@article{Brenti1994,
author = {Brenti, Francesco},
doi = {10.1007/BF01231537},
file = {:home/gbarkley/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Brenti - 1994 - A combinatorial formula for Kazhdan-Lusztig polynomials.pdf:pdf},
issn = {1432-1297},
journal = {Inventiones mathematicae 1994 118:1},
keywords = {Mathematics,general},
month = {dec},
number = {1},
pages = {371--394},
publisher = {Springer},
title = {{A combinatorial formula for Kazhdan-Lusztig polynomials}},
url = {https://link.springer.com/article/10.1007/BF01231537},
volume = {118},
year = {1994}
}
@article{Kazhdan1979,
author = {Kazhdan, David and Lusztig, George},
doi = {10.1007/BF01390031},
file = {:home/gbarkley/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Kazhdan, Lusztig - 1979 - Representations of Coxeter groups and Hecke algebras.pdf:pdf},
issn = {1432-1297},
journal = {Inventiones mathematicae 1979 53:2},
keywords = {Mathematics,general},
month = {jun},
number = {2},
pages = {165--184},
publisher = {Springer},
title = {{Representations of Coxeter groups and Hecke algebras}},
url = {https://link.springer.com/article/10.1007/BF01390031},
volume = {53},
year = {1979}
}
@article{Carrell1994,
author = {Carrell, James B},
doi = {10.1090/pspum/056.1/1278700},
file = {:home/gbarkley/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Carrell - 1994 - The Bruhat Graph of a Coxeter Group, a Conjecture of Deodhar, and Rational Smoothness of Schubert Varieties.pdf:pdf},
title = {{The Bruhat Graph of a Coxeter Group, a Conjecture of Deodhar, and Rational Smoothness of Schubert Varieties}},
url = {http://dx.doi.org/10.1090/pspum/056.1/1278700},
volume = {56},
year = {1994}
}
@techreport{Dyer1993,
author = {Dyer, M J},
file = {:home/gbarkley/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Dyer - 1993 - COMPOSITIO MATHEMATICA Hecke algebras and shellings of Bruhat intervals(2).pdf:pdf},
pages = {91--115},
title = {{COMPOSITIO MATHEMATICA Hecke algebras and shellings of Bruhat intervals}},
url = {http://www.numdam.org/conditions},
volume = {89},
year = {1993}
}
@article{Patimo2021,
archivePrefix = {arXiv},
arxivId = {1811.08184},
author = {Patimo, Leonardo},
doi = {10.1093/imrn/rnz255},
eprint = {1811.08184},
file = {:home/gbarkley/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Patimo - 2021 - A combinatorial formula for the coefficient of q in Kazhdan-Lusztig polynomials.pdf:pdf},
issn = {16870247},
journal = {International Mathematics Research Notices},
number = {5},
pages = {3203--3223},
title = {{A combinatorial formula for the coefficient of q in Kazhdan-Lusztig polynomials}},
volume = {2021},
year = {2021}
}
@article{Deodhar1985,
author = {Deodhar, Vinay V.},
doi = {10.1080/00927878508823227},
file = {:home/gbarkley/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Deodhar - 1985 - Local Poincare Duality and Non-Singularity of Schubert Varieties.pdf:pdf},
issn = {15324125},
journal = {Communications in Algebra},
number = {6},
pages = {1379--1388},
title = {{Local Poincare Duality and Non-Singularity of Schubert Varieties}},
volume = {13},
year = {1985}
}
@article{Burrull2022,
abstract = {The combinatorial invariance conjecture (due independently to Lusztig and Dyer) predicts that if $[x,y]$ and $[x^{\prime},y^{\prime}]$ are isomorphic Bruhat posets (of possibly different Coxeter systems), then the corresponding Kazhdan–Lusztig polynomials are equal, that is, $P_{x,y}(q)=P_{x^{\prime},y^{\prime}}(q)$. We prove this conjecture for the affine Weyl group of type $\widetilde {A}_2$. This is the first infinite group with non-trivial Kazhdan–Lusztig polynomials where the conjecture is proved.},
archivePrefix = {arXiv},
arxivId = {2105.04609},
author = {Burrull, Gaston and Libedinsky, Nicolas and Plaza, David},
doi = {10.1093/IMRN/RNAC105},
file = {:home/gbarkley/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Burrull, Libedinsky, Plaza - 2022 - Combinatorial Invariance Conjecture for $widetilde {A}_2$.pdf:pdf},
issn = {1073-7928},
journal = {International Mathematics Research Notices},
month = {may},
publisher = {Oxford University Press (OUP)},
title = {{Combinatorial Invariance Conjecture for $\widetilde {A}_2$}},
url = {https://academic.oup.com/imrn/advance-article/doi/10.1093/imrn/rnac105/6582898},
year = {2022}
}
@book{Bjorner2005,
