Preprint
Matching algebras and their Dunkl subalgebras
arXiv (Cornell University)
28 Sep 2026
Abstract
For a finite simple graphG = (V, E) , we consider the quotient algebraš(G)of the polynomial ring in the variablesuā(fore ā E ) by the ideal generated by all productsuā u_(f)for non-disjoint edgeseandf(including all squaresuāĀ² ). This quotient is called the matching algebra ofG , since it has a basis indexed by the matchings ofG . In this quotient, we define a subalgebraš(G)generated by the signed incidence sumsĪøįµ„=ā_(e=(v,w))uā-ā_(e=(w,v))uāfor allv ā V(where all edges ofGare oriented arbitrarily); we call this the Dunkl matching algebra. We show that, as a graded vector space,š(G)is dual to the span of all polynomialsp_(M) = ā_((i,j) ā M) (xįµ¢ - xā±¼) , whereMranges over all matchings ofG . For the complete graphK_(n) , the latter span is a direct sum of two-row Specht modules (one in each degree); thus its Hilbert series is that of the Catalan triangle, and, in characteristic zero, the Dunkl matching algebra can be presented by linear and quadratic relations. For arbitrary graphs, we formulate the saturation problem of deciding when the selected matching Specht generatorsp_(M)in a given degreekspan the full two-row Specht moduleS^((n-k,k)) . We show that saturation in degreekforcesk -connectivity, that saturation in degree2is equivalent to2 -connectivity, and thatk -linked graphs are saturated in degreek . We also prove saturation in every possible degree whenever the complement ofGis a matching. We show that the Dunkl matching algebra equals the full matching algebra exactly for forests, and give an explicit Hilbert series formula for unicyclic graphs.
Metrics
1 Record Views
Details
- Title
- Matching algebras and their Dunkl subalgebras
- Creators
- Darij Grinberg - Drexel UniversityAnatol KirillovBoris Shapiro - Drexel University
- Publication Details
- arXiv (Cornell University)
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991022237409904721