Standard Monomials Research Articles

Gröbner–Shirshov Bases for Temperley–Lieb Algebras of Type F

For the Temperley–Lieb algebras of type Fn with n≥4, we construct their Gröbner–Shirshov bases. Explicitly, the corresponding finite sets consisting of the standard monomials of type Fn, which are exactly the fully commutative elements of Fn, are enumerated when n = 4, 5, and 6.

Some properties of generalized cluster algebras of geometric type

We study the lower bound algebras generated by the generalized projective cluster variables of acyclic generalized cluster algebras of geometric type. We prove that this lower bound algebra coincides with the corresponding generalized cluster algebra under the coprimality condition. As a corollary, we obtain the dual PBW bases of these generalized cluster algebras. Moreover, we show that if the standard monomials of a generalized cluster algebra of geometric type are linearly independent, then the directed graph associated to the initial generalized seed of this algebra does not have 3-cycles.

Eulerian ideals

Let G be a simple graph and , where is the homomorphism that sends an edge to the product of its vertices. The ideal is Cohen–Macaulay, one-dimensional and binomial. If G is bipartite, it is known that the Castelnuovo–Mumford regularity of is equal to the maximum cardinality of a set of edges having no more than half of the edges of any Eulerian subgraph of G. Here, with respect to the grevlex order associated to an ordering of the edge set of G, we describe a Gröbner basis for , and we characterize the standard monomials of the ideal in terms of even sets of vertices marked with a parity. Using these results, we give a combinatorial interpretation of the degree of , via the set of even sets of vertices of G; and we show that the Castelnuovo–Mumford regularity of , for any graph, is the maximum cardinality of a set of edges having no more than half of the edges of any even Eulerian subgraph of G or, equivalently, the maximum cardinality of a minimum fixed parity T-join.

Standard monomials of 1-skeleton ideals of graphs and generalized signless Laplacians

For a graph G on the vertex set {0,1,…,n}, the G-parking function ideal MG is a monomial ideal in the polynomial ring R=K[x1,…,xn] such that the vector space dimension of R/MG is given by the determinant of its reduced Laplacian. For any integer k, the k-skeleton ideal MG(k) is the subideal of MG, where the monomial generators correspond to nonempty subsets of [n] of size at most k+1. For a simple graph G, Dochtermann conjectured that the vector space dimension of R/MG(1) is bounded below by the determinant of the reduced signless Laplacian. We show that the Dochtermann conjecture holds for any (multi) graph G. More generally, we prove that this bound holds for ideals JH defined by a larger class of symmetric positive semidefinite n×n matrices H.

The dual of an evaluation code

The aim of this work is to study the dual and the algebraic dual of an evaluation code using standard monomials and indicator functions. We show that the dual of an evaluation code is the evaluation code of the algebraic dual. We develop an algorithm for computing a basis for the algebraic dual. Let $$C_1$$ and $$C_2$$ be linear codes spanned by standard monomials. We give a combinatorial condition for the monomial equivalence of $$C_1$$ and the dual $$C_2^\perp $$ . Moreover, we give an explicit description of a generator matrix of $$C_2^\perp $$ in terms of that of $$C_1$$ and coefficients of indicator functions. For Reed–Muller-type codes we give a duality criterion in terms of the v-number and the Hilbert function of a vanishing ideal. As an application, we provide an explicit duality for Reed–Muller-type codes corresponding to Gorenstein ideals. In addition, when the evaluation code is monomial and the set of evaluation points is a degenerate affine space, we classify when the dual is a monomial code.

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.

