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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Rhombic tilings and Bott–Samelson varieties
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by Laura Escobar, Oliver Pechenik, Bridget Eileen Tenner and Alexander Yong PDF
Proc. Amer. Math. Soc. 146 (2018), 1921-1935 Request permission

Abstract:

S. Elnitsky (1997) gave an elegant bijection between rhombic tilings of $2n$-gons and commutation classes of reduced words in the symmetric group on $n$ letters. P. Magyar (1998) found an important construction of the Bott–Samelson varieties introduced by H. C. Hansen (1973) and M. Demazure (1974). We explain a natural connection between S. Elnitsky’s and P. Magyar’s results. This suggests using tilings to encapsulate Bott–Samelson data (in type $A$). It also indicates a geometric perspective on S. Elnitsky’s bijection. We also extend this construction by assigning desingularizations of Schubert varieties to the zonotopal tilings considered by B. Tenner (2006).
References
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Additional Information
  • Laura Escobar
  • Affiliation: Department of Mathematics, University of Illinois at Urbana–Champaign, Urbana, Illinois 61801
  • MR Author ID: 990678
  • ORCID: 0000-0002-7970-4152
  • Email: lescobar@illinois.edu
  • Oliver Pechenik
  • Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854
  • Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
  • MR Author ID: 863417
  • ORCID: 0000-0002-7090-2072
  • Email: pechenik@umich.edu
  • Bridget Eileen Tenner
  • Affiliation: Department of Mathematical Sciences, DePaul University, Chicago, Illinois 60614
  • MR Author ID: 776323
  • Email: bridget@math.depaul.edu
  • Alexander Yong
  • Affiliation: Department of Mathematics, University of Illinois at Urbana–Champaign, Urbana, Illinois 61801
  • MR Author ID: 693975
  • Email: ayong@uiuc.edu
  • Received by editor(s): July 13, 2016
  • Received by editor(s) in revised form: July 6, 2017
  • Published electronically: December 26, 2017
  • Additional Notes: The second author was supported by an NSF Graduate Research Fellowship.
    The third author was partially supported by a Simons Foundation Collaboration Grant for Mathematicians.
    The fourth author was supported by an NSF grant.
  • Communicated by: Patricia Hersh
  • © Copyright 2017 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 146 (2018), 1921-1935
  • MSC (2010): Primary 05B45, 05E15, 14M15
  • DOI: https://doi.org/10.1090/proc/13869
  • MathSciNet review: 3767346