@article {SM-advances,
    AUTHOR = {Brenti, Francesco and Caselli, Fabrizio and Marietti, Mario},
     TITLE = {Special matchings and {K}azhdan-{L}usztig polynomials},
   JOURNAL = {Adv. Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {202},
      YEAR = {2006},
    NUMBER = {2},
     PAGES = {555--601},
      ISSN = {0001-8708},
   MRCLASS = {05E15 (20C08)},
  MRNUMBER = {2222360},
MRREVIEWER = {Jian-yi Shi},
       DOI = {10.1016/j.aim.2005.01.011},
       URL = {https://doi-org.ezp-prod1.hul.harvard.edu/10.1016/j.aim.2005.01.011},
}

@article {SM-lower-classification,
    AUTHOR = {Caselli, Fabrizio and Marietti, Mario},
     TITLE = {A simple characterization of special matchings in lower
              {B}ruhat intervals},
   JOURNAL = {Discrete Math.},
  FJOURNAL = {Discrete Mathematics},
    VOLUME = {341},
      YEAR = {2018},
    NUMBER = {3},
     PAGES = {851--862},
      ISSN = {0012-365X},
   MRCLASS = {05E15 (20F55)},
  MRNUMBER = {3754399},
MRREVIEWER = {Alexander Diaz-Lopez},
       DOI = {10.1016/j.disc.2017.12.005},
       URL = {https://doi-org.ezp-prod1.hul.harvard.edu/10.1016/j.disc.2017.12.005},
}

@article {SM-first-lower-classification,
    AUTHOR = {Caselli, Fabrizio and Marietti, Mario},
     TITLE = {Special matchings in {C}oxeter groups},
   JOURNAL = {European J. Combin.},
  FJOURNAL = {European Journal of Combinatorics},
    VOLUME = {61},
      YEAR = {2017},
     PAGES = {151--166},
      ISSN = {0195-6698},
   MRCLASS = {20F55 (05E15 06A07)},
  MRNUMBER = {3588715},
MRREVIEWER = {Christian Stump},
       DOI = {10.1016/j.ejc.2016.10.007},
       URL = {https://doi-org.ezp-prod1.hul.harvard.edu/10.1016/j.ejc.2016.10.007},
}

@book {bjorner-brenti,
    AUTHOR = {Bj\"{o}rner, 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 {self-dual,
    AUTHOR = {Gaetz, Christian and Gao, Yibo},
     TITLE = {Self-dual intervals in the {B}ruhat order},
   JOURNAL = {Selecta Math. (N.S.)},
  FJOURNAL = {Selecta Mathematica. New Series},
    VOLUME = {26},
      YEAR = {2020},
    NUMBER = {5},
     PAGES = {Paper No. 77, 23},
      ISSN = {1022-1824},
   MRCLASS = {05E99 (14M15)},
  MRNUMBER = {4177574},
MRREVIEWER = {Marko Radovanovi\'{c}},
       DOI = {10.1007/s00029-020-00608-z},
       URL = {https://doi-org.ezp-prod1.hul.harvard.edu/10.1007/s00029-020-00608-z},
}

@article {lakshmibai-sandhya,
    AUTHOR = {Lakshmibai, V. and Sandhya, B.},
     TITLE = {Criterion for smoothness of {S}chubert varieties in {${\rm
              Sl}(n)/B$}},
   JOURNAL = {Proc. Indian Acad. Sci. Math. Sci.},
  FJOURNAL = {Indian Academy of Sciences. Proceedings. Mathematical
              Sciences},
    VOLUME = {100},
      YEAR = {1990},
    NUMBER = {1},
     PAGES = {45--52},
      ISSN = {0253-4142},
   MRCLASS = {14M15 (14L35)},
  MRNUMBER = {1051089},
MRREVIEWER = {H. H. Andersen},
       DOI = {10.1007/BF02881113},
       URL = {https://doi-org.ezp-prod1.hul.harvard.edu/10.1007/BF02881113},
}

