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On the Characteristic Polynomial of Cartan Matrices and Chebyshev Polynomials

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On the Characteristic Polynomial of Cartan Matrices and Chebyshev Polynomials ON THE CHARACTERISTIC POLYNOMIAL OF CARTAN MATRICES AND CHEBYSHEV POLYNOMIALS PANTELIS A. DAMIANOU Abstract. We explore some interesting features of the characteristic poly- nomial of the Cartan matrix of a complex simple Lie algebra. The charac- teristic polynomial is closely related with the Chebyshev polynomials of first and second kind. In addition, we give explicit formulas for the characteris- tic polynomial of the Coxeter adjacency matrix, we compute the associated polynomials and use them to derive the Coxeter polynomial of the underlying graph. We determine the expression of the Coxeter and associated polynomials as a product of cyclotomic factors. We use this data to propose an algorithm for factoring Chebyshev polynomials over the integers. Finally, we prove an interesting sine formula which involves the exponents, the Coxeter number and the determinant of the Cartan matrix. 1. Introduction The aim of this paper is to explore the intimate connection between Chebyshev polynomials and root systems of complex simple Lie algebras. Chebyshev polyno- mials are used to generate the characteristic and associated polynomials of Cartan and adjacency matrices and conversely one can use machinery from Lie theory to derive properties of Chebyshev polynomials. Many of the results in this paper are well-known but we re-derive them in an elementary fashion using properties of Chebyshev polynomials. In essence, this paper is mostly a survey but it also in- cludes some new results. For example, the explicit factorization of the Chebyshev polynomials in terms of the polynomials ψj (x) is new. Cartan matrices appear in the classification of simple Lie algebras over the com- plex numbers. A Cartan matrix is associated to each such Lie algebra. It is an ℓ ℓ square matrix where ℓ is the rank of the Lie algebra. The Cartan matrix encodes× all the properties of the simple Lie algebra it represents. Let g be a simple complex Lie algebra, h a Cartan subalgebra and Π = α1,...αℓ a basis of simple roots for the root system ∆ of h in g. The elements of{ the Cartan} matrix C are given by arXiv:1110.6620v4 [math.RT] 2 Oct 2014 (αi, αj ) (1) cij := 2 (αj , αj ) where the inner product is induced by the Killing form. The ℓ ℓ-matrix C is invertible. It is called the Cartan matrix of g. The Cartan matrix× for a complex simple Lie algebra obeys the following properties: (1) C is symmetrizable. There exists a diagonal matrix D such that DC is symmetric. (2) cii = 2. (3) cij 0, 1, 2, 3 for i = j. ∈{ − − − } 6 (4) cij =0 cji = 0. ⇔ 1991 Mathematics Subject Classification. 33C45, 17B20, 20F55. Key words and phrases. Cartan matrix, Chebyshev polynomials, simple Lie algebras, Coxeter polynomial, Cyclotomic polynomials. 1 2 The detailed machinery for constructing the Cartan matrix from the root system can be found e.g. in [15, p. 55] or [18, p. 111]. In the following example, we give full details for the case of sl(4, C) which is of type A3. Example 1. Let E be the hyperplane of R4 for which the coordinates sum to 0 (i.e. vectors orthogonal to (1, 1, 1, 1)). Let ∆ be the set of vectors in E of length √2 with integer coordinates. There are 12 such vectors in all. We use the standard inner product in R4 and the standard orthonormal basis ǫ ,ǫ ,ǫ ,ǫ . Then, it is { 1 2 3 4} easy to see that ∆ = ǫi ǫj i = j . The vectors { − | 6 } α = ǫ ǫ 1 1 − 2 α2 = ǫ2 ǫ3 α = ǫ − ǫ 3 3 − 4 form a basis of the root system in the sense that each vector in ∆ is a linear combination of these three vectors with integer coefficients, either all nonnegative or all nonpositive. For example, ǫ1 ǫ3 = α1 + α2, ǫ2 ǫ4 = α2 + α3 and ǫ1 ǫ4 = − − + − α1 + α2 + α3. Therefore Π = α1, α2, α3 , and the set of positive roots ∆ is given by { } ∆+ = α , α , α , α + α , α + α , α + α + α . { 1 2 3 1 2 2 3 1 2 3} Define the matrix C using (1). It is clear that cii = 2 and (αi, αi+1) ci,i+1 =2 = 1 i =1, 2 . (αi+1, αi+1) − Similar calculations lead to the following form of the Cartan matrix 2 1 0 − C = 1 2 1 . −0 1− 2 − The complex simple Lie algebras are classified as: Al,Bl, Cl,Dl, E6, E7, E8, F4, G2 . Traditionally, Al,Bl, Cl,Dl are called the classical Lie algebras while E6, E7, E8, F4, G2 are called the exceptional Lie algebras. Moreover, for any Cartan matrix there ex- ists just one simple complex