Stirling partitions of the symmetric group and Laplace operators for the orthogonal Lie algebra Original Research Article

Several constructions of Laplace operators for the canonical realization of the orthogonal Lie algebra are discussed. All of them are related with the Capelli-type determinant of a matrix formed by the generators of this Lie algebra. Combinatorial properties of the projection map image used in the definition of the Capelli-type determinant are studied. It is proved that the fibers of this projection form a partition of the Bruhat order on image into Boolean intervals such that the number of intervals with 2k elements is the Stirling number of the first kind c(N − 1,k).