Gaudin functions, and Euler-Poincaré characteristics

Alain Lascoux (1)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge

Given two positive integers n,r, we define the Gaudin function of level r to be quotient of the numerator of the determinant det(1/ ((x_i-y_j)(x_i-ty_j) ... (x_i-t^r y_j)), i,j=1..n, by the two Vandermonde in x and y. We show that it can be characterized by specializing the x-variables into the y-variables, multiplied by powers of t. This allows us to obtain the Gaudin function of level 1 (due to Korepin and Izergin) as the image of a resultant under the the Euler-Poincaré characteristics of the flag manifold. As a corollary, we recover a result of Warnaar about the generating function of Macdonald polynomials.

Mots clés: Gaudin functions; Izergin-Korepin determinant; Macdonald polynomials; Integrable Systems

Domaines: Combinatoire [math.CO]

Alain Lascoux. Gaudin functions, and Euler-Poincaré characteristics. 2007. ⟨hal-00171095⟩