A class of graphs: Bridging structures in commutative algebra
Fahimeh Khosh-Ahang Ghasr
TL;DR
This work addresses when the generators of the determinantal facet ideal $J_\Delta$ form a reduced Gröbner basis, and extends this perspective to generalized interval-graph structures via $d$-dimensional interval and closed concepts. It shows that the full set of $(d+1)$-minors of a matrix $X$ forms a reduced Gröbner basis for the determinantal ideal $I_{d+1}(X)$, and characterizes when $\mathcal{B}=\{\mathrm{det}(i_1, \dots, i_{d+1})\}$ is a Gröbner basis for $J_\Delta$ in terms of $\Delta$ being closed or unit interval (with a converse under poor-closed). The paper then develops and analyzes strong/global/proper interval graph notions, proving several equivalent definitions and showing that $G$ being $d$-unit interval, $d$-proper interval, or $d$-global interval corresponds to sortable $\mathrm{Ind}_d(G)$ and to favorable algebraic properties (e.g., Koszul and normal Cohen–Macaulay coordinate rings for $d$-unit interval graphs). Collectively, these results deepen the interplay between combinatorial graph structures and determinantal/Gröbner-basis techniques, extending known results on binomial edge and determinantal facet ideals and enabling concrete classifications for cycles and forests.
Abstract
We characterize classes of simplicial complexes $Δ$ for which the generators of the determinantal facet ideal $J_Δ$ form a reduced Gröbner basis. This result provides a new proof for a known fact about determinantal ideals in a special case and extends several existing results concerning binomial edge ideals and determinantal facet ideals. We also introduce and examine generalizations of interval graphs and proper interval graphs, aiming to establish equivalent characterizations for these classes. These formulations enable several algebraic applications, including constructing examples of Cohen--Macaulay normal rings, identifying Gröbner bases for binomial edge ideals of proper interval graphs, and deriving equivalent conditions for the sortability of $t$-independence complexes. These results demonstrate how the proposed graph-theoretic structures enrich the interplay between combinatorics and commutative algebra, building upon and extending existing connections in the literature.