ON SOME PROPERTIES OF CHEBYSHEV POLYNOMIALS AND THEIR APPLICATIONS

In this paper we investigate certain normalized versions Sk,F (x), S̃k,F (x) of Chebyshev polynomials of the second kind and the fourth kind over a field F of positive characteristic. Under the assumption that (charF, 2m + 1) = 1, we show that S̃m,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 S̃m,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 S̃k,R(x) or Sk,R(x). Mathematics Subject Classification (2010): 20C08, 12D10


Introduction
The Chebyshev polynomials are a sequence of important orthogonal polynomials over Z which are related to de Moivre's formula and which can be defined recursively.They have found many important applications in diverse areas of mathematics such as ordinary and partial differential equations, analysis and approximation theory.In past three decades these polynomials also come up in several places in nearby areas of representation theory.For example, they appear in the criterion for semisimplicity of Temperley-Lieb and Jones algebras [4], [5], in giving the dimension of a centralizer algebra of a Temperley-Lieb algebra, and in calculating the decomposition of a Brauer algebra module into Temperley-Lieb algebra modules [2], and in constructing irreducible representations of the semisimple Hecke algebra associated to the dihedral groups [3].However, it seems to us that all of these applications only use property of Chebyshev polynomials over the complex This work was carried out under the support of the National Natural Science Foundation of China (NSFC 11525102).
numbers field (or any field of characteristic 0).To the best of our knowledge, the property of Chebyshev polynomials over field of positive characteristic are not wellstudied and exploited in the literatures before.In this paper we shall study some of their properties for the normalized versions S n (x) := U n (x/2), S n (x) := W n (x/2) of Chebyshev polynomials of the second kind and of the fourth kind over certain fields of positive characteristic, where U n (x) and W n (x) are the Chebyshev polynomials of the second kind and of the fourth kind respectively.As an application, we shall construct an explicit cellular basis for the Hecke algebra H q,R (W n ) associated to the dihedral group W n := I 2 (n) of order 2n over any commutative ring R which is a Z[ 1 n , 2 cos( 2π n ), u ±1/2 ]-algebra.The content is organised as follows.In Section 2 we investigate the Chebyshev polynomials over fields of positive characteristic.We show (in Lemmas 2.8 and 2.9) that under the assumption that (char F, 2m + 1) = 1 (respectively, (char F, 2m) = 1), the normalised Chebyshev polynomial S m,F (x) of the fourth kind (respectively, S m−1,F (x) of the second kind) has no multiple roots over any one of its splitting fields.As a result, we show that for any n ∈ N and 1 In Section 3, we apply these results to construct a cellular basis of the Hecke algebra associated to the dihedral group W n := I 2 (n) of order 2n.We show (in Lemmas 2.10 and 2.11) that for any commutative ring R which is a Z[ 1 n , 2 cos( 2π n ), u ±1/2 ]-algebra, certain linear combinations of some Kazhdan-Lusztig bases can form a cellular basis of the Hecke algebra H q,R (W n ) over R, see Theorems 3.14 and 3.22.The coefficients of each Kazhdan-Lusztig bases are given by a scalar multiple of some evaluation of certain explicit Chebyshev polynomials.

