Eulerian numbers indexed naturally by reduced words of w. The description implies that the aw are positive for all permutations w\in S n of length n-1, thereby answering a question of Harada, Horiguchi, Masuda, and Park. We use the same expression to establish the invariance of aw under taking inverses and conjugation by the longest word and subsequently establish an intriguing cyclic sum rule for the numbers. We then move toward a deeper combinatorial understanding for the aw by exploiting in addition the relation to Postnikov's divided symmetrization. Finally, we are able to give a combinatorial interpretation for aw when w is vexillary, in terms of certain tableau descents. It is based in part on a relation between the aw and principal specializations of Schubert polynomials. Along the way, we prove results and raise questions of independent interest about the combinatorics of permutations, Schubert polynomials, and related objects. We also sketch how to extend our approach to other Lie types, highlighting an identity of Klyachko in particular.}, author = {Philippe Nadeau and Vasu Tewari}, doi = {10.1093/imrn/rnab337}, issue = {5}, journal = {International Mathematics Research Notices}, title = {The Permutahedral Variety, Mixed Eulerian Numbers, and Principal Specializations of Schubert Polynomials}, volume = {2023}, year = {2023}, } @article{ADGH18, abstract = {In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, allowing us to conclude that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain flat families of Hessenberg varieties, whose generic fibers are regular semisimple Hessenberg varieties and whose special fiber is a regular nilpotent Hessenberg variety, have reduced fibres. In the second half of the paper we present several applications of these results. First, we construct certain flags of subvarieties of a regular nilpotent Hessenberg variety, obtained by intersecting with Schubert varieties, with well-behaved geometric properties. Second, we give a computationally effective formula for the degree of a regular nilpotent Hessenberg variety with respect to a Plücker embedding. Third, we explicitly compute some Newton–Okounkov bodies of the two-dimensional Peterson variety.}, author = {Hiraku Abe and Lauren DeDieu and Federico Galetto and Megumi Harada}, doi = {10.1007/s00029-018-0405-3}, issue = {3}, journal = {Selecta Mathematica, New Series}, title = {Geometry of Hessenberg varieties with applications to Newton–Okounkov bodies}, volume = {24}, year = {2018}, } @article{ITW20, abstract = {Hessenberg varieties are subvarieties of the flag variety parametrized by a linear operator X and a nondecreasing function h. The family of Hessenberg varieties for regular X is particularly important: they are used in quantum cohomology, in combinatorial and geometric representation theory, in Schubert calculus and affine Schubert calculus. We show that the classes of a regular Hessenberg variety in the cohomology and K-theory of the flag variety are given by making certain substitutions in the Schubert polynomial (respectively Grothendieck polynomial) for a permutation that depends only on h. Our formula and our methods are different from a recent result of Abe, Fujita, and Zeng that gives the class of a regular Hessenberg variety with more restrictions on h than here.}, author = {Erik Insko and Julianna Tymoczko and Alexander Woo}, doi = {10.1016/j.jpaa.2019.106230}, issue = {5}, journal = {Journal of Pure and Applied Algebra}, title = {A formula for the cohomology and K-class of a regular Hessenberg variety}, volume = {224}, year = {2020}, } @article{Lian24, AUTHOR = {Lian, Carl}, TITLE = {The {HHMP} decomposition of the permutohedron and degenerations of torus orbits in flag varieties}, JOURNAL = {Int. Math. Res. Not. IMRN}, FJOURNAL = {International Mathematics Research Notices. IMRN}, YEAR = {2024}, NUMBER = {20}, PAGES = {13380--13399}, DOI = {10.1093/imrn/rnae204}, URL = {https://doi.org/10.1093/imrn/rnae204}, } @article {KW15, AUTHOR = {Kodama, Yuji and Williams, Lauren}, TITLE = {The full {K}ostant-{T}oda hierarchy on the positive flag variety}, JOURNAL = {Comm. Math. Phys.}, FJOURNAL = {Communications in Mathematical Physics}, VOLUME = {335}, YEAR = {2015}, NUMBER = {1}, PAGES = {247--283}, MRCLASS = {17B80 (14M15 22E46 52B12)}, MRNUMBER = {3314504}, MRREVIEWER = {Rutwig\ Campoamor-Stursberg}, DOI = {10.1007/s00220-014-2203-x}, URL = {https://doi.org/10.1007/s00220-014-2203-x}, } @book {BB05, AUTHOR = {Bj\"orner, Anders and Brenti, Francesco}, TITLE = {Combinatorics of {C}oxeter groups}, SERIES = {Graduate Texts in Mathematics}, VOLUME = {231}, PUBLISHER = {Springer, New York}, YEAR = {2005}, PAGES = {xiv+363}, ISBN = {978-3540-442387; 3-540-44238-3}, MRCLASS = {05-01 (05E15 20F55)}, MRNUMBER = {2133266}, MRREVIEWER = {Jian-yi\ Shi}, } @article {JLLO23, AUTHOR = {Joswig, Michael and Loho, Georg and Luber, Dante and Olarte, Jorge Alberto}, TITLE = {Generalized permutahedra and positive flag {D}ressians}, JOURNAL = {Int. Math. Res. Not. IMRN}, FJOURNAL = {International Mathematics Research Notices. IMRN}, YEAR = {2023}, NUMBER = {19}, PAGES = {16748--16777}, MRCLASS = {52B40 (05B35 14T15 52B05)}, MRNUMBER = {4651899}, DOI = {10.1093/imrn/rnac349}, URL = {https://doi.org/10.1093/imrn/rnac349}, } @misc{B23, title={Totally nonnegative tropical flags and the totally nonnegative flag {D}ressian}, author={Jonathan Boretsky}, year={2023}, eprinttype = {arXiv}, eprint = {2208.09128}, } @article {SW21, AUTHOR = {Speyer, David and Williams, Lauren K.}, TITLE = {The positive {D}ressian equals the positive tropical {G}rassmannian}, JOURNAL = {Trans. Amer. Math. Soc. Ser. B}, FJOURNAL = {Transactions of the American Mathematical Society. Series B}, VOLUME = {8}, YEAR = {2021}, PAGES = {330--353}, MRCLASS = {14T15 (05E14 14M15 52C40)}, MRNUMBER = {4241765}, MRREVIEWER = {Margherita\ Barile}, DOI = {10.1090/btran/67}, URL = {https://doi.org/10.1090/btran/67}, } @article {NT23Re, AUTHOR = {Nadeau, Philippe and Tewari, Vasu}, TITLE = {Remixed {E}ulerian numbers}, JOURNAL = {Forum Math. Sigma}, FJOURNAL = {Forum of Mathematics. Sigma}, VOLUME = {11}, YEAR = {2023}, PAGES = {Paper No. e65, 26}, MRCLASS = {05A15 (52B05)}, MRNUMBER = {4621503}, MRREVIEWER = {Wayne\ M.\ Dymacek}, DOI = {10.1017/fms.2023.57}, URL = {https://doi.org/10.1017/fms.2023.57}, }