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Rhombic tilings and Bott-Samelson varieties


Authors: Laura Escobar, Oliver Pechenik, Bridget Eileen Tenner and Alexander Yong
Journal: Proc. Amer. Math. Soc. 146 (2018), 1921-1935
MSC (2010): Primary 05B45, 05E15, 14M15
DOI: https://doi.org/10.1090/proc/13869
Published electronically: December 26, 2017
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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).


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Additional Information

Laura Escobar
Affiliation: Department of Mathematics, University of Illinois at Urbana–Champaign, Urbana, Illinois 61801
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
Email: pechenik@umich.edu

Bridget Eileen Tenner
Affiliation: Department of Mathematical Sciences, DePaul University, Chicago, Illinois 60614
Email: bridget@math.depaul.edu

Alexander Yong
Affiliation: Department of Mathematics, University of Illinois at Urbana–Champaign, Urbana, Illinois 61801
Email: ayong@uiuc.edu

DOI: https://doi.org/10.1090/proc/13869
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
Article copyright: © Copyright 2017 American Mathematical Society

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