The Waring formula and fusion rings

The potential that generates the cohomology ring of the Grassmannian is giv en 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 i ts coefficients. As a consequence, the fusion potential for su(N)(K) is obt ained. This potential is the explicit Chebyshev polynomial in several varia bles of the first kind. We also derive the fusion potential for sp(N)(K) fr om a reciprocal algebraic equation. This potential is identified with anoth er Chebyshev polynomial in several variables. We display a connection betwe en 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 symmetri c functions. For sp(N)(K), however, the generalized Fibonacci numbers are o btained from new sequences. (C) 2001 Elsevier Science B.V. All rights reser ved.