We will discuss representations of the Smith algebra which are free of finite rank over a subalgebra which plays a role analogous to that of the (enveloping algebra of the) Cartan subalgebra of the simple Lie algebra \( \mathfrak{sl}_2 \). In the case of rank 1, we obtain a full description...

The chain polynomial of a finite lattice \( \mathcal{L} \) is given by \( p_\mathcal{L} (x) = \sum_{k \geq 0} c_k(\mathcal{L})x^k \), where \( c_k(\mathcal{L}) \) is the number of chains of length \( k \) in \( \mathcal{L} \). The chain polynomials of posets in several important classes have...