abstract = {This book is a carefully written exposition of Coxeter groups, an area of mathematics which appears in algebra, geometry, and combinatorics. In this book, the combinatorics of Coxeter groups has mainly to do with reduced expressions, partial order of group elements, enumeration, associated graphs and combinatorial cell complexes, and connections with combinatorial representation theory. While Coxeter groups have already been exposited from algebraic and geometric perspectives, this book will be presenting the combinatorial aspects of Coxeter groups. The authors have included an exposition of Coxeter groups along with a rich variety of exercises, ranging from easy to very difficult, giving the book the unique character of serving as both a textbook and a monograph.},
author = {Bj{\"o}rner, Anders and Brenti, Francesco},
booktitle = {Graduate Texts in Mathematics},
doi = {10.1007/3-540-27596-7},
file = {:home/gbarkley/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Bjorner, Brenti - 2005 - Combinatorics of Coxeter Groups.pdf:pdf},
publisher = {Springer},
title = {{Combinatorics of Coxeter Groups}},
year = {2005}
}
@article{Tsukerman2015,
abstract = {Let u and v be permutations on n letters, with u≤v in Bruhat order. A Bruhat interval polytope Qu,v is the convex hull of all permutation vectors z=. (z(1), z(2), . . .. , z(n)) with u≤z≤v. Note that when u=e and v=w0 are the shortest and longest elements of the symmetric group, Qe,w0 is the classical permutohedron. Bruhat interval polytopes were studied recently in [15] by Kodama and the second author, in the context of the Toda lattice and the moment map on the flag variety.In this paper we study combinatorial aspects of Bruhat interval polytopes. For example, we give an inequality description and a dimension formula for Bruhat interval polytopes, and prove that every face of a Bruhat interval polytope is a Bruhat interval polytope. A key tool in the proof of the latter statement is a generalization of the well-known lifting property for Coxeter groups. Motivated by the relationship between the lifting property and R-polynomials, we also give a generalization of the standard recurrence for R-polynomials. Finally, we define a more general class of polytopes called Bruhat interval polytopes for G/P, which are moment map images of (closures of) totally positive cells in (G/P)≥0, and are a special class of Coxeter matroid polytopes. Using tools from total positivity and the Gelfand-Serganova stratification, we show that the face of any Bruhat interval polytope for G/P is again a Bruhat interval polytope for G/P.},
author = {Tsukerman, E. and Williams, L.},
doi = {10.1016/J.AIM.2015.07.030},
file = {:home/gbarkley/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Tsukerman, Williams - 2015 - Bruhat interval polytopes.pdf:pdf},
issn = {0001-8708},
journal = {Advances in Mathematics},
keywords = {Bruhat order,Flag varieties,Kazhdan-Lusztig polynomials,Lifting property,Matroid stratification,Polytopes,R-polynomials,Total positivity},
month = {nov},
pages = {766--810},
publisher = {Academic Press},
title = {{Bruhat interval polytopes}},
volume = {285},
year = {2015}
}
@article{Brenti2006,
abstract = {In 1979 Kazhdan and Lusztig defined, for every Coxeter group W, a family of polynomials, indexed by pairs of elements of W, which have become known as the Kazhdan-Lusztig polynomials of W, and which have proven to be of importance in several areas of mathematics. In this paper, we show that the combinatorial concept of a special matching plays a fundamental role in the computation of these polynomials. Our results also imply, and generalize, the recent one in [Adv. in Math. 180 (2003) 146-175] on the combinatorial invariance of Kazhdan-Lusztig polynomials. {\textcopyright} 2005 Elsevier Inc. All rights reserved.},
author = {Brenti, Francesco and Caselli, Fabrizio and Marietti, Mario},
doi = {10.1016/j.aim.2005.01.011},
file = {:home/gbarkley/math/FPSAC2023/1-s2.0-S0001870805000939-main (1).pdf:pdf},
issn = {00018708},
journal = {Advances in Mathematics},
keywords = {Bruhat order,Coxeter groups,Kazhdan-Lusztig polynomials,Special matching},
number = {2},
pages = {555--601},
title = {{Special matchings and Kazhdan-Lusztig polynomials}},
volume = {202},
year = {2006}
}
@article{Brenti2004,
