Formule avec préambule

+ + Formule avec préambule + +

+ Let $I_+$ denote the ideal generated by + the $S_n$ -invariant homogeneous + polynomials of positive degree in $+ {\mathbb{C}} + \C[x_1,\dots, x_n,y_1,\dots, y_n]$ and set +

+ $+ {\mathbb{C}} + R_n :=\C[\mathbf{x},\mathbf{y}] / I_+. +$ +

+ It is known for sometime that the bi-graded Frobenius character of + $R_n$ is given by the transformation of + the elementary symmetric function $e_{n}$ + under Bergeron-Garsia's "nabla" operator, $\nabla
+ e_n$ . In other words, the symmetric function + $\nabla e_n$ has an underlying + $S_{n}$ -representation. Roughly stated, + $\nabla$ is a + $+ + [1]{\mathbb{#1}} + \mbb{Q}$ -linear operator defined on the ring + of symmetric functions $\Lambda$ in + such a way that the modified Macdonald symmetric functions are + the eigenfunctions of $\nabla$ with + prescribed eigenvalues. +