Lagrange inversion and Schur functions (Q1975184)

The author considers the Macdonald involution \(\psi\) of the ring \(\Lambda\) of symmetric functions defined by the conditions that \(\psi(h_{\lambda})=h_{\lambda}^{\ast}= h_{\lambda_1}^{\ast}h_{\lambda_2}^{\ast}\cdots\) and \(tH^{\ast}(t)=t+h_1^{\ast}t^2+h_2^{\ast}t^3+\cdots\) is the compositional inverse of \(tH(-t)\), where \(H(t)=\sum_{n=0}^{\infty}h_nt^n\) and \(h_{\lambda}\) is the complete symmetric function indexed by the partition \(\lambda\). This involution is related to the composition of power series in the same way as the standard involution is related to the multiplication of power series. The author uses the Lagrange inversion formula and gives new combinatorial proofs of the explicit formulas for \(e_n^{\ast}\) and \(h_n^{\ast}\). Then he shows that the involution \(\psi\) maps any skew Schur function to a Schur positive or a Schur negative symmetric function which is the main result of the paper. The author also establishes a \(q\)-analogue of this result proving a special case of the recent conjecture of Bergeron and Garsia involving their operator \(\nabla\) (which acts on \(\Lambda[q,t]\) and has a modified version of the Macdonald polynomials as eigenfunctions).