Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

HIGHEST WEIGHT VECTORS OF IRREDUCIBLE REPRESENTATIONS OF THE QUANTUM SUPERALGEBRA μq(gl(m, n))

  • Moon, Dong-Ho (Department of Applied Mathematics Sejong University)
  • Published : 2003.01.01
Download PDF
⟨ Previous Next ⟩

Abstract

The Iwahori-Hecke algebra $H_{k}$ ( $q^2$) of type A acts on the k-fold tensor product space of the natural representation of the quantum superalgebra (equation omitted)$_{q}$(gl(m, n)). We show the Hecke algebra $H_{k}$ ( $q^2$) and the quantum superalgebra (equation omitted)$_{q}$(gl(m n)) have commuting actions on the tensor product space, and determine the centralizer of each other. Using this result together with Gyoja's q-analogue of the Young symmetrizers, we construct highest weight vectors of irreducible summands of the tensor product space.

Keywords

References

  1. J. Amer. Math. Soc. v.13 no.2 Crystal bases for the quantum superal-gebra ${U_q}$(gl(m,n)) G. Benkart;S. Kang;M. Kashiwara https://doi.org/10.1090/S0894-0347-00-00321-0
  2. Nova. J. Algebra Geom. v.2 no.4 Stability in modules for general linear Lie superalgebras G. Benkart;C. Lee
  3. Adv. in Math. v.64 no.2 Hook Young diagrams with applications to combinatorics and to representations of Lie superalgebras A. Berele;A. Regev https://doi.org/10.1016/0001-8708(87)90007-7
  4. Representation theory of finite groups and associative algebras C. W. Curtis;I. Reiner
  5. With applications to finite groups and orders, Pure and Applied Mathematics v.Ⅰ Methods of representation theory C. W. Curtis;I. Reiner
  6. Proc. London Math. Soc.(3) v.54 no.1 Blocks and idempotents of Hecke algebras of general linear groups R. Dipper;G. James https://doi.org/10.1112/plms/s3-54.1.57
  7. Publ. Res. Inst. Math. Sci. v.31 no.2 Euler-Poincare characteristic and polynomial representations of Iwahori-Hecke algebras G. Duchamp;D. Krob;A. Lascoux;B. Leclerc;T. Scharf;J.Y. Thibon https://doi.org/10.2977/prims/1195164438
  8. Lett. Math. Phys. v.23 no.2 On the defining relations of quantum superalgebras R. Floreanini;D. Leites;L. Vinet https://doi.org/10.1007/BF00703725
  9. Preuss. Akad. Wiss. Sitz. v.3 Uber die Charaktere der symmetricschen Gruppe F. Frobenius
  10. Osaka. J. Math. v.23 no.4 A q-analogue of Young symmetrizer A. Gyoja
  11. Lett. Math. Phys. v.11 no.3 A q-analog of Ц(gl(n+1)), Hecke algebra, and the Yang-Baxter equation M. Jimbo https://doi.org/10.1007/BF00400222
  12. Comm. Math. Phys. v.141 no.3 Universal R-matrix for quantized(super)algebras S. M. Khoroshkin;V. N. Tolstoy https://doi.org/10.1007/BF02102819
  13. Adv. Math. v.125 no.1 A ribbon Hopf algebra approach to the irreducible representations of centralizer algebras : the Brauer, Birman-Wenzl, and type A Iwahori-Hecke algebras R. Leduc;A. Ram https://doi.org/10.1006/aima.1997.1602
  14. J. Algebra v.71 no.2 On a theorem of Benson and Curtis G. Lusztig https://doi.org/10.1016/0021-8693(81)90188-5
  15. J. Phys. v.A 26 no.24 The structure of n-variable polynomial rings as Hecke algebra modules P. Martin
  16. Invent. Math. v.106 no.3 A Frobenius formula for the characters of the Hecke algebras A. Ram https://doi.org/10.1007/BF01243921
  17. Lett. Math. Phys. v.24 no.3 Serre-type relations for special linear Lie superalgebras M. Scheunert https://doi.org/10.1007/BF00402892
  18. J. Math. Phys. v.34 no.8 The presentation and q deformation of special linear Lie superalgebras M. Scheunert https://doi.org/10.1063/1.530059
  19. Ph. D. thesis v.1 Uber eine klasse von matrizen, die sich einer gegeben matrix zuordenen lassen I. Schur
  20. Preuss. Akad. Wiss. Sitz. v.3 Uber die rationalen Darstellungen der allgemeinen linearen Gruppe I. Schur
  21. Invent. Math. v.92 no.2 Hecke algebras of type $A_n$ and subfactors H. Wenzl https://doi.org/10.1007/BF01404457
  22. Mathematical Exposition v.21 On Quantitative substitutional analysisⅠ-Ⅸ, 1901-1952 A. Young

Cited by

  1. Presenting Schur superalgebras vol.262, pp.2, 2013, https://doi.org/10.2140/pjm.2013.262.285
  2. Quantum Schur superalgebras and Kazhdan–Lusztig combinatorics vol.215, pp.11, 2011, https://doi.org/10.1016/j.jpaa.2011.03.015
  3. Fusion Formulas and Fusion Procedure for the Yang-Baxter Equation 2017, https://doi.org/10.1007/s10468-017-9692-1
  4. Mixed Tensor Representations of Quantum Superalgebra q(gl(m,n)) vol.35, pp.3, 2007, https://doi.org/10.1080/00927870601115682
  5. Characters of Iwahori–Hecke algebras pp.1565-8511, 2019, https://doi.org/10.1007/s11856-018-1779-9