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Monomial basis

About: Monomial basis is a research topic. Over the lifetime, 706 publications have been published within this topic receiving 10600 citations.


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TL;DR: In this paper, an upper cluster algebra, defined as an intersection of certain Laurent polynomial rings, is proposed and shown to coincide with its upper counterpart and is finitely generated.
Abstract: We continue the study of cluster algebras initiated in math.RT/0104151 and math.RA/0208229. We develop a new approach based on the notion of an upper cluster algebra, defined as an intersection of certain Laurent polynomial rings. Strengthening the Laurent phenomenon from math.RT/0104151, we show that, under an assumption of "acyclicity", a cluster algebra coincides with its "upper" counterpart, and is finitely generated. In this case, we also describe its defining ideal, and construct a standard monomial basis. We prove that the coordinate ring of any double Bruhat cell in a semisimple complex Lie group is naturally isomorphic to the upper cluster algebra explicitly defined in terms of relevant combinatorial data.

397 citations

Journal ArticleDOI
Gerald Allen Reisner1
TL;DR: In this paper, the authors characterize the ideals, I, for which I is CohenMacaulay in terms of topological properties of a simplicial complex associated with I. The main result is that the property of I being Cohenblacaulay, for a fixed choice of monomials, is dependent upon k (see end of Section 1 for specific examples).

373 citations

Journal ArticleDOI

254 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that the Stanley-Reisner ideal of a simplicial complex has a linear resolution if and only if its Alexander dual is Cohen-Macaulay.
Abstract: A componentwise linear ideal is a graded ideal $I$ of a polynomial ring such that, for each degree $q$, the ideal generated by all homogeneous polynomials of degree $q$ belonging to $I$ has a linear resolution. Examples of componentwise linear ideals include stable monomial ideals and Gotzmann ideals. The graded Betti numbers of a componentwise linear ideal can be determined by the graded Betti numbers of its components. Combinatorics on squarefree componentwise linear ideals will be especially studied. It turns out that the Stanley-Reisner ideal $I_{\Delta}$ arising from a simplicial complex $\Delta$ is componentwise linear if and only if the Alexander dual of $\Delta$ is sequentially Cohen-Macaulay. This result generalizes the theorem by Eagon and Reiner which says that the Stanley-Reisner ideal of a simplicial complex has a linear resolution if and only if its Alexander dual is Cohen-Macaulay.

243 citations

Journal ArticleDOI
TL;DR: The fitting basis is used to obtain a new PES for H3O(+) based on roughly 62 000 ab initio energies and is illustrated for several classes of molecules.
Abstract: We describe a procedure to develop a fitting basis for molecular potential energy surfaces (PESs) that is invariant with respect to permutation of like atoms. The method is based on a straightforward symmetrization of a primitive monomial basis and illustrated for several classes of molecules. A numerically efficient method to evaluate the resulting expression for the PES is also described. The fitting basis is used to obtain a new PES for H3O(+) based on roughly 62 000 ab initio energies.

226 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20236
202223
202128
202019
201920
201813