Chebyshev polynomials over fields of positive characteristic
The purpose of this section is to study certain normalised Chebyshev polynomials over fields of positive characteristic.Let x be an indeterminate over Z.
Definition 2.1.The Chebyshev polynomials {T k (x)} k≥0 of the first kind are defined recursively by: The Chebyshev polynomials {U k (x)} k≥0 of the second kind are defined recursively by: The Chebyshev polynomials {W k (x)} k≥0 of the fourth kind are defined recursively by: Definition 2.2.For each k ≥ 0, we define We shall call S k (x), S k (x) the normalized Chebyshev polynomials of the second kind and of the fourth kind respectively.
One can check that S 0 (x) = 1, S 1 (x) = x and while S 0 (x) = 1, S 1 (x) = x + 1 and Furthermore, for any k ≥ 0, and Lemma 2.4.[8, (2.30a),(2.30c),(2.30d)]For any k ≥ 0, we have that Let R be a commutative ring.For any Lemma 2.5.For any integer k ≥ 1, we have that Lemma 2.6.[1,8] Let k ∈ Z ≥0 and F be an arbitrary field.We have that In particular, if F is a field with char F = 2 and m ∈ N, then Now the lemma follows from the above equality, (2) and the fact that The Chebyshev polynomials over the complex numbers field C have many nice properties.For example, it is well-known that for any integer k ≥ 1, (1) S k (2 cos θ) = sin (k+1)θ sin θ ; (2) the roots of the polynomial (3) the roots of the polynomial The following lemma gives a positive characteristic analogue of the property c) for the Chebyshev polynomials over fields of positive characteristic.
Lemma 2.8.Let F be an arbitrary field and k ∈ Z ≥0 such that either char F = 0 or char F is coprime to 2k + 1.Then the polynomial S k,F (x) over F has k-distinct roots in any one of its splitting field.
Proof.If k = 0, then S k,F (x) = x + 1 has a unique root −1.So there is nothing to prove.Henceforth we assume that k ≥ 1.
We first assume that char F = 2. Then 2 ).To prove the lemma, it suffices to show that W k,F (x) over F has k-distinct roots in any one of its splitting field.
As a result, we see that to prove the lemma, it suffices to show that the F -polynomial U 2k,F (x) has 2k-distinct roots in any one of its splitting field.
On the other hand, by definition, for any m ≥ 1, It follows that (T m+1,F (x), T m,F (x)) is always a factor of (T m,F (x), T m−1,F (x)).
Now assume that char F = 2. Suppose that k = 2m + 1 is an odd integer.
Lemma 2.9.Let F be an arbitrary field and k ∈ Z ≥1 such that either char F = 0 or char F is coprime to 2k.Then the polynomial S k−1,F (x) over F has (k − 1)-distinct roots in any one of its splitting field.
Proof.If k = 1 then S k−1,F (x) = 1 and there is nothing to prove.If k = 2, then x has a unique root 0 as required.Henceforth we assume that k ≥ 3.
By assumption, 2 To prove the lemma, it suffices to show that roots in any one of its splitting field.
By Lemma 2.6, we have that By assumption, k On the other hand, by definition, for any m ≥ 1, It follows that (T m+1,F (x), T m,F (x)) is always a factor of (T m,F (x), T m−1,F (x)).
Inductively, we can deduce that (T m+1,F (x), T m,F (x)) is a factor of This proves that (T m+1,F (x), T m,F (x)) = 1 for any m ≥ 0. As a result, we can as required.This completes the proof of the lemma.
Henceforth, we set Lemma 2.10.Suppose that n = 2m + 1.Then for any In particular, for each Let m be a maximal ideal of A which contains 2 cos(2kπ/n) − 2 cos(2lπ/n).Let k := A/m be the residue field and τ be the canonical homomorphism A → A/m = k.By construction, we see that 2m + 1 is invertible in k.
Applying Lemma 2.8, we can deduce that S m,k (x) has no multiple roots.On the other hand, it is clear that It follows that the elements in {τ (2 cos(2jπ/n))|1 ≤ j ≤ m} must be pairwise distinct.In particular, τ It remains to prove the second half of the lemma.
Lemma 2.11.Suppose that n = 2m.Then for any In particular, for each 1 Proof.The first half of the lemma can be proved in a similar way as the proof of Lemma 2.10.It remains to prove the second half of the lemma.

It follows that
as required, where the second last equality follows from Lemma 2.4.

