Rectangular low level case of modular branching problem for GL_n(K)

In this paper, we find an explicit combinatorial criterion for the existence of a nonzero GL_{n-1}(K)-high weight vector of weight (\lambda_1,...,\lambda_{i-1},\lambda_i-d,\lambda_{i+1},..., \lambda_{n-1}), where d<char K and K is an algebraically closed filed, in the irreducible rational GL_n(K)-module L_n(\lambda_1,...,\lambda_n) with highest weight (\lambda_1,...,\lambda_n). For this purpose, new modular lowering operators are introduced.