  We first prove an identity involving symmetric polynomials. This identity
leads us into exploring the geometry of Lagrangian Grassmannians. As an insight
applications, we obtain a formula for the integral over the Lagrangian
Grassmannian of a characteristic class of the tautological sub-bundle.
Moreover, a relation to that over the ordinary Grassmannian and its application
to the degree formula for the Lagrangian Grassmannian are given. Finally, we
present further applications to the computation of Schubert structure constants
and three-point, degree $1$, genus $0$ Gromov--Witten invariants of the
Lagrangian Grassmannian. Some examples together with explicit computations are
presented.
