ON SOME PROPERTIES OF CHEBYSHEV POLYNOMIALS AND THEIR APPLICATIONS
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,
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- [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.
- ISSN: 1306-6048
- Yayın Aralığı: Yılda 2 Sayı
- Başlangıç: 2007
- Yayıncı: Abdullah HARMANCI