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  • Undergraduate Poster Abstracts
  • FRI-G27 PEAK SETS OF COXETER GROUPS OF CLASSICAL LIE TYPES

    • Darleen Perez-Lavin ;
    • Erik Insko ;
    • Pamela Harris ;
    • Alex Diaz ;

    FRI-G27

    PEAK SETS OF COXETER GROUPS OF CLASSICAL LIE TYPES

    Darleen Perez-Lavin1, Erik Insko1, Pamela Harris2, Alex Diaz3.

    1Florida Gulf Coast University, Fort Myers, FL, 2United States Military Academy, West Point, NY, 3University of Notre Dame, Notre Dame, IN.

    We say a permutation π = π1 π 2 ... πn in the symmetric group Sn has a peak at index i if
    π{i-1}  < πi > π{i+1} and we let P(π) = {π in [n] | {i is a peak of π}}. Given a set S of positive integers, we let P(Sn) denote the subset of Sn consisting of all permutations π, where P(π) = S. In 2013, Billey, Burdzy, and Sagan proved |P(Sn)| = p(n)2{n-|S|-1}, where p(n) is a polynomial of degree max(S) - 1. In 2014, Castro-Velez et al., considered the Coxeter group of type B, n as the group of signed permutations on n letters and showed that |P[B(Sn)]|=p(n)2{2n-|S|-1}. We partitioned the set P(Sn) ⊂ Sn studied by Billey, Burdzy, and Sagan into subsets P[(Sn), a(k)] ⊂ Sn of permutations with peak set S that ends with an ascent to a fixed integer k and provided polynomial formulas for the cardinalities of these subsets. After embedding the Coxeter groups of Lie type Cn and D;n into S{2n}, we partitioned these groups into bundles of permutations π1... πn π{n+1}... π{2n} that have the same relative order as some permutation σ1 σ2 ... σn in Sn. This allowed us to count the number of permutations in types Cn and Dn with a given peak set S by reducing the enumeration to calculations in type A and sums across the rows of Pascal's triangle.