@article {kazhdan-lusztig-polynomials,
    AUTHOR = {Kazhdan, David and Lusztig, George},
     TITLE = {Representations of {C}oxeter groups and {H}ecke algebras},
   JOURNAL = {Invent. Math.},
  FJOURNAL = {Inventiones Mathematicae},
    VOLUME = {53},
      YEAR = {1979},
    NUMBER = {2},
     PAGES = {165--184},
      ISSN = {0020-9910},
   MRCLASS = {20H15 (17B35 20G05 22E47)},
  MRNUMBER = {560412},
MRREVIEWER = {Vinay V. Deodhar},
       DOI = {10.1007/BF01390031},
       URL = {https://doi-org.ezp-prod1.hul.harvard.edu/10.1007/BF01390031},
}

@article {billey-postnikov,
    AUTHOR = {Billey, Sara and Postnikov, Alexander},
     TITLE = {Smoothness of {S}chubert varieties via patterns in root
              subsystems},
   JOURNAL = {Adv. in Appl. Math.},
  FJOURNAL = {Advances in Applied Mathematics},
    VOLUME = {34},
      YEAR = {2005},
    NUMBER = {3},
     PAGES = {447--466},
      ISSN = {0196-8858},
   MRCLASS = {14M15 (17B20)},
  MRNUMBER = {2123545},
MRREVIEWER = {Philip A. Foth},
       DOI = {10.1016/j.aam.2004.08.003},
       URL = {https://doi-org.ezp-prod1.hul.harvard.edu/10.1016/j.aam.2004.08.003},
}

@article {richmond-slofstra-fiber-bundle,
    AUTHOR = {Richmond, Edward and Slofstra, William},
     TITLE = {Billey-{P}ostnikov decompositions and the fibre bundle
              structure of {S}chubert varieties},
   JOURNAL = {Math. Ann.},
  FJOURNAL = {Mathematische Annalen},
    VOLUME = {366},
      YEAR = {2016},
    NUMBER = {1-2},
     PAGES = {31--55},
      ISSN = {0025-5831},
   MRCLASS = {14M15},
  MRNUMBER = {3552231},
MRREVIEWER = {Justin Brown},
       DOI = {10.1007/s00208-015-1299-4},
       URL = {https://doi-org.ezp-prod1.hul.harvard.edu/10.1007/s00208-015-1299-4},
}

@article {richmond-slofstra-enumeration,
    AUTHOR = {Richmond, Edward and Slofstra, William},
     TITLE = {Staircase diagrams and enumeration of smooth {S}chubert
              varieties},
   JOURNAL = {J. Combin. Theory Ser. A},
  FJOURNAL = {Journal of Combinatorial Theory. Series A},
    VOLUME = {150},
      YEAR = {2017},
     PAGES = {328--376},
      ISSN = {0097-3165},
   MRCLASS = {14M15 (05E18)},
  MRNUMBER = {3645580},
MRREVIEWER = {Peter Crooks},
       DOI = {10.1016/j.jcta.2017.03.009},
       URL = {https://doi-org.ezp-prod1.hul.harvard.edu/10.1016/j.jcta.2017.03.009},
}

@article {bjorner-ekedahl,
    AUTHOR = {Bj\"{o}rner, Anders and Ekedahl, Torsten},
     TITLE = {On the shape of {B}ruhat intervals},
   JOURNAL = {Ann. of Math. (2)},
  FJOURNAL = {Annals of Mathematics. Second Series},
    VOLUME = {170},
      YEAR = {2009},
    NUMBER = {2},
     PAGES = {799--817},
      ISSN = {0003-486X},
   MRCLASS = {05E99 (05A05 06A11 14F20 14F43 14M15 20F55)},
  MRNUMBER = {2552108},
MRREVIEWER = {Frank Sottile},
       DOI = {10.4007/annals.2009.170.799},
       URL = {https://doi-org.ezp-prod1.hul.harvard.edu/10.4007/annals.2009.170.799},
}

