Symmetric Functions as Characters of Hyperoctahedral Group

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dc.contributor.advisorZabrocki, Mike
dc.contributor.authorIslami, Arash
dc.date.accessioned2021-03-08T17:24:01Z
dc.date.available2021-03-08T17:24:01Z
dc.date.copyright2020-10
dc.date.issued2021-03-08
dc.date.updated2021-03-08T17:24:01Z
dc.degree.disciplineMathematics & Statistics
dc.degree.levelDoctoral
dc.degree.namePhD - Doctor of Philosophy
dc.description.abstractIn Symmetric Group Characters as Symmetric Functions, the authors Orellana and Zabrocki showed that there is a non-homogenous basis of the ring of symmetric functions such that when these elements are evaluated at the eigenvalues of a permutation matrix, they are the values of the irreducible characters of the symmetric group. They found a formula by developing a slightly simpler basis of the symmetric functions called the induced trivial character basis which represent the trivial characters induced from a subgroup to the symmetric group. We will be introducing an analogous idea of the irreducible character basis of symmetric group to the hyperoctahedral group, namely, hyperoctahedral group - irreducible character basis. To do this, we have defined an intermediate basis, hyperoctahedral hyperoctahedral group - induced trivial character basis that is constructed in tensor square of ring of symmetric functions in type A. We will provide an algebraic proof that these bases have the analogous properties when evaluated at pairs of eigenvalues of permutation matrices. Moreover, a combinatorial interpretation of this calculation will be provided using fillings of pairs of diagrams in two alphabets. We will define hyperoctahedral group - irreducible character basis by change of basis using Kostka coefficients and show that these bases are families of symmetric functions that are the values of the irreducible characters of the hyperoctahedral group when evaluated at pair of multi-sets of eigenvalues of permutation matrices. We will also define a computational tool, the Frobenius map on tensor square of ring of symmetric functions such that the image is the character of hyper octahedral group that is plethysm of Schur functions in two alphabets. The Kronecker product of two irreducible representations of hyperoctahedral group has the stability property, that is, there exists the reduced Kronecker coefficients that stabilizes in expansion of this Kronecker product for n sufficiently large. We will show that the structure coefficients in this expansion is the same as the coefficients in expansion of the regular product of any two hyperoctahedral group - irreducible character basis.
dc.identifier.urihttp://hdl.handle.net/10315/38194
dc.languageen
dc.rightsAuthor owns copyright, except where explicitly noted. Please contact the author directly with licensing requests.
dc.subjectMathematics
dc.subject.keywordsHyperoctahedral group
dc.subject.keywordsIrreducible character
dc.subject.keywordsIrreducible character basis
dc.subject.keywordsHyperoctahedral group irreducible character
dc.subject.keywordsHyperoctahedral group irreducible character basis
dc.subject.keywordsRing of symmetric functions
dc.subject.keywordsSymmetric functions
dc.subject.keywordsPermutation matrices
dc.subject.keywordsSigned permutation matrices
dc.titleSymmetric Functions as Characters of Hyperoctahedral Group
dc.typeElectronic Thesis or Dissertation

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