Shapovalov elements and Hasse diagrams

Shapovalov elements of quantum groups are special polynomials in negative simple root vectors with coefficients in the rational Cartan subalgebra that relate singular vectors in reducible Verma modules with their highest vectors. We give explicit expressions for Shapovalov elements of nonexceptional quantum groups in terms of matrix elements of quantum $L$-operators using calculations on Hasse diagrams associated with auxiliary representations.