Topics in Fomin-Kirillov Algebra

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dc.contributor.advisorBergeron, Nantel
dc.contributor.authorHomayouni, Sirous
dc.date.accessioned2022-03-03T14:07:05Z
dc.date.available2022-03-03T14:07:05Z
dc.date.copyright2021-12
dc.date.issued2022-03-03
dc.date.updated2022-03-03T14:07:05Z
dc.degree.disciplineMathematics & Statistics
dc.degree.levelDoctoral
dc.degree.namePhD - Doctor of Philosophy
dc.description.abstractWe introduce the notion of being 'z-star' for homogeneous polynomials. By proving a theorem plus developing a conjecture we state that, with a graded lexicographic monomial ordering, the reduced Grobner basis, for the ideal I generated by the relations of Fomin-Kirilov algebra FK(n), consists of 'z-star' polynomials. For general n, we find the character of the Fomin-Kirillov a lgebra, for some finite usual degrees with general set partition degree. We find the decomposition of this character in irreducible characters where this decomposition stabilizes at some enough big n. We develop a quotient of FK(n), denoted by FKCn (n), by making the quotient of the free algebra generated by the edges of an n-cycle, compared to the associated complete graph, where the ideal is generated by the relations of FK(n) except for letting the missing edges equal to zero and keeping only the edges of the polygon. We find the character map of algebra FKCn (n) and prove that the dimension of it equals the Lucas Number Ln and its Hilbert series is q-Lucas polynomial. We consider the commutative quotient of Fomin-Kirilov algebra, denoted by FK(n) and find its Grobner basis.
dc.identifier.urihttp://hdl.handle.net/10315/39127
dc.languageen
dc.rightsAuthor owns copyright, except where explicitly noted. Please contact the author directly with licensing requests.
dc.subjectMathematics
dc.subject.keywordsFomin-Kirillov Algebra
dc.subject.keywordsQuotient algebra
dc.subject.keywordsGro¨bner basis
dc.subject.keywordsDimension
dc.subject.keywordsNon-commutative Buchberger criterion
dc.subject.keywordsLexicographic monomial ordering
dc.subject.keywordsq-Lucas polynomial
dc.subject.keywordsq-Fibonacci polynomial
dc.subject.keywordsS-polynomial
dc.subject.keywordsSchur function
dc.subject.keywordsSchubert polynomial
dc.subject.keywordsz-star polynomials
dc.subject.keywordsCharacter map
dc.subject.keywordsFrobenius image
dc.subject.keywordsRepresentation decomposition
dc.subject.keywordsConjugacy class decomposition
dc.subject.keywordsSet partition type decomposition
dc.subject.keywordsPermutation degree
dc.subject.keywordsSet partition degree
dc.subject.keywordsHomogeneous decomposition
dc.subject.keywordsHomogeneous ideal
dc.subject.keywordsFree Algebra
dc.subject.keywordsGenerator
dc.subject.keywordsMatchings in a graph
dc.subject.keywordsn-cycle graph
dc.subject.keywordsLucas number. Irreducible character
dc.subject.keywordsDecomposition into irreducible characters. Graded ring
dc.subject.keywordsIrreducible Gro¨bner basis. Graded algebra.
dc.subject.keywordsIrreducible Grobner basis. Graded algebra.
dc.titleTopics in Fomin-Kirillov Algebra
dc.typeElectronic Thesis or Dissertation

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