@article {SM-original,
    AUTHOR = {Brenti, Francesco},
     TITLE = {The intersection cohomology of {S}chubert varieties is a
              combinatorial invariant},
   JOURNAL = {European J. Combin.},
  FJOURNAL = {European Journal of Combinatorics},
    VOLUME = {25},
      YEAR = {2004},
    NUMBER = {8},
     PAGES = {1151--1167},
      ISSN = {0195-6698},
   MRCLASS = {14M15 (05E15 06A07)},
  MRNUMBER = {2095476},
MRREVIEWER = {Harry Tamvakis},
       DOI = {10.1016/j.ejc.2003.10.011},
       URL = {https://doi-org.ezp-prod1.hul.harvard.edu/10.1016/j.ejc.2003.10.011},
}

@article {Waterhouse,
    AUTHOR = {Waterhouse, William C.},
     TITLE = {Automorphisms of the {B}ruhat order on {C}oxeter groups},
   JOURNAL = {Bull. London Math. Soc.},
  FJOURNAL = {The Bulletin of the London Mathematical Society},
    VOLUME = {21},
      YEAR = {1989},
    NUMBER = {3},
     PAGES = {243--248},
      ISSN = {0024-6093},
   MRCLASS = {20F05 (06D30 06F10 20H15)},
  MRNUMBER = {986365},
MRREVIEWER = {S. Milne},
       DOI = {10.1112/blms/21.3.243},
       URL = {https://doi-org.ezp-prod1.hul.harvard.edu/10.1112/blms/21.3.243},
}

@article {SM-diamonds,
    AUTHOR = {Brenti, Francesco and Caselli, Fabrizio and Marietti, Mario},
     TITLE = {Diamonds and {H}ecke algebra representations},
   JOURNAL = {Int. Math. Res. Not.},
  FJOURNAL = {International Mathematics Research Notices},
      YEAR = {2006},
     PAGES = {Art. ID 29407, 34},
      ISSN = {1073-7928},
   MRCLASS = {05E10 (20C08)},
  MRNUMBER = {2250022},
MRREVIEWER = {David E. Anderson},
       DOI = {10.1155/IMRN/2006/29407},
       URL = {https://doi-org.ezp-prod1.hul.harvard.edu/10.1155/IMRN/2006/29407},
}

@article {SM-zircon,
    AUTHOR = {Marietti, Mario},
     TITLE = {Algebraic and combinatorial properties of zircons},
   JOURNAL = {J. Algebraic Combin.},
  FJOURNAL = {Journal of Algebraic Combinatorics. An International Journal},
    VOLUME = {26},
      YEAR = {2007},
    NUMBER = {3},
     PAGES = {363--382},
      ISSN = {0925-9899},
   MRCLASS = {06A07 (05E15 20F55)},
  MRNUMBER = {2348102},
MRREVIEWER = {Alessandro Conflitti},
       DOI = {10.1007/s10801-007-0061-8},
       URL = {https://doi-org.ezp-prod1.hul.harvard.edu/10.1007/s10801-007-0061-8},
}

@article {SM-topology,
    AUTHOR = {Abdallah, Nancy and Hansson, Mikael and Hultman, Axel},
     TITLE = {Topology of posets with special partial matchings},
   JOURNAL = {Adv. Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {348},
      YEAR = {2019},
     PAGES = {255--276},
      ISSN = {0001-8708},
   MRCLASS = {06A07 (05C70 05E45 55U05)},
  MRNUMBER = {3925932},
MRREVIEWER = {Konrad P. Pi\'{o}ro},
       DOI = {10.1016/j.aim.2019.02.031},
       URL = {https://doi-org.ezp-prod1.hul.harvard.edu/10.1016/j.aim.2019.02.031},
}

@book {billey-book,
    AUTHOR = {Billey, Sara and Lakshmibai, V.},
     TITLE = {Singular loci of {S}chubert varieties},
    SERIES = {Progress in Mathematics},
    VOLUME = {182},
 PUBLISHER = {Birkh\"{a}user Boston, Inc., Boston, MA},
      YEAR = {2000},
     PAGES = {xii+251},
      ISBN = {0-8176-4092-4},
   MRCLASS = {14M15 (14L35 20F55 20G05)},
  MRNUMBER = {1782635},
MRREVIEWER = {Michel Brion},
       DOI = {10.1007/978-1-4612-1324-6},
       URL = {https://doi-org.ezp-prod1.hul.harvard.edu/10.1007/978-1-4612-1324-6},
}