Cellular basis of
In this section we shall use the main result of the last section to construct an explicit cellular basis of the Hecke algebras H q,R (W n ) associated to the dihedral group We first briefly recall some well-known basic knowledge about the Kazhdan-Lusztig basis and Kazhdan-Lusztig polynomials for the Hecke algebras H u (W ) associated to a Coxeter group W .
Let u 1/2 be an indeterminate over Z and where s ∈ S, w ∈ W , and " * " is the anti-isomorphism of H u (W ) which is defined on generators by T * w := T w −1 for any w ∈ W .Let a → a be the involution of the ring A which is defined by u 1/2 = u −1/2 .This extends to an involution h → h of the ring H u (W ), defined by Kazhdan and Lusztig proved (in [7]) that for each w ∈ W there exists a unique where P y,w ∈ A is a polynomial on u of degree ≤ 1 2 ( (w) − (y) − 1) for y < w, P w,w = 1, and "≤" is the Bruhat partial order on W . Furthermore, {C w |w ∈ W } forms an A-basis of H u (W ) and is called the Kazhdan-Lusztig basis of H u (W ), and the polynomial P y,w (u) is the well-known Kazhdan-Lusztig polynomial.For any field F which is an A-algebra with u 1/2 specialized to q 1/2 ∈ F , let H q,F (W ) be the Iwahori-Hecke algebra associated to W which is defined over F and with Hecke parameter q.Then the elements in the set {T w ⊗ A 1 F |w ∈ W } (respectively, in the set {C w ⊗ A 1 F |w ∈ W }) form an F -basis of H q,F (W ).In the past decades these bases and polynomials have played important roles in many aspects of modern representation theory, cf.[6] and [7].
We now recall the definition of finite dihedral group.Definition 3.1.Let W n := I 2 (n) be the finite dihedral group of order 2n, which is presented by the generators: s, t, and the following relations: Let H u (W n ) be the corresponding Iwahori-Hecke algebra over A with Hecke parameter u.As an A-algebra, H u (W n ) has a presentation with generators T s , T t and the following relations: For any field F which is an A-algebra with u 1/2 specialized to q 1/2 ∈ F , we shall often abbreviate T w ⊗ A 1 F and C w ⊗ A 1 F as T w and C w respectively.
Proof.We use induction on (w) to prove P y,w = 1.If (w) = 0, then y ≤ w implies that y = w = 1 and hence P y,w = P w,w = 1 in this case.
Suppose that P y,w = 1 holds for any y ≤ w and any w ∈ W n with (w) < k.Now assume that w ∈ W n and (w) = k.Let y ≤ w.We want to show that P y,w = 1.
Since W n = I 2 (n) is generated by {s, t}.Without loss of generality, we can assume that w = sv > v.By [7, 2.2.c], where c = 1 if sy < y; or c = 0 if sy > y, and "≺" is as defined in [7, Definition  Case 3. y ≤ v, sy ≤ v. Since w = sv is of the form stst • • • , it follows that this case happens if and only if y ≤ tv < v.By induction hypothesis, P z,v = 1 for any z ≤ v, hence z = tv is the unique element in W n such that y ≤ z ≺ v, sz < z and µ(z, v) = 0 and the definition of µ(z, v).In this case, µ(z, v) = 1.Hence This completes the proof of the lemma.Lemma 3.3.Let w ∈ I 2 (n).If w ∈ {1, t}, then we have that The same is true if we interchange the role of s and t.
Proof.By definition, Similarly, It follows that as required.
Let z ∈ W n such that z ≺ w and sz < z.In particular, z = 1.Since W n is the dihedral group with generating set {s, t} and P z,w = 1 by Lemma 3.2, it follows (cf.[7, Lemma 2.6(iii)]) that our assumption forces that µ(z, w) = 1, w = tz and hence z = tw as required. Let in C, and We set Lemma 3.4.(see [3]) The Hecke algebra H u,K (W n ) is split semisimple.Furthermore, if n is even, then Irr(H u,K (W n )) consists of the following four one-dimension representations and (n − 2)/2 two-dimensional representations: If n is odd, then Irr(H u,K (W n )) consists of the following two one-dimension representations and (n − 1)/2 two-dimensional representations: In [5], Graham and Lehrer introduced the notions of cellular bases and cellular algebras which capture the common feature of many important examples (including the Kazhdan-Lusztig basis and the Murphy basis for the type A Iwahori-Hecke algebras).A cellular structure on an algebra enables one to obtain a general description and systematic understanding of its irreducible representations and block theory by some unified linear algebra argument, which is very useful especially in the non-semisimple situation.It turns out that many important algebras in Lie theory fit in the framework of cellular algebras, see [5] and [9].
