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Lisa Carbone

  1. Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits.

    Wed, 03/03/2010 - 16:01 — SomNambul
    Authors: Lisa Carbone, Leigh Cobbs, Sjuvon Chung, Robert McRae, Debajyoti Nandi, Yusra Naqvi, Diego Penta
    Subjects: Representation Theory
    Abstract

    We give a criterion for a Dynkin diagram, equivalently a generalized Cartan
    matrix, to be symmetrizable. This criterion is easily checked on the Dynkin
    diagram. We obtain a simple proof that the maximal rank of a Dynkin diagram of
    compact hyperbolic type is 5, while the maximal rank of a symmetrizable Dynkin
    diagram of compact hyperbolic type is 4. Building on earlier classification
    results of Kac, Kobayashi-Morita, Li and Sa\c{c}lio\~{g}lu, we present the 238
    hyperbolic Dynkin diagrams in ranks 3-10, 142 of which are symmetrizable.

  2. Fundamental domains for congruence subgroups of SL2 in positive characteristic.

    Wed, 09/02/2009 - 12:02 — SomNambul
    Authors: Lisa Carbone, Leigh Cobbs, Scott H. Murray
    Subjects: Group Theory
    Abstract

    Morgenstern ([M]) claimed to have constructed fundamental domains for
    congruence subgroups of the lattice group Gamma=PGL_2(F_q[t]), and subgraphs
    providing the first known examples of linear families of bounded concentrators.
    His method was to construct the fundamental domain for a congruence subgroup as
    a `ramified covering' of the fundamental domain for Gamma on the Bruhat-Tits
    tree X of G=PGL_2(F_q((t^-1))). We prove that Morgenstern's constructions do
    not yield the desired ramified coverings, and in particular yield graphs that
    are not connected in characteristic 2.

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