Abstract
Coxeter groups are of significant interest to communities in combinatorics, algebra, and geometry. Their structures and properties are both deeply beautiful and still not entirely understood. We explore some of the many enumerative aspects of these objects. We count elements with desirable properties, we give size orderings to certain features of all group elements, and we relate some of these statistics in ways that give bounds and rankings—if not exact values—to their sizes. Our primary tools come from leveraging different group presentations against each other, and interpreting element properties at the level of generator representations. In this article, we present a sampling of results in this area, putting them into context and hopefully inspiring future research.
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References
N. Bergeron, C. Ceballos, and J.-P. Labbé, Fan realizations of type A subword complexes and multi-associahedra of rank 3, Discrete Computational Geom.54 (2015), 195–231.
A. Björner and F. Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics 231, Springer, New York, 2005.
S. Elnitsky, Rhombic tilings of polygons and classes of reduced words in Coxeter groups, J. Comb. Theory, Ser. A77 (1997), 193–221.
C. K. Fan, Schubert varieties and short braidedness, Transform. Groups3 (1998), 51–56.
S. Fishel, E. Milićević, R. Patrias, and B. E. Tenner, Enumerations relating braid and commutation classes, European J. Comb.74 (2018), 11–26.
S. Kitaev, Patterns in Permutations and Words, Monographs in Theoretical Computer Science, Springer-Verlag, Berlin, 2011.
S. Linton, N. Rus~kuc, V. Vatter, Permutation Patterns, London Mathematical Society Lecture Note Series 376, Cambridge University Press, Cambridge, 2010.
K. Ragnarsson and B. E. Tenner, Homotopy type of the boolean complex of a Coxeter system, Adv. Math.222 (2009), 409–430.
K. Ragnarsson and B. E. Tenner, Homology of the boolean complex, J. Algebraic Comb.34 (2011), 617–639.
R. P. Stanley, On the number of reduced decompositions of elements of Coxeter groups, European J. Comb.5 (1984), 359–372.
B. E. Tenner, Reduced decompositions and permutation patterns, J. Algebraic Comb.24 (2006), 263–284.
B. E. Tenner, Pattern avoidance and the Bruhat order, J. Comb. Theory, Ser. A114 (2007), 888–905.
B. E. Tenner, Reduced word manipulation: patterns and enumeration, J. Algebraic Comb.46 (2017), 189–217.
J. West, Generating trees and forbidden subsequences, Discrete Math.157 (1996), 363–374.
H. Wilf, The patterns of permutations, Discrete Math.257 (2002), 575–583.
D. M. Zollinger, Equivalence classes of reduced words, Master’s thesis, University of Minnesota, 1994.
Acknowledgement
Research partially supported by Simons Foundation Collaboration Grant for Mathematicians 277603.
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Tenner, B.E. (2020). Enumerating in Coxeter Groups (Survey). In: Acu, B., Danielli, D., Lewicka, M., Pati, A., Saraswathy RV, Teboh-Ewungkem, M. (eds) Advances in Mathematical Sciences. Association for Women in Mathematics Series, vol 21. Springer, Cham. https://doi.org/10.1007/978-3-030-42687-3_5
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DOI: https://doi.org/10.1007/978-3-030-42687-3_5
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