Homogeneous Algebras, Statistics and Combinatorics

Abstract

After some generalities on homogeneous algebras, we give a formula connecting the Poincare 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 D degrees 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; p and q being related to the Artin–Schelter parameters. It is claimed that this has a counterpart for any integer D⩾2.