ON SOME PROPERTIES OF CHEBYSHEV POLYNOMIALS AND THEIR APPLICATIONS

Year 2017, Volume: 21 Issue: 21, 137 - 163, 17.01.2017

Jun Hu Yabo Wu

https://doi.org/10.24330/ieja.296263

https://izlik.org/JA52ZA66DJ

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).

Keywords

Chebyshev polynomials , dihedral group , Hecke algebras

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.