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).