Skeleton ideals of certain graphs, standard monomials and spherical parking functions

Chanchal Kumar¹ ; Gargi Lather² ; Sonica³

¹Indian Institute of Science Eductaion and Research, Mohali
²Indian Institute of Science Education and Research, Mohali
³MCM DAV College for Women, Sector-36A, Chandigarh-160036

The electronic journal of combinatorics, Tome 28 (2021) no. 1

Résumé

Let $G$ be a graph on the vertex set $V = \{ 0, 1,\ldots,n\}$ with root $0$. Postnikov and Shapiro were the first to consider a monomial ideal $\mathcal{M}_G$, called the $G$-parking function ideal, in the polynomial ring $ R = {\mathbb{K}}[x_1,\ldots,x_n]$ over a field $\mathbb{K}$ and explained its connection to the chip-firing game on graphs. The standard monomials of the Artinian quotient $\frac{R}{\mathcal{M}_G}$ correspond bijectively to $G$-parking functions. Dochtermann introduced and studied skeleton ideals of the graph $G$, which are subideals of the $G$-parking function ideal with an additional parameter $k ~(0\le k \le n-1)$. A $k$-skeleton ideal $\mathcal{M}_G^{(k)}$ of the graph $G$ is generated by monomials corresponding to non-empty subsets of the set of non-root vertices $[n]$ of size at most $k+1$. Dochtermann obtained many interesting homological and combinatorial properties of these skeleton ideals. In this paper, we study the $k$-skeleton ideals of graphs and for certain classes of graphs provide explicit formulas and combinatorial interpretation of standard monomials and the Betti numbers.

Détail

DOI : 10.37236/9874

Classification : 05E40, 13D02
Mots-clés : \(G\)-parking functions, \(k\)-skeleton ideal