THE DESCRIPTION OF THE CAPELLI ELEMENTS WITH A PARAMETER FOR CLASSICAL LIE ALGEBRAS IN TERMS OF SYMMETRIZED DETERMINANT

Abstract

We construct the Capelli elements C_k(u) = C_(1^k)(u) (k = 1, . . . , n) with a parameter u for the symplectic Lie algebras and orthogonal Lie algebras. They correspond to factorial Schur functions with parameter u attached to the column partitions (1^k). We also give explicit formulas for C_k(u) arising from the expansion of C_n(u) = C_(1ⁿ)(u) with respect to the parameter u. We use the Jacobi-Trudi formula for the factorial Schur functions R_λ(x; u) to construct the higher Capelli elements C_λ(u). They are expressed as determinants of matrices whose entries are Capelli elements C_k(u) attached to the column partitions.