abstract = {We give an explicit and entirely poset-theoretic way to compute, for any permutation v, all the Kazhdan-Lusztig polynomials Px,y for x,y≤v, starting from the Bruhat interval [e,v] as an abstract poset. This proves, in particular, that the intersection cohomology of Schubert varieties depends only on the inclusion relations between the closures of its Schubert cells. {\textcopyright} 2003 Elsevier Ltd. All rights reserved.},
author = {Brenti, Francesco},
doi = {10.1016/J.EJC.2003.10.011},
file = {:home/gbarkley/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Brenti - 2004 - The intersection cohomology of Schubert varieties is a combinatorial invariant.pdf:pdf},
issn = {0195-6698},
journal = {European Journal of Combinatorics},
month = {nov},
number = {8},
pages = {1151--1167},
publisher = {Academic Press},
title = {{The intersection cohomology of Schubert varieties is a combinatorial invariant}},
volume = {25},
year = {2004}
}
@article{Delanoy2006,
abstract = {We show that for Bruhat intervals starting from the identity in Coxeter groups the conjecture of Lusztig and Dyer holds, that is, the R-polynomials and the Kazhdan-Lusztig polynomials defined on [e,u] only depend on the isomorphism type of [e,u]. To achieve this we use the purely poset-theoretic notion of special matching. Our approach is essentially a synthesis of the explicit formula for special matchings discovered by Brenti and the general special matching machinery developed by Du Cloux. {\textcopyright} Springer Science+Business Media, LLC 2006.},
author = {Delanoy, Ewan},
doi = {10.1007/s10801-006-0014-7},
file = {:home/gbarkley/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Delanoy - 2006 - Combinatorial invariance of Kazhdan-Lusztig polynomials on intervals starting from the identity.pdf:pdf},
issn = {15729192},
journal = {Journal of Algebraic Combinatorics},
keywords = {Coxeter group,Kazhdan-Lusztig polynomials,Special matching},
number = {4},
pages = {437--463},
title = {{Combinatorial invariance of Kazhdan-Lusztig polynomials on intervals starting from the identity}},
volume = {24},
year = {2006}
}
@phdthesis{Dyer1987,
author = {Dyer, Matthew},
file = {:home/gbarkley/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Dyer - 1987 - Hecke algebras and reflections in Coxeter groups.pdf:pdf},
school = {University of Sydney},
title = {{Hecke algebras and reflections in Coxeter groups}},
year = {1987}
}
@article{Blundell2021,
abstract = {Kazhdan-Lusztig polynomials are important and mysterious objects in
representation theory. Here we present a new formula for their computation for
symmetric groups based on the Bruhat graph. Our approach suggests a solution to
the combinatorial invariance conjecture for symmetric groups, a well-known
conjecture formulated by Lusztig and Dyer in the 1980s.},
archivePrefix = {arXiv},
arxivId = {2111.15161},
author = {Blundell, Charles and Buesing, Lars and Davies, Alex and Veli{\v{c}}kovi{\'{c}}, Petar and Williamson, Geordie},
doi = {10.48550/arxiv.2111.15161},
eprint = {2111.15161},
file = {:home/gbarkley/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Blundell et al. - 2021 - Towards combinatorial invariance for Kazhdan-Lusztig polynomials.pdf:pdf},
month = {nov},
title = {{Towards combinatorial invariance for Kazhdan-Lusztig polynomials}},
url = {https://arxiv.org/abs/2111.15161v1},
year = {2021}
}
@article {Blundell2023,
    AUTHOR = {Blundell, Charles and Buesing, Lars and Davies, Alex and
              Veli\v{c}kovi\'{c}, Petar and Williamson, Geordie},
     TITLE = {Towards combinatorial invariance for {K}azhdan-{L}usztig
              polynomials},
   JOURNAL = {Representation Theory},
  FJOURNAL = {Representation Theory. An Electronic Journal of the American
              Mathematical Society},
    VOLUME = {26},
      YEAR = {2022},
     PAGES = {1145--1191},
   MRCLASS = {05E10 (17B10)},
  MRNUMBER = {4510816},
MRREVIEWER = {\c{S}erban B\u{a}rc\u{a}nescu},
       DOI = {10.1090/ert/624},
       URL = {https://doi.org/10.1090/ert/624},
}
@article{DuCloux2003,
