[Paper Review] On the characteristic polynomial of Cartan matrices and Chebyshev polynomials

Pantelis A. Damianou|arXiv (Cornell University)|Oct 30, 2011

Algebraic structures and combinatorial models22 references3 citations

TL;DR

This paper establishes a deep connection between the characteristic polynomial of Cartan matrices in complex simple Lie algebras and Chebyshev polynomials of the first and second kind. It derives explicit formulas for the Coxeter adjacency matrix's characteristic polynomial, expresses the Coxeter polynomial as a product of cyclotomic polynomials, and proves a novel sine formula linking the determinant of the Cartan matrix to the exponents and Coxeter number via trigonometric products.

ABSTRACT

We explore some interesting features of the characteristic polynomial of the Cartan matrix of a simple Lie algebra. The characteristic polynomial is closely related with the Chebyshev polynomials of first and second kind. In addition, we give explicit formulas for the characteristic 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 formula which involves products of sines, the exponents, the Coxeter number and the determinant of the Cartan matrix.

Motivation & Objective

• To explore the structural relationship between the characteristic polynomial of Cartan matrices and Chebyshev polynomials of the first and second kind.
• To derive explicit formulas for the characteristic polynomial of the Coxeter adjacency matrix and its associated polynomials.
• To express the Coxeter polynomial as a product of cyclotomic polynomials using the exponents and Coxeter number of the Lie algebra.
• To develop an algorithm for factoring Chebyshev polynomials over the integers using Lie-theoretic data.
• To prove a new sine formula that relates the determinant of the Cartan matrix to the exponents, Coxeter number, and trigonometric functions of the root system.

Proposed method

• The paper uses the standard definition of Cartan matrices from root systems and relates their characteristic polynomial to Chebyshev polynomials via a transformation: $ p_n(x) = q_n\left(\frac{x}{2} - 1\right) $, where $ q_n $ corresponds to $ U_n $, $ 2T_n $, or $ 4xT_{n-1} $ depending on the Lie type.
• The Coxeter adjacency matrix is defined as $ A = 2I - C $, and its characteristic polynomial is derived from the Cartan matrix $ C $, with roots related to $ 2\cos\left(\frac{m_i\pi}{h}\right) $, where $ m_i $ are the exponents and $ h $ the Coxeter number.
• The associated polynomial $ a_n(x) $ is shown to factor into products of cyclotomic polynomials $ \Phi_d(x) $, with the full Coxeter polynomial $ Q_n(x) $ expressed as a product of $ \Phi_d $ for divisors $ d $ of the Coxeter number.
• The paper constructs an algorithm to factor Chebyshev polynomials over $ \mathbb{Z} $ by leveraging the known factorization of $ a_n(x) $ into $ \psi_j(x) $, which are related to cyclotomic polynomials.
• A key technical tool is Lemma 2, which identifies the roots of $ a_n(x) $ as $ 2\cos\left(\frac{m_i\pi}{h}\right) $, linking spectral data of the adjacency matrix to Lie-theoretic invariants.
• The sine formula is derived by evaluating $ a_n(2) $, using trigonometric identities and the duality of exponents $ m_i + m_{\ell+1-i} = h $, ultimately equating $ \det C $ to $ 2^{2\ell} \prod_{i=1}^{\ell} \sin^2\left(\frac{m_i\pi}{2h}\right) $.

Experimental results

Research questions

• RQ1 How are the characteristic polynomials of Cartan matrices related to Chebyshev polynomials of the first and second kind across different Lie types?
• RQ2 Can the Coxeter polynomial of a Dynkin diagram be expressed as a product of cyclotomic polynomials, and if so, how are the cyclotomic factors determined by the exponents and Coxeter number?
• RQ3 What is the explicit factorization of Chebyshev polynomials over the integers, and can it be derived from Lie-theoretic data such as the exponents of the root system?
• RQ4 How do the eigenvalues of the Cartan matrix pair symmetrically around 2, and what does this imply for the characteristic polynomial?
• RQ5 Is there a closed-form trigonometric identity that relates the determinant of the Cartan matrix to the exponents and the Coxeter number?

Key findings

• The characteristic polynomial of the Cartan matrix for type $ A_n $ is $ p_n(x) = U_n\left(\frac{x}{2} - 1\right) $, where $ U_n $ is the Chebyshev polynomial of the second kind.
• For types $ B_n $ and $ C_n $, the characteristic polynomial satisfies $ p_n(x) = 2T_n\left(\frac{x}{2} - 1\right) $, with $ T_n $ being the Chebyshev polynomial of the first kind.
• The associated polynomial $ a_n(x) $ for $ E_6 $ factors as $ (x+1)(x-1)(x^4 - 4x^2 + 1) $, and its full factorization is $ \psi_3(x)\psi_6(x)\psi_{24}(x) $, with the Coxeter polynomial $ Q_6(x) = \Phi_3(x)\Phi_6(x)\Phi_{24}(x) $.
• For $ E_7 $, the associated polynomial $ a_7(x) = x(x^6 - 6x^4 + 9x^2 - 3) $ factors as $ \psi_4(x)\psi_{36}(x) $, and the Coxeter polynomial is $ Q_7(x) = \Phi_4(x)\Phi_{36}(x) $.
• For $ E_8 $, the associated polynomial is $ a_8(x) = \psi_{60}(x) $, and the Coxeter polynomial is $ Q_8(x) = \Phi_{60}(x) $, with the exponents being the integers less than 30 and coprime to 30.
• The paper proves the sine formula: $ \det C = 2^{2\ell} \prod_{i=1}^{\ell} \sin^2\left( \frac{m_i\pi}{2h} \right) $, where $ m_i $ are the exponents and $ h $ the Coxeter number, verified by evaluating $ a_n(2) = \det C $.