@article {Adve.Robichaux.Yong,
    AUTHOR = {Adve, A. and Robichaux, C. and Yong, A.},
     TITLE = {An efficient algorithm for deciding vanishing of {S}chubert
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   JOURNAL = {Adv. Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {383},
      YEAR = {2021},
     PAGES = {Paper No. 107669, 38},
      ISSN = {0001-8708},
   MRCLASS = {14N15 (05E05 52B05 68Q17 68W30)},
  MRNUMBER = {4232542},
MRREVIEWER = {Trygve Johnsen},
       DOI = {10.1016/j.aim.2021.107669},
       URL = {https://doi.org/10.1016/j.aim.2021.107669},
}

@book {Bjorner.Brenti,
    AUTHOR = {Bj\"{o}rner, A. and Brenti, F.},
     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 {Deodhar,
    AUTHOR = {Deodhar, V. V.},
     TITLE = {Some characterizations of {B}ruhat ordering on a {C}oxeter
              group and determination of the relative {M}\"{o}bius function},
   JOURNAL = {Invent. Math.},
  FJOURNAL = {Inventiones Mathematicae},
    VOLUME = {39},
      YEAR = {1977},
    NUMBER = {2},
     PAGES = {187--198},
      ISSN = {0020-9910},
   MRCLASS = {20H15},
  MRNUMBER = {435249},
MRREVIEWER = {James E. Humphreys},
       DOI = {10.1007/BF01390109},
       URL = {https://doi.org/10.1007/BF01390109},
}

@article {Fan,
    AUTHOR = {Fan, N. J. Y. and Guo, P. L. and Peng, S. C. Y. and
              Sun, Sophie C. C.},
     TITLE = {Lattice points in the {N}ewton polytopes of key polynomials},
   JOURNAL = {SIAM J. Discrete Math.},
  FJOURNAL = {SIAM Journal on Discrete Mathematics},
    VOLUME = {34},
      YEAR = {2020},
    NUMBER = {2},
     PAGES = {1281--1289},
      ISSN = {0895-4801},
   MRCLASS = {52B20},
  MRNUMBER = {4109582},
MRREVIEWER = {Chuanming Zong},
       DOI = {10.1137/19M1305562},
       URL = {https://doi.org/10.1137/19M1305562},
}

@article {Fink,
    AUTHOR = {Fink, A. and M\'{e}sz\'{a}ros, K. and St. Dizier, A.},
     TITLE = {Schubert polynomials as integer point transforms of
              generalized permutahedra},
   JOURNAL = {Adv. Math.},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {332},
      YEAR = {2018},
     PAGES = {465--475},
      ISSN = {0001-8708},
   MRCLASS = {05E18 (14M25 14N15 52B12 52B20)},
  MRNUMBER = {3810259},
MRREVIEWER = {Vladimir N. Gruji\'{c}},
       DOI = {10.1016/j.aim.2018.05.028},
       URL = {https://doi.org/10.1016/j.aim.2018.05.028},
}

@book {Fulton,
    AUTHOR = {Fulton, W.},
     TITLE = {Young tableaux},
    SERIES = {London Mathematical Society Student Texts},
    VOLUME = {35},
      NOTE = {With applications to representation theory and geometry},
 PUBLISHER = {Cambridge University Press, Cambridge},
      YEAR = {1997},
     PAGES = {x+260},
      ISBN = {0-521-56144-2; 0-521-56724-6},
   MRCLASS = {05E10 (05E05 05E15 14M15 20G05)},
  MRNUMBER = {1464693},
MRREVIEWER = {Tadeusz J\'{o}zefiak},
}

@article {Gaetz,
    AUTHOR = {Gaetz, C.},
     TITLE = {Spherical {S}chubert varieties and pattern avoidance},
   JOURNAL = {Selecta Math. (N.S.)},
  FJOURNAL = {Selecta Mathematica. New Series},
    VOLUME = {28},
      YEAR = {2022},
    NUMBER = {2},
     PAGES = {Paper No. 44, 9},
      ISSN = {1022-1824},
   MRCLASS = {14M27 (05E14 14M15)},
  MRNUMBER = {4376215},
       DOI = {10.1007/s00029-022-00760-8},
       URL = {https://doi.org/10.1007/s00029-022-00760-8},
}

@book {LR08,
    AUTHOR = {Lakshmibai, V. and Raghavan, K. N.},
     TITLE = {Standard Monomial Theory},
    SERIES = {Encyclopaedia of Mathematical Sciences},
    VOLUME = {137},
      NOTE = {Invariant theoretic approach,
              Invariant Theory and Algebraic Transformation Groups, 8},
 PUBLISHER = {Springer-Verlag, Berlin},
      YEAR = {2008},
     PAGES = {xiv+265},
      ISBN = {978-3-540-76756-5},
   MRCLASS = {14M15 (14L24 14L30 14M12 14M17)},
  MRNUMBER = {2388163},
MRREVIEWER = {Dmitry A. Timash\"{e}v},
}

