Some new identities for Schur functions (Q5956778)

Let \(s_{\lambda}(X)\) be the Schur function and NEWLINE\[NEWLINE f_{\lambda}(a,b) = \prod_{j\;\text{odd}} {a^{c_j+1}-b^{c_j}+1 \over a-b} \prod_{j\;\text{even}}{1-(ab)^{c_j+1} \over 1 - ab}NEWLINE\]NEWLINE where \(c_j\) is the number of columns of length \(j\) in \(\lambda\). \textit{M. Ishikawa} and \textit{M. Wakayama} [J. Comb. Theory, Ser. A 88, No. 1, 136-157 (1999; Zbl 0947.15008)] proved that NEWLINE\[NEWLINE \sum_{\lambda} f_{\lambda}(a,b)s_{\lambda}(X) = \Phi(X;a,b)NEWLINE\]NEWLINE where NEWLINE\[NEWLINE \Phi(X;a,b) := \prod_i(1-ax_i)^{-1}(1-bx_i)^{-1}\prod_{j<k}(1-x_jx_k)^{-1}.NEWLINE\]NEWLINE Using Macdonald's approach, the authors prove the conjectured formula for the coefficients \(f_{\lambda}(a,b,c)\) in the equality NEWLINE\[NEWLINE \sum_{\lambda} f_{\lambda}(a,b,c) s_{\lambda}(X) = \Phi(X;a,b,) \prod_i(1-cx_i)^{-1}. NEWLINE\]NEWLINE They also find and prove an explicit formula for the coefficients \(\beta(\xi,a,b)\) in NEWLINE\[NEWLINE \sum_{\lambda \subseteq (m^n)} f_{\lambda}(a,b)s_{\lambda}(X) = \sum_{\xi\in\{\pm\}^n}\beta(\xi,a,b)\Phi(X^{\xi};a,b) \prod_i x_i^{m(1-\xi_i)/2}NEWLINE\]NEWLINE where \(X^{\xi}=(x_1^{\xi_1},\ldots,x_n^{\xi_n})\).