On constructing boundaries for boundary value problems defined by continuous 2-D autonomous systems

Abstract In this paper, we provide a constructive way of specifying initial/boundary data for a given continuous 2-D autonomous system described by a set of linear partial differential equations (PDEs) with real constant coefficients. One of the ways of specifying initial/boundary data is by specifying the values of various derivatives of the solution trajectories at the origin; the derivatives correspond to a standard monomial set obtained using Grobner basis. However, such an initial/boundary data often lacks physical interpretation. In this paper, we consider subsets of the domain having some algebraic structure (in the form of subspaces and strips of finite width around such subspaces) such that trajectories restricted to these subsets, often called characteristic sets, serve as initial/boundary conditions for the given autonomous system of linear PDEs. We provide a systematic way to construct such characteristic sets with the help of Grobner bases and Oberst-Riquier algorithm. Thus we bridge the gap between initial/boundary conditions involving standard monomials and more conventional initial/boundary conditions in the form of restrictions on characteristic sets. We also show that every scalar system of PDEs admits such a characteristic set given by a rectangular strip of finite width around a subspace whose dimension equals the Krull dimension of the system’s quotient ring.

Standard monomials and extremal point sets

We say that a set system F⊆2[n] shatters a set S⊆[n] if every possible subset of S appears as the intersection of S with some element of F, and we denote by Sh(F) the family of sets shattered by F. According to the Sauer–Shelah lemma we know that in general, every set system F shatters at least |F| sets, and we call a set system shattering-extremal if |Sh(F)|=|F|. In Mészáros (2010) and Rónyai and Mészáros (2011), among other things, an algebraic characterization of shattering-extremality was given, which offered the possibility to generalize the notion to general finite point sets. Here we extend the results obtained for set systems to this more general setting, and as an application, strengthen a result of Li, Zhang and Dong from Li et al. (2012).

Truncated normal forms for solving polynomial systems: Generalized and efficient algorithms

We consider the problem of finding the isolated common roots of a set of polynomial functions defining a zero-dimensional ideal I in a ring R of polynomials over C. Normal form algorithms provide an algebraic approach to solve this problem. The framework presented in Telen et al. (2018) uses truncated normal forms (TNFs) to compute the algebra structure of R/I and the solutions of I. This framework allows for the use of much more general bases than the standard monomials for R/I. This is exploited in this paper to introduce the use of two special (non-monomial) types of basis functions with nice properties. This allows us, for instance, to adapt the basis functions to the expected location of the roots of I. We also propose algorithms for efficient computation of TNFs and a generalization of the construction of TNFs in the case of non-generic zero-dimensional systems. The potential of the TNF method and usefulness of the new results are exposed by many experiments.

Gröbner-Shirshov bases for non-crystallographic Coxeter groups.

For the group algebra of the finite non-crystallographic Coxeter group of type H4, 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

Monomial ideals induced by permutations avoiding patterns

Let S (or T) be the set of permutations of $$[n]=\{1,\ldots ,n\}$$ avoiding 123 and 132 patterns (or avoiding 123, 132 and 213 patterns). The monomial ideals $$I_S = \langle {\mathbf {x}}^{\sigma } = \prod _{i=1}^n x_i^{\sigma (i)} : \sigma \in S \rangle $$ and $$I_T = \langle {\mathbf {x}}^{\sigma } : \sigma \in T \rangle $$ in the polynomial ring $$R = k[x_1,\ldots ,x_n]$$ over a field k have many interesting properties. The Alexander dual $$I_S^{[{\mathbf {n}}]}$$ of $$I_S$$ with respect to $${\mathbf {n}}=(n,\ldots ,n)$$ has the minimal cellular resolution supported on the order complex $$\mathbf {\Delta }(\Sigma _n)$$ of a poset $$\Sigma _n$$ . The Alexander dual $$I_T^{[{\mathbf {n}}]}$$ also has the minimal cellular resolution supported on the order complex $$\mathbf {\Delta } ({\tilde{\Sigma }}_n)$$ of a poset $${\tilde{\Sigma }}_n$$ . The number of standard monomials of the Artinian quotient $$\frac{R}{I_S^{[{\mathbf {n}}]}}$$ is given by the number of irreducible (or indecomposable) permutations of $$[n+1]$$ , while the number of standard monomials of the Artinian quotient $$\frac{R}{I_T^{[{\mathbf {n}}]}}$$ is given by the number of permutations of $$[n+1]$$ having no substring $$\{l,l+1\}$$ .