@article {Newman,
    AUTHOR = {Newman, M. H. A.},
     TITLE = {On theories with a combinatorial definition of
              ``equivalence.''},
   JOURNAL = {Ann. of Math. (2)},
  FJOURNAL = {Annals of Mathematics. Second Series},
    VOLUME = {43},
      YEAR = {1942},
     PAGES = {223--243},
      ISSN = {0003-486X},
   MRCLASS = {02.0X},
  MRNUMBER = {7372},
MRREVIEWER = {O. Ore},
       DOI = {10.2307/1968867},
       URL = {https://doi.org/10.2307/1968867},
}

@article {Perrin,
    AUTHOR = {Perrin, N.},
     TITLE = {On the geometry of spherical varieties},
   JOURNAL = {Transform. Groups},
  FJOURNAL = {Transformation Groups},
    VOLUME = {19},
      YEAR = {2014},
    NUMBER = {1},
     PAGES = {171--223},
      ISSN = {1083-4362},
   MRCLASS = {14M27 (14-02)},
  MRNUMBER = {3177371},
MRREVIEWER = {Paolo Bravi},
       DOI = {10.1007/s00031-014-9254-0},
       URL = {https://doi.org/10.1007/s00031-014-9254-0},
}

@article {STEM,
    AUTHOR = {Stembridge, J. R.},
     TITLE = {Multiplicity-free products of {S}chur functions},
   JOURNAL = {Ann. Comb.},
  FJOURNAL = {Annals of Combinatorics},
    VOLUME = {5},
      YEAR = {2001},
    NUMBER = {2},
     PAGES = {113--121},
      ISSN = {0218-0006},
   MRCLASS = {05E05 (05E15 20G05)},
  MRNUMBER = {1904379},
MRREVIEWER = {Timothy Y. Chow},
       DOI = {10.1007/s00026-001-8008-6},
       URL = {https://doi.org/10.1007/s00026-001-8008-6},
}

@article {Tenner,
    AUTHOR = {Tenner, B. E.},
     TITLE = {Pattern avoidance and the {B}ruhat order},
   JOURNAL = {J. Combin. Theory Ser. A},
  FJOURNAL = {Journal of Combinatorial Theory. Series A},
    VOLUME = {114},
      YEAR = {2007},
    NUMBER = {5},
     PAGES = {888--905},
      ISSN = {0097-3165},
   MRCLASS = {05E15 (06A07 20F55)},
  MRNUMBER = {2333139},
MRREVIEWER = {Alessandro Conflitti},
       DOI = {10.1016/j.jcta.2006.10.003},
       URL = {https://doi.org/10.1016/j.jcta.2006.10.003},
}

@misc{Lascoux:polynomials,
  author = {Lascoux, A.},
  title = {{Polynomials}},
  url = {https://www.math.uwaterloo.ca/~opecheni/lascoux.pdf},
  year = {2013}, 
  note = "Online; Accessed March 29, 2022"
}

@article {Kohnert,
    AUTHOR = {Kohnert, A.},
     TITLE = {Weintrauben, {P}olynome, {T}ableaux},
      NOTE = {Dissertation, Universit\"{a}t Bayreuth, Bayreuth, 1990},
   JOURNAL = {Bayreuth. Math. Schr.},
  FJOURNAL = {Bayreuther Mathematische Schriften},
    NUMBER = {38},
      YEAR = {1991},
     PAGES = {1--97},
      ISSN = {0172-1062},
   MRCLASS = {05E10 (05A10 05A17 14M15 20C30)},
  MRNUMBER = {1132534},
MRREVIEWER = {Volker Strehl},
}

@misc{Gao.Hodges.Yong,
    AUTHOR = {Gao, Y. and Hodges, R. and Yong, A},
    EPRINT = {2104.10101},
    EPRINTTYPE = {arxiv},
    TITLE = {Classification of Levi-spherical Schubert varieties},
    YEAR = {2021}}

@misc{Brewster.Hodges.Yong,
    AUTHOR = {Brewster, D. and Hodges, R. and Yong, A},
    EPRINT = {2012.09749},
    EPRINTTYPE = {arxiv},
    TITLE = {Proper permutations, Schubert geometry, and randomness},
    YEAR = {2020}}

@misc{Hodges.Yong1,
    AUTHOR = {{Hodges}, R. and {Yong}, A.},
    EPRINT = {2007.09238},
    EPRINTTYPE = {arxiv},
    TITLE = {Coxeter combinatorics and spherical Schubert geometry},
    YEAR = {2020}}

@misc{Hodges.Yong2,
    AUTHOR = {{Hodges}, R. and {Yong}, A.},
    EPRINT = {2007.09229},
    EPRINTTYPE = {arxiv},
    TITLE = {Multiplicity-free key polynomials},
    YEAR = {2020}}