Comparing and characterizing some constructions of canonical bases from Coxeter systems View Full Text


Ontology type: schema:Chapter      Open Access: True


Chapter Info

DATE

2015

AUTHORS

Eric Marberg

ABSTRACT

The Iwahori–Hecke algebra \(\mathcal{H}\) of a Coxeter system (W, S) has a “standard basis” indexed by the elements of W and a “bar involution” given by a certain antilinear map. Together, these form an example of what Webster calls a pre-canonical structure, relative to which the well-known Kazhdan–Lusztig basis of \(\mathcal{H}\) is a canonical basis. Lusztig and Vogan defined a representation of a modified Iwahori–Hecke algebra on the free \(\mathbb{Z}[v,v^{-1}]\)-module generated by the set of twisted involutions in W, and showed that this module has a unique pre-canonical structure compatible with the \(\mathcal{H}\)-module structure, which admits its own canonical basis which can be viewed as a generalization of the Kazhdan–Lusztig basis. One can modify the definition of Lusztig and Vogan’s module to obtain other pre-canonical structures, each of which admits a unique canonical basis indexed by twisted involutions. We classify all of the pre-canonical structures which arise in this manner, and explain the relationships between their resulting canonical bases. Some of these canonical bases are equivalent in a trivial fashion to Lusztig and Vogan’s construction, while others appear to be unrelated. Along the way, we also clarify the differences between Webster’s notion of a canonical basis and the related concepts of an IC basis and a P-kernel. More... »

PAGES

399-436

References to SciGraph publications

  • 1979-06. Representations of Coxeter groups and Hecke algebras in INVENTIONES MATHEMATICAE
  • 2004-11. The Bruhat Order on the Involutions of the Symmetric Group in JOURNAL OF ALGEBRAIC COMBINATORICS
  • 2008-09. Twisted identities in Coxeter groups in JOURNAL OF ALGEBRAIC COMBINATORICS
  • Book

    TITLE

    Representations of Reductive Groups

    ISBN

    978-3-319-23442-7
    978-3-319-23443-4

    Author Affiliations

    Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/978-3-319-23443-4_14

    DOI

    http://dx.doi.org/10.1007/978-3-319-23443-4_14

    DIMENSIONS

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