GENERATING FUNCTION FOR SCHUR POLYNOMIALS

Abstract

For the generating function $$ G_n(\mathbi{x},\mathbi{t})=\sum_{\lambda} \mathbi{s}_{\lambda}(x_1,x_2,\ldots, x_n) t_1^{\lambda_1 } t_2^{\lambda_2 } \cdots t_n^{\lambda_n}, $$ where the Sсhur polynomials $\mathbi{s}_{\lambda}(x_1,x_2,\ldots, x_n) $ are indexed by partitions $ \lambda $ of length no more than $ n $ the explicit form for $ n = 2,3 $ is calculated and a recurrent relation for an arbitrary $ n $ is found. It is proved that $ G_n (\mathbi {x}, \mathbi {t}) $ is a rational function $$G_n(\boldsymbol{x}, \boldsymbol{t})=\frac{P(\boldsymbol{x}, \boldsymbol{t})}{Q(\boldsymbol{x}, \boldsymbol{t})},$$ the numerator and denominator of which belong to the kernel of the differential operator $$ \mathcal{D}_n=\sum_{i=1}^n x_i \frac{\partial}{\partial x_i}- \sum_{i=1}^n t_i \frac{\partial}{\partial t_i}. $$ For the numerator $ P (\boldsymbol {x}, \boldsymbol {t}) $ we find its specialization at $ t_1 = t_2 = \cdots = t_n = 1. $