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Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields

Andersen, Kasper LU ; Carbone, Lisa and Penta, Diego (2010) In Journal of Combinatorics and Number Theory 2(3). p.245-278
Abstract
Let A be the generalized Cartan matrix of rank 2 Kac-Moody algebra g. We write g = g(a, b) when A has non-diagonal entries −a and −b. To each such A, its Weyl group and corresponding root lattice, we associate a ‘Fibonacci type ’ integer sequence. These sequences are derived from the coordinates of the real root vectors in the root space. Each element of each sequence can be expressed as a polynomial in the non-diagonal entries of the generalized Cartan matrix, whose coefficients are shallow diagonals of Pascal’s triangle. Among the Fibonacci type sequences are the bisected Lucas and Fibonacci sequences, the Kekule numbers for the benzoids, the integers whose squares are triangular numbers, Chebyshev polynomials of the second kind, as well... (More)
Let A be the generalized Cartan matrix of rank 2 Kac-Moody algebra g. We write g = g(a, b) when A has non-diagonal entries −a and −b. To each such A, its Weyl group and corresponding root lattice, we associate a ‘Fibonacci type ’ integer sequence. These sequences are derived from the coordinates of the real root vectors in the root space. Each element of each sequence can be expressed as a polynomial in the non-diagonal entries of the generalized Cartan matrix, whose coefficients are shallow diagonals of Pascal’s triangle. Among the Fibonacci type sequences are the bisected Lucas and Fibonacci sequences, the Kekule numbers for the benzoids, the integers whose squares are triangular numbers, Chebyshev polynomials of the second kind, as well as some other known sequences. Each sequence has an associated ‘hyperbolic golden ratio ’ Ψ which is a unit in a real quadratic field. We show that Ψ can be obtained as the limit of ratios of areas of hyperbolic triangles spanned by a triple of adjacent real roots in the root space. For the bisected Fibonacci sequence that occurs in our setting, Ψ coincides with the classical golden ratio ψ = 1 + φ for the bisection, where φ = 1.61803399.... It follows that ψ is an eigenvalue for the fundamental endomorphism w1w2 in the root system of g(5, 1), and ψ 2 is an eigenvalue for w1w2 in g(3, 3). This leads to an infinite family of identities involving continued fractions. We identify the generalized Lucas and Fibonacci sequences with Chebyshev polynomials of the 2nd kind, which in turn can be expressed as products. This leads us to an arithmetic proof that Kitchloo’s ‘generalized binomial coefficients’ occuring in the cohomology of flag varieties of rank 2 Kac-Moody groups over C are integers. (Less)
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author
publishing date
type
Contribution to journal
publication status
published
subject
in
Journal of Combinatorics and Number Theory
volume
2
issue
3
pages
245 - 278
publisher
Nova Science Publishers, Inc.
ISSN
1942-5600
language
English
LU publication?
no
id
ead0fa11-dbed-4fe7-9faf-db2077d25578 (old id 3412522)
date added to LUP
2013-04-09 16:53:34
date last changed
2018-05-29 10:07:49
@article{ead0fa11-dbed-4fe7-9faf-db2077d25578,
  abstract     = {Let A be the generalized Cartan matrix of rank 2 Kac-Moody algebra g. We write g = g(a, b) when A has non-diagonal entries −a and −b. To each such A, its Weyl group and corresponding root lattice, we associate a ‘Fibonacci type ’ integer sequence. These sequences are derived from the coordinates of the real root vectors in the root space. Each element of each sequence can be expressed as a polynomial in the non-diagonal entries of the generalized Cartan matrix, whose coefficients are shallow diagonals of Pascal’s triangle. Among the Fibonacci type sequences are the bisected Lucas and Fibonacci sequences, the Kekule numbers for the benzoids, the integers whose squares are triangular numbers, Chebyshev polynomials of the second kind, as well as some other known sequences. Each sequence has an associated ‘hyperbolic golden ratio ’ Ψ which is a unit in a real quadratic field. We show that Ψ can be obtained as the limit of ratios of areas of hyperbolic triangles spanned by a triple of adjacent real roots in the root space. For the bisected Fibonacci sequence that occurs in our setting, Ψ coincides with the classical golden ratio ψ = 1 + φ for the bisection, where φ = 1.61803399.... It follows that ψ is an eigenvalue for the fundamental endomorphism w1w2 in the root system of g(5, 1), and ψ 2 is an eigenvalue for w1w2 in g(3, 3). This leads to an infinite family of identities involving continued fractions. We identify the generalized Lucas and Fibonacci sequences with Chebyshev polynomials of the 2nd kind, which in turn can be expressed as products. This leads us to an arithmetic proof that Kitchloo’s ‘generalized binomial coefficients’ occuring in the cohomology of flag varieties of rank 2 Kac-Moody groups over C are integers.},
  author       = {Andersen, Kasper and Carbone, Lisa and Penta, Diego},
  issn         = {1942-5600},
  language     = {eng},
  number       = {3},
  pages        = {245--278},
  publisher    = {Nova Science Publishers, Inc.},
  series       = {Journal of Combinatorics and Number Theory},
  title        = {Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields},
  volume       = {2},
  year         = {2010},
}