@incollection {Carrell-smoothness,
    AUTHOR = {Carrell, James B.},
     TITLE = {The {B}ruhat graph of a {C}oxeter group, a conjecture of
              {D}eodhar, and rational smoothness of {S}chubert varieties},
 BOOKTITLE = {Algebraic groups and their generalizations: classical methods
              ({U}niversity {P}ark, {PA}, 1991)},
    SERIES = {Proc. Sympos. Pure Math.},
    VOLUME = {56},
     PAGES = {53--61},
 PUBLISHER = {Amer. Math. Soc., Providence, RI},
      YEAR = {1994},
   MRCLASS = {14M15 (14L35)},
  MRNUMBER = {1278700},
MRREVIEWER = {E. Aky\i ld\i z},
}

@article {Gasharov,
    AUTHOR = {Gasharov, Vesselin},
     TITLE = {Factoring the {P}oincar\'{e} polynomials for the {B}ruhat order on
              {$S_n$}},
   JOURNAL = {J. Combin. Theory Ser. A},
  FJOURNAL = {Journal of Combinatorial Theory. Series A},
    VOLUME = {83},
      YEAR = {1998},
    NUMBER = {1},
     PAGES = {159--164},
      ISSN = {0097-3165},
   MRCLASS = {05E05 (14M15)},
  MRNUMBER = {1629464},
MRREVIEWER = {Ernesto Vallejo},
       DOI = {10.1006/jcta.1997.2861},
       URL = {https://doi.org/10.1006/jcta.1997.2861},
}

@article {Oh-Postnikov-Yoo,
    AUTHOR = {Oh, Suho and Postnikov, Alexander and Yoo, Hwanchul},
     TITLE = {Bruhat order, smooth {S}chubert varieties, and hyperplane
              arrangements},
   JOURNAL = {J. Combin. Theory Ser. A},
  FJOURNAL = {Journal of Combinatorial Theory. Series A},
    VOLUME = {115},
      YEAR = {2008},
    NUMBER = {7},
     PAGES = {1156--1166},
      ISSN = {0097-3165},
   MRCLASS = {06A07 (05A05 05B35 14N15)},
  MRNUMBER = {2450335},
MRREVIEWER = {David E. Anderson},
       DOI = {10.1016/j.jcta.2008.01.003},
       URL = {https://doi.org/10.1016/j.jcta.2008.01.003},
}

@incollection {Tits-words,
    AUTHOR = {Tits, Jacques},
     TITLE = {Le probl\`eme des mots dans les groupes de {C}oxeter},
 BOOKTITLE = {Symposia {M}athematica ({INDAM}, {R}ome, 1967/68), {V}ol. 1},
     PAGES = {175--185},
 PUBLISHER = {Academic Press, London},
      YEAR = {1969},
   MRCLASS = {20.10},
  MRNUMBER = {0254129},
MRREVIEWER = {J. E. Humphreys},
}

@article {dyer-bruhat-graph,
    AUTHOR = {Dyer, Matthew},
     TITLE = {On the ``{B}ruhat graph'' of a {C}oxeter system},
   JOURNAL = {Compositio Math.},
  FJOURNAL = {Compositio Mathematica},
    VOLUME = {78},
      YEAR = {1991},
    NUMBER = {2},
     PAGES = {185--191},
      ISSN = {0010-437X},
   MRCLASS = {20F55 (06F15 20F05)},
  MRNUMBER = {1104786},
MRREVIEWER = {Robert B\'{e}dard},
       URL = {http://www.numdam.org/item?id=CM_1991__78_2_185_0},
}

@article {GKM,
    AUTHOR = {Goresky, Mark and Kottwitz, Robert and MacPherson, Robert},
     TITLE = {Equivariant cohomology, {K}oszul duality, and the localization
              theorem},
   JOURNAL = {Invent. Math.},
  FJOURNAL = {Inventiones Mathematicae},
    VOLUME = {131},
      YEAR = {1998},
    NUMBER = {1},
     PAGES = {25--83},
      ISSN = {0020-9910},
   MRCLASS = {55N91 (14F25 14F32 16E99 18G10 55N33)},
  MRNUMBER = {1489894},
MRREVIEWER = {Roy Joshua},
       DOI = {10.1007/s002220050197},
       URL = {https://doi-org.ezp-prod1.hul.harvard.edu/10.1007/s002220050197},
}

