Homogeneous Algebras, Statistics and Combinatorics

After some generalities on homogeneous algebras, we give a formula connecting the Poincaré series of a homogeneous algebra with the homology of the corresponding Koszul complex generalizing thereby a standard result for quadratic algebras. We then investigate two particular types of cubic algebras: The first one, called the parafermionic (parabosonic) algebra, is the algebra generated by the creation operators of the universal fermionic (bosonic) parastatics with Ddegrees of freedom while the second is the plactic algebra, i.e., the algebra of the plactic monoid with entries in {1, 2,..., D}. In the case D=2, we describe the relations with the cubic Artin–Schelter algebras. It is pointed out that the natural action of GL(2) on the parafermionic algebra for D=2 extends as an action of the quantum group GLp,q(2) on the generic cubic Artin–Schelter regular algebra of type S1; pand qbeing related to the Artin–Schelter parameters. It is claimed that this has a counterpart for any integer D⩾2.