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

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

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