Stirling partitions of the symmetric group and Laplace operators for the orthogonal Lie algebra
If \(S_N\) and \(S_{N-1}\) are symmetric groups equipped with the Bruhat order, then the standard inclusion \(i_N\colon S_{N-1}\to S_N\) is well-behaved as a mapping of posets. On the other hand projections \(\pi_N\colon S_N\to S_{N-1}\) need to be defined with more care if \(i_N\circ \pi_N\) is to be the identity map as well as order-preserving. Accordingly, the author defines a particular such projection and uses its inverse to partition \(S_N\) into Boolean (algebra) intervals \(B_k\), generating Stirling partitions if the number of intervals \(B_k\) is the (signless) Stirling number (of the first kind). From the combinatorial properties he is then able to provide descriptive information concerning generalized determinants and related forms on certain Lie and enveloping algebras, thereby giving a combinatorial perspective as developed by the author and others in a considerable sequence of previous papers along with the present one in mathematical journals as well as journals of physics.
scientific article; zbMATH DE number 1135944
scientific article