In the remaining part of this section, we shall use the main result of the last section to construct an explicit cellular basis of the Hecke algebras H q,R (W n ) associated to the dihedral group W n = I 2 (n) over any commutative ring R which is a Z[1/n, 2 cos(2π/n), u ±1/2 ]-algebra.Our bases will be some linear combinations of certain Kazhdan-Lusztig bases with coefficients given by evaluation of some explicit Chebyshev polynomials.Our construction is motivated by the work in [3], where Fakiolas gave a decomposition of the regular module of the semisimple Hecke algebra H u,K (W ) over the field Q[cos(π/n)](u 1/2 ) into a direct sum of irreducible submodules.First, let's recall the definition of cellular algebras.
Definition 3.5.(see [5]) Let R be a commutative domain and A be an R-algebra which is free as an R-module.Let (Λ, ) be a finite poset.Suppose that for each λ ∈ Λ there is a finite indexing set T (λ), and for each pair (s, t) with s, t ∈ T (λ) there is an element c λ st ∈ A such that the elements in the following set form an R-basis of A.
The basis {c λ st λ ∈ Λ and s, t ∈ T (λ)} is called a cellular basis of A if (C1) the R-linear map * : A → A determined by c λ st * = c λ ts for all λ ∈ Λ and all s, t ∈ T (λ), is an algebra anti-isomorphism of A; and (C2) for any λ ∈ Λ, s, b ∈ T (λ) and a ∈ A, there exists an element r b ∈ R such that for all t ∈ T (λ) where A λ denotes the R-submodule of A spanned by the elements in the subset If the R-algebra A has a cellular basis over R then A is called a cellular algebra over R.
Assumption 3.6.Let R be a commutative ring such that there is a ring ho- For any commutative ring R, we use R × to denote the set of invertible elements in R. Proof.This follows directly from Lemmas 2.10 and 2.11.
We are going to construct an explicit cellular basis for the Hecke algebra H q,R (W n ) over R associated to W n = I 2 (n).We shall consider the case when n = 2m + 1 and the case when n = 2m separately.
Case 1: Suppose that n = 2m + 1.In this case, by our assumption, Lemma 2.10 and Corollary 3.7, By definition, we have that a k ∈ R for any j, k.Furthermore, applying (1) and the fact that S m,R (p j ) = 0 we can get that a (j) m+1 = 0 and For each 2 ≤ k ≤ m − 1, by definition we have that The same equality still holds when k = 1.
Definition 3.9.(compare [3, Section 4]) For each 1 ≤ j ≤ m, we define Lemma 3.10.With the notations as above, the elements in the set {u j , v j |1 ≤ j ≤ m} are R-linearly independent in H q,R (W n ) and form an R-basis of the space spanned by {C (st Proof.Let Then we can write X = DY , where (1) 3 where (1) 2 (1) 3 (2) 3 It is easy to see that det D = det A • det B and det B = det A. Therefore, to prove the lemma, it suffices to show that det(B) is invertible in R.
By the discussion above Lemma 3.9, we have that for any 1 ≤ k ≤ m, a Since S k (x) = x k + lower degree terms, it follows from an easy induction that where the first step follows from Lemma 2.10.This completes the proof of the lemma.
In this case, we define m (j) 12 For each λ ∈ Λ, let H λ q,R and H λ q,R be the R-submodule of H q,R (W n ) generated by the elements in the set {m which is defined on generators by T * w = T w −1 for all w ∈ I 2 (n).3) with the data of the anti-isomorphism " * ", the poset (Λ, ), and the set T (λ) for each λ ∈ Λ, the set {m (λ) st } forms a cellular basis of H q,R (W n ).
Proof. 1) follows from the definition and a direct verification.Since ) follows from Lemmas 3.10, 3.12, and Corollary 3.7.
It remains to prove 3).To this end, it suffices to verify the cellular axiom C2) in Definition 3.5.
Let j ∈ Λ, s ∈ T (j).To verify the cellular axiom C2), it suffices to show that for each u ∈ T (j), there exist r u , r u ∈ R, such that for any t ∈ T (j), ut (mod H j q,R ) .
By Lemma 3.3 we have that as required.
Next, we want to compute T s m (j) 21 .By definition, we have that k + 2a We claim that a 1 , and for each 1 k + 2a Once this is proved, we shall get that 21 + q 1/2 (2 + p j )m (j) 11 (mod (H j q,R )) , and we are done.
In fact, by Definition 3.8 and the paragraph above Definition 3.9, we have that a where the last inequality follows from Corollary 3.7.

1. 2 ]
. Note that y ≤ w = sv implies that either y ≤ v or sy ≤ v. Therefore there are three possibilities: Case 1. y v, sy ≤ v. Since w = sv is of the form stst • • • , it follows that this case happens if and only if y = sv = w.Thus P y,w = P w,w = 1 as required.

Case 2 .
y ≤ v, sy v. Since w = sv is of the form stst • • • , it follows that this case happens if and only if y = v.Thus P y,w = P v,w = P sw,w = 1 by [7, Lemma 2.6(iii)], as required.

Theorem 3 . 2 )
14. Suppose that n = 2m + 1.We keep the Assumption 3.6 on R and n and the Definition 3.13.Then 1) for any λ ∈ Λ, s, t ∈ T (λ), we have that (m the elements in the set m (λ) st λ ∈ Λ, s, t ∈ T (λ) are R-linearly independent and form an R-basis of H q,R (W n );