Matching algebras and their Dunkl subalgebras
Abstract
For a finite simple graph G = (V, E), we consider the quotient algebra ℳ(G) of the polynomial ring in the variables ue (for e ∈ E) by the ideal generated by all products ue uf for non-disjoint edges e and f (including all squares ue2). This quotient is called the matching algebra of G, since it has a basis indexed by the matchings of G. In this quotient, we define a subalgebra 𝒟(G) generated by the signed incidence sums θv=∑e=(v,w)ue−∑e=(w,v)ue for all v ∈ V (where all edges of G are oriented arbitrarily); we call this the Dunkl matching algebra. We show that, as a graded vector space, 𝒟(G) is dual to the span of all polynomials pM = ∏(i,j) ∈ M (xi − xj), where M ranges over all matchings of G. For the complete graph Kn, 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 generators pM in a given degree k span the full two-row Specht module S(n−k,k). We show that saturation in degree k forces k-connectivity, that saturation in degree $2$ is equivalent to $2$-connectivity, and that k-linked graphs are saturated in degree k. We also prove saturation in every possible degree whenever the complement of G is 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.
Related
- Borel subalgebras of Lie algebras of vector fields
- On nilpotent Lie algebras arising from parabolic subalgebras and quivers --- three-step cases
- Intermediate subalgebras of cross-product von Neumann algebras arising from weakly compact actions
- Conditioning on Subalgebras - Entropy Duality and Generalized Quantum Stein's Lemma in von Neumann Algebras
- One-dimensional partial actions of 8-dimensional Hopf algebras and right coideal subalgebras
BibTeX
@misc{grinberg2026matching,
title = {{Matching algebras and their Dunkl subalgebras}},
author = {Darij Grinberg and Anatol Kirillov and Boris Shapiro},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.34260},
eprint = {2609.34260},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34260},
note = {CC0},
}
RIS
TY - UNPB
TI - Matching algebras and their Dunkl subalgebras
AU - Darij Grinberg
AU - Anatol Kirillov
AU - Boris Shapiro
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2609.34260
UR - https://arxiv.org/abs/2609.34260
AB - For a finite simple graph G = (V, E), we consider the quotient algebra ℳ(G) of the polynomial ring in the variables u_e (for e ∈ E) by the ideal generated by all products u_e u_f for non-disjoint edges e and f (including all squares u_e^2). This quotient is called the matching algebra of G, since it has a basis indexed by the matchings of G. In this quotient, we define a subalgebra 𝒟(G) generated by the signed incidence sums θ_v=∑_e=(v,w)u_e−∑_e=(w,v)u_e for all v ∈ V (where all edges of G are oriented arbitrarily); we call this the Dunkl matching algebra. We show that, as a graded vector space, 𝒟(G) is dual to the span of all polynomials p_M = ∏_(i,j) ∈ M (x_i − x_j), where M ranges over all matchings of G. For the complete graph K_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 generators p_M in a given degree k span the full two-row Specht module S^(n−k,k). We show that saturation in degree k forces k-connectivity, that saturation in degree $2$ is equivalent to $2$-connectivity, and that k-linked graphs are saturated in degree k. We also prove saturation in every possible degree whenever the complement of G is 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.
ER -