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AIMS Mathematics

2020, Volume 5, Issue 6: 7562-7581. doi: 10.3934/math.2020484
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Research article

On Reidemeister torsion of flag manifolds of compact semisimple Lie groups

  • 1.
    Department of Mathematics, King Abdulaziz University, 21589 Jeddah, Makkah, Saudi Arabia
  • 2.
    Department of Mathematics, AIBU Izzet Baysal University, Bolu, Turkey
  • 3.
    Department of Mathematics, Hacettepe University, Ankara, Turkey
  • 4.
    Department of Mathematics, Daejin University, Kyunggi 11159, Korea
  • 5.
    Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea
  • Received: 25 June 2020 Accepted: 23 September 2020 Published: 24 September 2020

  • MSC : 57Q10, 16Z05, 14M15, 14N15, 22E67

  • In this paper we calculate Reidemeister torsion of flag manifold $K/T$ of compact semi-simple Lie group $K = SU_{n+1}$ using Reidemeister torsion formula and Schubert calculus, where $T$ is maximal torus of $K$. We find that this number is 1. Also we explicitly calculate ring structure of integral cohomology algebra of flag manifold $K/T$ of compact semi-simple Lie group $K = SU_{n+1 }$ using root data, and Groebner basis techniques.

    Citation: Cenap Özel, Habib Basbaydar, Yasar Sñzen, Erol Yilmaz, Jung Rye Lee, Choonkil Park. On Reidemeister torsion of flag manifolds of compact semisimple Lie groups[J]. AIMS Mathematics, 2020, 5(6): 7562-7581. doi: 10.3934/math.2020484

    Related Papers:

  • In this paper we calculate Reidemeister torsion of flag manifold $K/T$ of compact semi-simple Lie group $K = SU_{n+1}$ using Reidemeister torsion formula and Schubert calculus, where $T$ is maximal torus of $K$. We find that this number is 1. Also we explicitly calculate ring structure of integral cohomology algebra of flag manifold $K/T$ of compact semi-simple Lie group $K = SU_{n+1 }$ using root data, and Groebner basis techniques.


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    [1] H. Basbaydar, Calculation of Reidmesster torsion of flag manifolds of compact semi-simple Lie groups, Master Thesis, Abant Izzet Baysal University, 2013.
    [2] J. M. Bismut and H. Gillet, C. Soulé, Analytic torsion and holomorphic determinant bundles I. Bott-Chern forms and analytic torsion, Commu. Math. Phys., 115 (1988), 49-78.
    [3] J. M. Bismut and F. Labourie, Symplectic geometry and the Verlinde formulas, Surv. Differ. Geom., 5 (1999), 97-311.
    [4] L. A. Bokut and L. S. Shiao, Gröbner-Shirshov bases for coxeter groups, Comm. Algebra, 29 (2001), 4305-4319.
    [5] A. S. Buch, A. Kresch, H. Tamvakis, Gromov-Witten Invariants on Grassmannaians, J. Amer. Math. Soc., 16 (2003), 901-915.
    [6] A. S. Buch, A. Kresch, H. Tamvakis, Quantum Pieri rules for isotropic Grassmannians, Invent. Math., 178 (2009), 345-405.
    [7] T. A. Chapman, Hilbert cube manifolds and the invariance of Whitehead torsion, Bull. Am. Math. Soc., 79 (1973), 52-56.
    [8] T. A. Chapman, Topological invariance of Whitehead torsion, Am. J. Math., 96 (1974), 488-497.
    [9] D. Cox, J. Little, D. O'Shea, Ideals, Varieties and Algorithms, Springer-Verleg, New York, 1992.
    [10] G. de Rham, Reidemeister's torsion invariant and rotation of Sn,, Differential Analysis, Bombay Colloq, (1964), 27-36.
    [11] W. Franz, Über die Torsion einer Überdeckung, J. Reine Angew. Math., 173 (1935), 245-254.
    [12] W. Fulton and P. Pragacz, Schubert varieties and degeneracy loci, Lecture Notes in Mathematics, Vol. 1689, Springer-Verlag, Berlin, 1998.
    [13] P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Willey Library Edition, New York, 1994.
    [14] R. C. Kirby and L. C. Siebenmann, On triangulation of manifolds and Haupvermutung, Bull. Am. Math. Soc., 75 (1969), 742-749.
    [15] B. Kostant and S. Kumar, The nil Hecke ring and cohomology of G/P for a Kac-Moody group G, Adv. Math., 62 (1986), 187-237.
    [16] J. Milnor, A duality theorem for Reidemeister torsion, Ann. Math., 76 (1962), 137-147.
    [17] J. Milnor, Whitehead torsion, Bull. Am. Soc., 72 (1966), 358-426.
    [18] J. Milnor, Infinite cyclic covers, Topology of Manifolds, in Michigan, Prindle, Weber & Schmidt, Boston, (1967), 115-133.
    [19] C. Özel, On the cohomology ring of the infinite flag manifold LG/T, Turk. J. Math., 22 (1998), 415-448.
    [20] K. Reidemeister, Homotopieringe und Linsenräume, Abh. Math. Sem. Univ. Hamburg, 11 (1935), 102-109.
    [21] Y. Sözen, On Reidemeister torsion of a symplectic complex, Osaka J. Math., 45 (2008), 1-39.
    [22] Y. Sözen, On Fubini-Study form and Reidemeister torsion, Topol. Appl., 156 (2009), 951-955.
    [23] Y. Sözen, Symplectic chain complex and Reidemeister torsion of compact manifolds, Math. Scand., 111 (2012), 65-91.
    [24] Y. Sözen, A note on Reidemeister torsion and period matrix of Riemann surfaces, Math. Slovaca, (to appear).
    [25] H. Tamvakis, Quantum cohomology of isotropic Grassmannians, Geometric Methods in Algebra and Number Theory, Progress in Math, Birkhäuser, Boston, 235 (2005), 311-338.
    [26] H. Tamvakis, Gromov-Witten invariants and quantum cohomology of Grassmannians, Topics in Cohomological Studies of Algebraic Varieties, Trends in Math., Birkhäuser, Boston, (2005), 271-297.
    [27] E. Witten, On quantum gauge theories in two dimensions, Comm. Math. Phys., 141 (1991), 153-209.
    [28] W. Zhe-Xian, Introduction to Kač-Moody Algebra, World Scientific, Singapore, 1991.
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  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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