Skeleton Ideals of Certain Graphs, Standard Monomials and Spherical Parking Functions

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.

Gröbner–Shirshov bases for non-crystallographic Coxeter groups

For the group algebra of the finite non-crystallographic Coxeter group of type H 4, its Gröbner–Shirshov basis is constructed as well as the corresponding standard monomials, which describe explicitly all symmetries acting on the 120-cell and produce a natural operation table between the 14400 elements for the group.

Gröbner–Shirshov bases for Temperley–Lieb algebras of types B and D

For Temperley–Lieb algebras of types [Formula: see text] and [Formula: see text], we construct their Gröbner-Shirshov bases and the corresponding standard monomials

Gröbner-Shirshov Bases for Temperley-Lieb Algebras of Complex Reflection Groups

We construct a Gröbner-Shirshov basis of the Temperley-Lieb algebra T ( d , n ) of the complex reflection group G ( d , 1 , n ) , inducing the standard monomials expressed by the generators { E i } of T ( d , n ) . This result generalizes the one for the Coxeter group of type B n in the paper by Kim and Lee We also give a combinatorial interpretation of the standard monomials of T ( d , n ) , relating to the fully commutative elements of the complex reflection group G ( d , 1 , n ) . More generally, the Temperley-Lieb algebra T ( d , r , n ) of the complex reflection group G ( d , r , n ) is defined and its dimension is computed.

An integer sequence and standard monomials

For an (oriented) graph [Formula: see text] on the vertex set [Formula: see text] (rooted at [Formula: see text]), Postnikov and Shapiro (Trans. Amer. Math. Soc. 356 (2004) 3109–3142) associated a monomial ideal [Formula: see text] in the polynomial ring [Formula: see text] over a field [Formula: see text] such that the number of standard monomials of [Formula: see text] equals the number of (oriented) spanning trees of [Formula: see text] and hence, [Formula: see text], where [Formula: see text] is the truncated Laplace matrix of [Formula: see text]. The standard monomials of [Formula: see text] correspond bijectively to the [Formula: see text]-parking functions. In this paper, we study a monomial ideal [Formula: see text] in [Formula: see text] having rich combinatorial properties. We show that the minimal free resolution of the monomial ideal [Formula: see text] is the cellular resolution supported on a subcomplex of the first barycentric subdivision [Formula: see text] of an [Formula: see text] simplex [Formula: see text]. The integer sequence [Formula: see text] has many interesting properties. In particular, we obtain a formula, [Formula: see text], with [Formula: see text] for [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text], similar to [Formula: see text].

Standard monomials for the Weyl group F4

For the exceptional Weyl group of type [Formula: see text], its Gröbner–Shirshov basis and the corresponding standard monomials are constructed.