@article {Brenti-combinatorial-formula,
    AUTHOR = {Brenti, Francesco},
     TITLE = {A combinatorial formula for {K}azhdan-{L}usztig polynomials},
   JOURNAL = {Invent. Math.},
  FJOURNAL = {Inventiones Mathematicae},
    VOLUME = {118},
      YEAR = {1994},
    NUMBER = {2},
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      ISSN = {0020-9910},
   MRCLASS = {20F55},
  MRNUMBER = {1292116},
MRREVIEWER = {Toshiyuki Tanisaki},
       DOI = {10.1007/BF01231537},
       URL = {https://doi-org.ezp-prod1.hul.harvard.edu/10.1007/BF01231537},
}

@article {Dyer-hecke-algebras,
    AUTHOR = {Dyer, M. J.},
     TITLE = {Hecke algebras and shellings of {B}ruhat intervals},
   JOURNAL = {Compositio Math.},
  FJOURNAL = {Compositio Mathematica},
    VOLUME = {89},
      YEAR = {1993},
    NUMBER = {1},
     PAGES = {91--115},
      ISSN = {0010-437X},
   MRCLASS = {20F55 (06A08)},
  MRNUMBER = {1248893},
MRREVIEWER = {Robert B\'{e}dard},
       URL = {http://www.numdam.org/item?id=CM_1993__89_1_91_0},
}

@misc{blundell2021towards,
  title={Towards combinatorial invariance for {K}azhdan-{L}usztig polynomials},
  author={Blundell, Charles and Buesing, Lars and Davies, Alex and Veli{\v{c}}kovi{\'c}, Petar and Williamson, Geordie},
  eprint={2111.15161},
  year={2021}
}

@article{davies2021advancing,
  title={Advancing mathematics by guiding human intuition with AI},
  author={Davies, Alex and Veli{\v{c}}kovi{\'c}, Petar and Buesing, Lars and Blackwell, Sam and Zheng, Daniel and Toma{\v{s}}ev, Nenad and Tanburn, Richard and Battaglia, Peter and Blundell, Charles and Juh{\'a}sz, Andr{\'a}s and others},
  journal={Nature},
  volume={600},
  number={7887},
  pages={70--74},
  year={2021},
  publisher={Nature Publishing Group}
}

@article {guillemin-holm-zara,
    AUTHOR = {Guillemin, V. and Holm, T. and Zara, C.},
     TITLE = {A {GKM} description of the equivariant cohomology ring of a
              homogeneous space},
   JOURNAL = {J. Algebraic Combin.},
  FJOURNAL = {Journal of Algebraic Combinatorics. An International Journal},
    VOLUME = {23},
      YEAR = {2006},
    NUMBER = {1},
     PAGES = {21--41},
      ISSN = {0925-9899},
   MRCLASS = {55N91 (22F30 57S10)},
  MRNUMBER = {2218848},
MRREVIEWER = {Tomasz Rybicki},
       DOI = {10.1007/s10801-006-6027-4},
       URL = {https://doi-org.ezp-prod1.hul.harvard.edu/10.1007/s10801-006-6027-4},
}

@article {VT-original,
    AUTHOR = {Gaetz, Christian and Gao, Yibo},
     TITLE = {On automorphisms of undirected {B}ruhat graphs},
   JOURNAL = {Math. Z.},
  FJOURNAL = {Mathematische Zeitschrift},
    VOLUME = {303},
      YEAR = {2023},
    NUMBER = {2},
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   MRCLASS = {05E18 (05E16 20B27 20F55)},
  MRNUMBER = {4530180},
       DOI = {10.1007/s00209-022-03194-2},
       URL = {https://doi.org/10.1007/s00209-022-03194-2},
}