Search Advanced

AIMS Mathematics

2020, Volume 5, Issue 6: 7562-7581. doi: 10.3934/math.2020484
Previous Article Next Article
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.


    加载中


    [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.
  • Reader Comments
  • © 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)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
AIMS Mathematics
1.8 3.4

Metrics

Article views(2910) PDF downloads(82) Cited by(0)

Preview PDF
Download XML
Export Citation
Article outline

Other Articles By Authors

Related pages

Tools

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog