The Graph Algebra I: Representation-Theoretic Structure

Abstract: The paper studies the graph algebra whose monomial basis is naturally indexed by simple graphs on a fixed set of vertices. This algebra is at the same time the algebra of pseudo-Boolean functions on the Boolean cube and a natural object of algebraic combinatorics, related to the Boolean lattice of subsets of the edge set of the complete graph. The main aim of the paper is to study two compatible representation-theoretic structures on this algebra: the action of the Lie algebra $\mathfrak{sl}_2$, arising from the operators of adding and deleting one edge, and the action of the pair group $S_n^{(2)}$, induced by the renumbering of vertices. It is proved that the graph algebra with this $\mathfrak{sl}_2$-action is isomorphic to a tensor power of the standard two-dimensional $\mathfrak{sl}_2$-module, and on this basis its decomposition into irreducible $\mathfrak{sl}_2$-modules is obtained. Primitive spaces, that is, the kernels of the edge-deletion operator on rank components, are also described, and it is shown that they have a natural interpretation in terms of two-row Specht modules. It is then established that the $\mathfrak{sl}_2$-action commutes with the action of the pair group. It follows that the space of graph invariants also inherits the structure of an $\mathfrak{sl}_2$-module. Using Schur--Weyl duality, primitive invariants are described through the fixed parts of the restrictions of two-row Specht modules from the full symmetric group on the edge set to the pair group. As a consequence, the classical enumeration of non-isomorphic graphs by the number of edges receives a representation-theoretic refinement: the orbital components entering the Burnside--Polya formula decompose into natural primitive contributions associated with the $\mathfrak{sl}_2$-structure and two-row Specht modules.