The Waring formula and fusion rings |
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| Institution: | 1. Biomedical Research Theme, School of Human Sciences, University of Derby, Derby, UK;2. Healthy Living for Pandemic Event Protection (HL – PIVOT) Network, Chicago, IL, United States of America;3. MRC-Versus Arthritis Centre for Musculoskeletal Ageing Research and National Institute for Health Research Nottingham Biomedical Research Centre, School of Medicine, University of Nottingham, Derby, UK;4. Department of Physical Therapy, College of Applied Sciences, University of Illinois at Chicago, Chicago, IL, United States of America |
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| Abstract: | The potential that generates the cohomology ring of the Grassmannian is given in terms of the elementary symmetric functions using the Waring formula that computes the power sum of roots of an algebraic equation in terms of its coefficients. As a consequence, the fusion potential for su(N)K is obtained. This potential is the explicit Chebyshev polynomial in several variables of the first kind. We also derive the fusion potential for sp(N)K from a reciprocal algebraic equation. This potential is identified with another Chebyshev polynomial in several variables. We display a connection between these fusion potentials and generalized Fibonacci and Lucas numbers. In the case of su(N)K the generating function for the generalized Fibonacci numbers obtained are in agreement with Lascoux using the theory of symmetric functions. For sp(N)K, however, the generalized Fibonacci numbers are obtained from new sequences. |
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