Central decomposition of S^z within the symmetry algebra

Establish that the total spin operator’s third component S^z admits a decomposition S^z = p + Σ_{i=1}^N α_i h_i, where h_i are Cartan generators of the symmetry algebra C_N and p is a central element commuting with all Chevalley generators (i.e., [p, e_i] = [p, f_i] = [p, h_i] = 0), and determine the coefficients α_i.

Background

Building on the conjectured sp_{2N} symmetry, the authors further posit that S^z can be expressed as a sum of Cartan elements plus a central element p that commutes with the entire algebra. Section 3 encodes this as a formal conjecture while examples in Section 4 illustrate specific α_i combinations.

The introduction highlights that the symmetry algebra is wider and includes a central element associated with S^z, motivating the decomposition conjectured in Section 3.