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.

Block Stanley Decompositions II. Greedy Algorithms, Applications, and Open Problems

Stanley decompositions are used in applied mathematics (dynamical systems) and \(\mathfrak {sl}_2\) invariant theory as finite descriptions of the set of standard monomials of a monomial ideal. The block notation for Stanley decompositions has proved itself in this context as a shorter notation and one that is useful in formulating algorithms such as the “box product”. Since the box product appears only in dynamical systems literature, we sketch its purpose and the role of block notation in this application. Then we present a greedy algorithm that produces incompressible block decompositions (called “organized”) from the monomial ideal; these are desirable for their likely brevity. Several open problems are proposed. We also continue to simplify the statement of the Soleyman–Jahan condition for a Stanley decomposition to be prime (come from a prime filtration) and for a block decomposition to be subprime, and present a greedy algorithm to produce “stacked decompositions”, which are subprime.

Ill‐conditioning in the virtual element method: Stabilizations and bases

In this article, we investigate the behavior of the condition number of the stiffness matrix resulting from the approximation of a 2D Poisson problem by means of the virtual element method. It turns out that ill‐conditioning appears when considering high‐order methods or in presence of “bad‐shaped” (for instance nonuniformly star‐shaped, with small edges…) sequences of polygons. We show that in order to improve such condition number one can modify the definition of the internal moments by choosing proper polynomial functions that are not the standard monomials. We also give numerical evidence that at least for a 2D problem, standard choices for the stabilization give similar results in terms of condition number.

A note on linear Sperner families

In an earlier work we described Grobner bases of the ideal of polynomials over a field, which vanish on the set of characteristic vectors $$\mathbf {v}\in \{0,1\}^n$$ of the complete d uniform set family over the ground set [n]. In particular, it turns out that the standard monomials of the above ideal are ballot monomials. We give here a partial extension of this fact. A set family is a linear Sperner system if the characteristic vectors satisfy a linear equation $$a_1v_1+\cdots +a_nv_n=k$$ , where the $$a_i$$ and k are positive integers. We prove that the lexicographic standard monomials for linear Sperner systems are also ballot monomials, provided that $$0<a_1\le a_2\le \cdots \le a_n$$ . As an application, we confirm a conjecture of Frankl in the special case of linear Sperner systems.

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 temperley-lieb algebras

We deal with Temperley-Lieb algebras of type B, extending the result in [3, §6]. By completing the relations coming from a presentation of the Temperley-Lieb algebra, we find its Gröbner-Shirshov basis to obtain the corresponding set of standard monomials. The explicit multiplication table between the monomials follows naturally.

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.

Accessible proof of standard monomial basis for coordinatization of Schubert sets of flags

The main results of this paper are accessible with only basic linear algebra. Given an increasing sequence of dimensions, a flag in a vector space is an increasing sequence of subspaces with those dimensions. The set of all such flags (the flag manifold) can be projectively coordinatized using products of minors of a matrix. These products are indexed by tableaux on a Young diagram. A basis of “standard monomials” for the vector space generated by such projective coordinates over the entire flag manifold has long been known. A Schubert variety is a subset of flags specified by a permutation. Lakshmibai, Musili, and Seshadri gave a standard monomial basis for the smaller vector space generated by the projective coordinates restricted to a Schubert variety. Reiner and Shimozono made this theory more explicit by giving a straightening algorithm for the products of the minors in terms of the right key of a Young tableau. Since then, Willis introduced scanning tableaux as a more direct way to obtain right keys. This paper uses scanning tableaux to give more-direct proofs of the spanning and the linear independence of the standard monomials. In the appendix it is noted that this basis is a weight basis for the dual of a Demazure module for a Borel subgroup of GLn. This paper contains a complete proof that the characters of these modules (the key polynomials) can be expressed as the sums of the weights for the tableaux used to index the standard monomial bases.