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ON SOME PROPERTIES OF CHEBYSHEV POLYNOMIALS AND THEIR APPLICATIONS

Year 2017, Volume 21, Issue 21, 137 - 163, 17.01.2017
https://doi.org/10.24330/ieja.296263

Abstract

In this paper we investigate certain normalized versions Sk,F (x), Sek,F (x) of Chebyshev polynomials of the second kind and the fourth kind over a field F of positive characteristic. Under the assumption that (char F, 2m + 1) = 1, we show that Sem,F (x) has no multiple roots in any one of its splitting fields. The same is true if we replace 2m + 1 by 2m and Sem,F (x) by Sm−1,F (x). As an application, for any commutative ring R which is a Z[1/n, 2 cos(2π/n), u±1/2 ]-algebra, we construct an explicit cellular basis for the Hecke algebra associated to the dihedral groups I2(n) of order 2n and defined over R by using linear combinations of some Kazhdan-Lusztig bases with coefficients given by certain evaluations of Sek,R(x) or Sk,R(x).

References

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  • Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York, 1966.
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  • J. Algebra, 126(2) (1989), 466–492.
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  • [8] J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman &
  • Hall/CRC, Boca Raton, FL, 2003.
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  • 173(1) (1995), 97–121.

Year 2017, Volume 21, Issue 21, 137 - 163, 17.01.2017
https://doi.org/10.24330/ieja.296263

Abstract

References

  • [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with
  • Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York, 1966.
  • [2] G. Benkart and D. Moon, Tensor product representations of Temperley-Lieb
  • algebras and Chebyshev polynomials, in: Representations of Algebras and Related
  • Topics, in: Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 45 (2005), 57–80.
  • [3] A. P. Fakiolas, The Lusztig isomorphism for Hecke algebras of dihedral type,
  • J. Algebra, 126(2) (1989), 466–492.
  • [4] F. M. Goodman, P. de la Harpe and V. F. R. Jones, Coxeter Graphs and
  • Towers of Algebras, Mathematical Sciences Research Institute Publications, 14, Springer-Verlag, New York, 1989.
  • [5] J. J. Graham and G. I. Lehrer, Cellular algebras, Invent. Math., 123(1) (1996), 1–34.
  • [6] J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge Studies
  • in Advanced Mathematics, 29, Cambridge Univ. Press, Cambridge, UK, 1990.
  • [7] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke
  • algebras, Invent. Math., 53(2) (1979), 165–184.
  • [8] J. C. Mason and D. C. Handscomb, Chebyshev Polynomials, Chapman &
  • Hall/CRC, Boca Raton, FL, 2003.
  • [9] E. Murphy, The representations of Hecke algebras of type An, J. Algebra,
  • 173(1) (1995), 97–121.

Details

Journal Section Articles
Authors

Jun Hu This is me


Yabo Wu This is me

Publication Date January 17, 2017
Published in Issue Year 2017, Volume 21, Issue 21

Cite

Bibtex @research article { ieja296263, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {1710 Sokak, No:41, Batikent/Ankara}, publisher = {Abdullah HARMANCI}, year = {2017}, volume = {21}, number = {21}, pages = {137 - 163}, doi = {10.24330/ieja.296263}, title = {ON SOME PROPERTIES OF CHEBYSHEV POLYNOMIALS AND THEIR APPLICATIONS}, key = {cite}, author = {Hu, Jun and Wu, Yabo} }
APA Hu, J. & Wu, Y. (2017). ON SOME PROPERTIES OF CHEBYSHEV POLYNOMIALS AND THEIR APPLICATIONS . International Electronic Journal of Algebra , 21 (21) , 137-163 . DOI: 10.24330/ieja.296263
MLA Hu, J. , Wu, Y. "ON SOME PROPERTIES OF CHEBYSHEV POLYNOMIALS AND THEIR APPLICATIONS" . International Electronic Journal of Algebra 21 (2017 ): 137-163 <https://dergipark.org.tr/en/pub/ieja/issue/27921/296263>
Chicago Hu, J. , Wu, Y. "ON SOME PROPERTIES OF CHEBYSHEV POLYNOMIALS AND THEIR APPLICATIONS". International Electronic Journal of Algebra 21 (2017 ): 137-163
RIS TY - JOUR T1 - ON SOME PROPERTIES OF CHEBYSHEV POLYNOMIALS AND THEIR APPLICATIONS AU - JunHu, YaboWu Y1 - 2017 PY - 2017 N1 - doi: 10.24330/ieja.296263 DO - 10.24330/ieja.296263 T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 137 EP - 163 VL - 21 IS - 21 SN - 1306-6048-1306-6048 M3 - doi: 10.24330/ieja.296263 UR - https://doi.org/10.24330/ieja.296263 Y2 - 2016 ER -
EndNote %0 International Electronic Journal of Algebra ON SOME PROPERTIES OF CHEBYSHEV POLYNOMIALS AND THEIR APPLICATIONS %A Jun Hu , Yabo Wu %T ON SOME PROPERTIES OF CHEBYSHEV POLYNOMIALS AND THEIR APPLICATIONS %D 2017 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 21 %N 21 %R doi: 10.24330/ieja.296263 %U 10.24330/ieja.296263
ISNAD Hu, Jun , Wu, Yabo . "ON SOME PROPERTIES OF CHEBYSHEV POLYNOMIALS AND THEIR APPLICATIONS". International Electronic Journal of Algebra 21 / 21 (January 2017): 137-163 . https://doi.org/10.24330/ieja.296263
AMA Hu J. , Wu Y. ON SOME PROPERTIES OF CHEBYSHEV POLYNOMIALS AND THEIR APPLICATIONS. IEJA. 2017; 21(21): 137-163.
Vancouver Hu J. , Wu Y. ON SOME PROPERTIES OF CHEBYSHEV POLYNOMIALS AND THEIR APPLICATIONS. International Electronic Journal of Algebra. 2017; 21(21): 137-163.
IEEE J. Hu and Y. Wu , "ON SOME PROPERTIES OF CHEBYSHEV POLYNOMIALS AND THEIR APPLICATIONS", International Electronic Journal of Algebra, vol. 21, no. 21, pp. 137-163, Jan. 2017, doi:10.24330/ieja.296263