literature
Literature
Authors
Jobs
Seminars
Conferences
DataBETA

Quantum and affine Schubert calculus and Macdonald polynomials

2014
132 pages
Supervisor:
Thesis: PhD
  • Drexel U.
(2014)
  • Published: 2014
DOI:
reference search0 citations

Citations per year

0 Citations
Abstract: (Drexel U.)
This thesis is on the theory of symmetric functions and quantum and affine Schubert calculus. Namely, it establishes that the theory of symmetric Macdonald polynomials aligns with quantum and affine Schubert calculus using a discovery that distinguished weak chains can be identified by chains in the strong (Bruhat) order poset on the type-A affine Weyl group. Through this discovery, there is a construction of two one-parameter families of functions that respectively transition positively with Hall-Littlewood polynomials and Macdonald's P-functions. Furthermore, these functions specialize to the representatives for Schubert classes of homology and cohomology of the affine Grassmannian. This shows that the theory of symmetric Macdonald polynomials connects with affine Schubert calculus. There is a generalization of the discovery of the strong order chains. This generalization connects the theory of Macdonald polynomials to quantum Schubert calculus. In particular, the approach leads to conjecture that all elements in a defining set of 3-point genus 0 Gromov-Witten invariants for flag manifolds can be formulated as strong covers.
  • Mathematics
  • Combinatorics and computer science
  • Geometry, Algebraic
  • [BB96]
    A. Björner and F. Brenti. Affine permutations of type
    • A. The Foata
  • [BBPZ12]
    Expansions of k-Schur Functions in the Affine nil-Coxeter Algebra. The
    • C. Berg
      ,
    • N. Bergeron
      ,
    • S. Pon
      ,
    • M. Zabrocki
      • Electron.J.Comb. 19 2012
  • [BBTZ12]
    Expansion of k-Schur functions for maximal k-rectangles within the affine nil-Coxeter algebra. preprint
    • C. Berg
      ,
    • N. Bergeron
      ,
    • H. Thomas
      ,
    • M. Zabrocki
  • [BKPT]
    The puzzle conjecture for the cohomology of two-step flag manifolds
    • A. Buch
      ,
    • A. Kresch
      ,
    • K. Purboo
      ,
    • H. Tamvakis
  • [BKT03]
    Gromov-Witten invariants on Grassmannians
    • A. Buch
      ,
    • A. Kresch
      ,
    • H. Tamvakis
      • J.Am.Math.Soc. 16 (2003) 901-915
  • [Bot58]
    The space of loops on a Lie group
    • R. Bott
      • Michigan Math.J. 5 (1958) 35-61
  • [BS98]
    Schubert polynomials, the Bruhat order, and the geometry of flag manifolds
    • N. Bergeron
      ,
    • F. Sottile
      • Duke Math.J. 95 (1998) 373-423
  • [BSSa]
    Combinatorial expansions for families of noncommutative k-Schur functions. math
    • C. Berg
      ,
    • F. Saliola
      ,
    • L. Serrano
  • [BSSb]
    The down operator and expansions of near rectangular k-Schur functions. math
    • C. Berg
      ,
    • F. Saliola
      ,
    • L. Serrano
  • [Buc]
    Mutations of puzzles and equivariant cohomology of two-step flag varieties
    • A. Buch
  • [Cos09]
    A Littlewood-Richardson
    • I. Coskun
      • Invent.Math. 176 (2009) 325-395
  • [DM12]
    The ABC’s of the affine Grassmannians and Hall-Littlewood polynomials. DMTCS Proceedings,. 114
    • A. Dalal
      ,
    • J. Morse
  • [DM13]
    Quantum and affine Schubert calculus and Macdonald polynomials. Submitted
    • A. Dalal
      ,
    • J. Morse
  • [FGP97]
    Quantum Schubert polynomials
    • S. Fomin
      ,
    • S. Gelfand
      ,
    • A. Postnikov
      • J.Am.Math.Soc. 10 (1997) 565-596
  • [Fis95]
    Statistics for special q, t-Kostka polynomials
    • S. Fishel
      • Proc.Am.Math.Soc. 123 (1995) 2961-2969
  • [FK13]
    Results and conjectures on the number of standard strong marked tableaux. DMTCS Proceedings
    • S. Fishel
      ,
    • M. Konvalinka
  • [GH96a]
    A remarkable q, t-Catalan sequence and q-Lagrange inversion
    • A. Garsia
      ,
    • M. Haiman
      • J.Algebr.Comb. 5 (1996) 191-244
  • [GH96b]
    Some natural bigraded S n-modules and q, t-Kostka coefficients. The Foata Festsrift Electron, 3/2
    • A.M. Garsia
      ,
    • M. Haiman
  • [GP92]
    On certain graded S n-modules and the q-Kostka polynomials
    • A.M. Garsia
      ,
    • C. Procesi
      • Adv.Math. 87 (1992) 82-138
  • [GR75]
    A Bruhat
    • H. Garland
      ,
    • S. Raghunathan
      • Proc.Nat.Acad.Sci. 72 (1975) 4716-4717
  • [GR96]
    Plethystic formulas and positivity for q, t-Kostka coefficients. Mathematical essays in honor of Gian-Carlo Rota (Cambridge, MA), pages 245-262
    • A.M. Garsia
      ,
    • J. Remmel
  • [Gre55]
    The characters of the finite general linear groups
    • J.A. Green
      • Trans.Am.Math.Soc. 80 (1955) 442-407
  • [GT96]
    Plethystic formulas for Macdonald q, t-Kostka coefficients
    • A.M. Garsia
      ,
    • G. Tesler
      • Adv.Math. 123 (1996) 144-222
  • [Hag05]
    The q, t-Catalan numbers and the space of diagonal harmonics. lecture notes
    • J. Haglund
  • 1-25 of 86
  • 1
  • 2
  • 3
  • 4
  • 25 / page