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  • Undergraduate Poster Abstracts
  • Mathematics (General)

    THU-G27 MORITA EQUIVALENCE OF DUAL W*-CORRESPONDENCES DERIVED FROM FINITE-DIRECTED GRAPHS

    • Rene Ardila ;
    • Paul Muhly ;

    THU-G27

    MORITA EQUIVALENCE OF DUAL W*-CORRESPONDENCES DERIVED FROM FINITE-DIRECTED GRAPHS

    Rene Ardila, Paul Muhly.

    The University of Iowa, Iowa City, IA.

    Given a von Neumann algebra A and a W*-correspondence E over A, Muhly and Solel constructed an algebra H(E) which they called the Hardy algebra of E. This algebra is a noncommutative generalization of the classical Hardy space H of bounded analytic functions on the open unit disc. The particular kind of W*-correspondences that we study are built from finite, directed graphs. In fact, any finite dimensional correspondence E over a finite dimensional algebra A can be studied as a graph correspondence. Given any faithful normal representation σ of A on a Hilbert space H, there is a dual correspondence Eσ over the commutant σ(A)' such that H(E) can be realized in terms of B(H)-valued functions on the open unit ball D(Eσ)* of (Eσ)*. For different representations σ and τ of the algebra A, the dual correspondences Eσ and Eτ are not isomorphic. Are they Morita equivalent? This is an important question because a better understanding of the dual correspondences implies a better understanding of the Hardy algebra. We identify what all the objects in this theory (A, σ(A), σ(A)', Eσ, (Eσ)*, H(E), etc.) become when the correspondence is determined by a finite, directed graph, and then we show that the answer to this question is affirmative. Furthermore, we explore the implications of this answer in the general theory of noncommutative functions. Our methods include the use of equivalence bimodules and correspondence isomorphisms.

    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.