Invariant bilinear forms under the operator group of order p³ with odd prime p

• Mahto, Dilchand ^[1] ; Tanti, Jagmohan ^[2]
  1. [1] Central University of Jharkhand

     Central University of Jharkhand

     India

     
  2. [2] Babasaheb Bhimrao Ambedkar University

     Babasaheb Bhimrao Ambedkar University

     India

     
• Localización: Proyecciones: Journal of Mathematics, ISSN 0716-0917, ISSN-e 0717-6279, Vol. 42, Nº. 6, 2023, págs. 1391-1415
• Idioma: inglés
• DOI: 10.22199/issn.0717-6279-5228
• Enlaces
  • Texto completo
• Resumen

  • For an odd prime p, we formulate the number of all degree n representations of a group of order p3. And calculating the dimension of space of invariant bilinear forms corresponding to degree n representation over a field F which contains a primitive p3 root of unity. Here we also explicitly discussed the existence of a non-degenerate invariant bilinear form of the same space.

• Referencias bibliográficas
  • M. Artin, Algebra, Prentice Hall Inc., 1991.
  • Y. Chen, “Matrix representations of the real numbers,” Linear Algebra Appl., vol. 536, pp. 174-185, 2018.
  • K. Conrad, Group of order p3, [On line] Available https://kconrad.math.uconn.edu/blurbs/grouptheory/groupsp3.pdf
  • D. S. Dummit and R. M. Foote, Abstract Algebra, Wiley, 2004.
  • G. Frobenius, “Uber die mit einer Matrix vertauschbaren matrizen”, Sitzungsber, pp. 3-15, 1910.
  • K. Gongopadhyay, R. S. Kulkarni, “On the existence of an invariant non-degenerate bilinear form under a linear map”, Linear Algebra Appl.,...
  • R. Gow and T.J. Laffey, “Pairs of alternating forms and products of two skew-symmetric matrices”, Linear Algebra Appl., vol. 63, pp. 119-132,...
  • K. Gongopadhyay, S. Mazumder and S. K. Sardar, “Conjugate Real Classes in General Linear Groups”, Journal of Algebra and Its Applications,...
  • K. Hoffman and R. Kunze, Linear Alebra, Prentice Hall Inc., 1961.
  • R. S. Kulkarni and J. Tanti, “Space of invariant bilinear forms”, Indian Academic of sciences, vol. 128, no. 4, pp. 47, 2018.
  • C. S. Pazzis, “When does a linear map belong to at least one orthogonal or symplectic group?”, Linear Algebra Appl., vol. 436, no. 5, pp....
  • H. Stenzel, “Uber die Darstellbarkeit einer Matrix als Produkt von zwei symmetrischer matrizen, als Produkt von zwei alternierenden matrizen...
  • J. P. Serre, Linear representations of finite groups, Springer-Verlag, 1977.
  • V. V. Sergeichuk, “Classification problems for systems of forms and linear map”, Izv. Akad. Nauk SSSR Ser. Mat., vol. 51, no. 6, pp. 1170-1190,...