Lie algebra up to isomorphism giving rise to it. The classification of simple complex Lie algebras is due to Killing and Cartan around 1890. Simple Lie algebras over C are classified by using the associated Dynkin diagram. It is a graph whose vertices correspond to the elements of Π. Each pair of vertices αi, αj are connected by 2 4(αi, αj ) mij = (αi, αi)(αj , αj ) edges, where mij 0, 1, 2, 3 . ∈{ } 3 Dynkin Diagrams for simple Lie algebras An ✐ ✐ ✐... ✐ ✐ Bn ✐ ✐ ✐... ✐ ✐ ≫ Cn ✐ ✐ ✐... ✐ ✐ ≪ D ✐ n ✐ ✐ ✐... ✐ ✐ ❅ ❅ ❅❅ ✐ E6 ✐ ✐ ✐ ✐ ✐ ✐ E7 ✐ ✐ ✐ ✐ ✐ ✐ ✐ E8 ✐✐✐✐✐✐✐ ✐ F4 ✐ ✐ ✐ ✐ G2 ≫ ✐ ✐ For more details on these classifications≪ see [15], [17]. The following result is useful in computing the characteristic polynomial of the Cartan matrix. 4 Proposition 1. Let C be the n n Cartan matrix of a simple Lie algebra over C. × Let pn(x) be its characteristic polynomial. Then x pn(x)= qn 1 2 − where qn is a polynomial related to Chebyshev polynomials as follows: An : qn = Un Bn, Cn : qn =2Tn Dn : qn =4xTn−1 where Tn and Un are the Chebyshev polynomials of first and second kind respec- tively. To a given Dynkin diagram Γ with n nodes we associate the Coxeter adjacency matrix which is the n n matrix A =2I C where C is the Cartan matrix. The characteristic polynomial× of Γ is that of A−. Similarly the norm of Γ is defined to be the norm of A. One defines, see [8], the spectral radius of Γ to be ρ(Γ) = max λ : λ is an eigenvalue of A . {| | } If the graph is a tree then the characteristic polynomial pA of the adjacency matrix is simply related to the characteristic polynomial of the Cartan matrix pC . In fact: (2) an(x)= pn(x + 2) . Using the fact that the spectrum of A is the same as the spectrum of A it follows easily that if λ is an eigenvalue of the adjacency matrix then 2 + λ−is an eigenvalue of the corresponding Cartan matrix. In this paper we use the following notation. Note that the subscript n, in all cases, is equal to the degree of the polynomial except in the case of Qn(x) which is of degree 2n. pn(x) will denote the characteristic polynomial of the Cartan matrix. • an(x) will denote the characteristic polynomial of the adjacency matrix. • Note that x a (x)= p (x +2)= q ( ) . n n n 2 Finally we define the associated polynomial • 1 Q (x)= xna (x + ) . n n x Qn(x) turns out to be an even, reciprocal polynomial of the form Qn(x) = 2 fn(x ), (see [2], [24]). The polynomial fn is the so called Coxeter polynomial of the underlying graph. For the definition and spectral properties of the Coxeter polynomial see, [1], [2], [3], [6], [16], [20], [22], [28], [30], [31]. The roots of Qn in the cases we consider are in the unit disk and therefore by a theorem of Kronecker, see [9], Qn(x) is a product of cyclotomic polynomials. We determine the factorization of fn as a product of cyclotomic polynomials. This factorization in turn determines the factorization of Qn. The irreducible factors of Qn are in one-to-one correspondence with the irreducible factors of an(x). As a bi-product we obtain the factorization of the Chebyshev polynomials of the first and second kind over the integers. More precisely we prove the following result: 5 2π Theorem 1. Let ψn(x) be the minimal polynomial of the algebraic integer 2cos n . Then Un(x)= ψj (2x) . j|2Yn+2 j6=1,2 Let n =2αN where N is odd and let r =2α+2. Then 1 T (x)= ψ (2x) . n 2 rj jY|N The irreducible polynomials ψn were introduced by Lehmer in [23]. The factor- ization is consistent with previous results, e.g. [14], [27]. Using the factorization of the polynomials an(x) and pn(x) we obtain the fol- lowing interesting sine formula. See [11] for various appearances and interesting connections. Theorem 2. Let g be a complex simple Lie algebra of rank ℓ, h the Coxeter number, m1,m2,...,mℓ the exponents of g, and C the Cartan matrix. Then ℓ m π 22ℓ sin2 i = det C. 2h iY=1 2. Chebyshev polynomials To compute explicitly pn(x), we use the following result, see [6] and [19]. Proposition 2. Let C be the n n Cartan matrix of a simple Lie algebra over C. × Let pn(x) be its characteristic polynomial and define qn(x) = det(2xI + A). Then x x pn(x)= qn 1 , and an(x)= qn . 2 − 2 The polynomial qn is related to Chebyshev polynomials as follows: An : qn = Un Bn, Cn : qn =2Tn Dn : qn =4xTn−1 where Tn and Un are the Chebyshev polynomials of first and second kind respectively. Proof. We give an outline of the proof. Note that x x 2 qn 1 = det 2 − In + A 2 − 2 = det (xIn 2In + A) − = det (xIn 2In +2In C) − − = det (xIn C)= pn(x) .
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