abstract = {Let [e,w], [e,w'] be intervals from the origin in two Coxeter groups W, W'. We prove that any poset isomorphism $\varphi$ from [e,w] to [e,w'] preserves Kazhdan-Lusztig polynomials, in the sense that $P_{\varphi(x),\varphi(y)}=P_{x,y}$ for all $x\leq y$ in $[e,w]$, in the case where each irreducible factor of one of the groups $W$, $W'$ has a Coxeter graph which is a tree, or is affine of type $\tilde A_n$. In particular, the result holds for all finite or affine Coxeter groups. We obtain this result as a consequence of a detailed analysis of all poset isomorphisms from [e,w] to [e,w'].},
author = {du Cloux, Fokko},
doi = {10.1016/S0001-8708(02)00100-7},
file = {:home/gbarkley/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/du Cloux - 2003 - Rigidity of Schubert closures and invariance of Kazhdan-Lusztig polynomials.pdf:pdf},
issn = {00018708},
journal = {Advances in Mathematics},
keywords = {Bruhat ordering,Kazhdan-Lusztig polynomials,Schubert varieties},
month = {dec},
number = {1},
pages = {146--175},
publisher = {Academic Press Inc.},
title = {{Rigidity of Schubert closures and invariance of Kazhdan-Lusztig polynomials}},
url = {https://hal.archives-ouvertes.fr/hal-00005543},
volume = {180},
year = {2003}
}
@article{Incitti2006,
abstract = {In this paper, we solve the conjecture about the combinatorial invariance of Kazhdan-Lusztig polynomials for the first open cases, showing that it is true for intervals of length 5 and 6 in the symmetric group. We also obtain explicit formulas for the R-polynomials and for the Kazhdan-Lusztig polynomials associated with any interval of length 5 in any Coxeter group, showing in particular what they look like in the symmetric group. {\textcopyright} 2005 Elsevier Inc. All rights reserved.},
author = {Incitti, Federico},
doi = {10.1016/J.JCTA.2005.12.003},
file = {:home/gbarkley/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Incitti - 2006 - On the combinatorial invariance of Kazhdan–Lusztig polynomials.pdf:pdf},
issn = {0097-3165},
journal = {Journal of Combinatorial Theory. Series A},
keywords = {Bruhat order,Combinatorial invariance conjecture,Coxeter group,Kazhdan-Lusztig polynomial,Symmetric group},
month = {oct},
number = {7},
pages = {1332--1350},
publisher = {Academic Press},
title = {{On the combinatorial invariance of Kazhdan–Lusztig polynomials}},
volume = {113},
year = {2006}
}
@article{Dyer1991a,
author = {Dyer, Matthew},
file = {:home/gbarkley/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Dyer - 1991 - On the “Bruhat graph” of a Coxeter system.pdf:pdf},
journal = {Compositio Mathematica},
number = {2},
pages = {185--191},
title = {{On the “Bruhat graph” of a Coxeter system}},
url = {http://www.numdam.org/item/CM_1991__78_2_185_0/},
volume = {78},
year = {1991}
}
@article{Dyer2012a,
annote = {Bruhat graph restriction 1.4},
author = {Dyer, M. J.},
doi = {10.1090/conm/139/1197833},
file = {:home/gbarkley/.local/share/data/Mendeley Ltd./Mendeley Desktop/Downloaded/Dyer - 2012 - Hecke algebras and shellings of Bruhat intervals. II. Twisted Bruhat orders.pdf:pdf},
pages = {141--165},
title = {{Hecke algebras and shellings of Bruhat intervals. II. Twisted Bruhat orders}},
volume = {139},
year = {2012}
}
@misc{Barkley2023,
      title={Combinatorial invariance for elementary intervals}, 
      author={Grant T. Barkley and Christian Gaetz},
      year={2023},
      eprint={2303.15577},
      archivePrefix={arXiv},
      primaryClass={math.CO}
}
@article {Brenti1997,
    AUTHOR = {Brenti, Francesco},
     TITLE = {Combinatorial properties of the {K}azhdan-{L}usztig
              {$R$}-polynomials for {$S_n$}},
   JOURNAL = {Advances in Mathematics},
  FJOURNAL = {Advances in Mathematics},
    VOLUME = {126},
      YEAR = {1997},
    NUMBER = {1},
     PAGES = {21--51},
      ISSN = {0001-8708},
   MRCLASS = {20C30 (05E15 06A07)},
  MRNUMBER = {1440252},
MRREVIEWER = {Robert B\'{e}dard},
       DOI = {10.1006/aima.1996.1607},
       URL = {https://doi.org/10.1006/aima.1996.1607},
}
@article {Incitti2007,
    AUTHOR = {Incitti, Federico},
     TITLE = {More on the combinatorial invariance of {K}azhdan-{L}usztig
              polynomials},
   JOURNAL = {Journal of Combinatorial Theory. Series A},
  FJOURNAL = {Journal of Combinatorial Theory. Series A},
    VOLUME = {114},
      YEAR = {2007},
    NUMBER = {3},
     PAGES = {461--482},
      ISSN = {0097-3165},
   MRCLASS = {05E15 (20F55)},
  MRNUMBER = {2310745},
MRREVIEWER = {Riccardo Biagioli},
       DOI = {10.1016/j.jcta.2006.06.009},
       URL = {https://doi.org/10.1016/j.jcta.2006.06.009},
}