Generalizing the Borel property

Francisco, Christopher A.; Mermin, Jeffrey; Schweig, Jay

doi:10.1112/jlms/jds071

Cited by 11 publications

(25 citation statements)

References 7 publications

(1 reference statement)

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“…Only if there is a longest b-chain going through a we can say this. Our notion of P -stable is therefore quite distinct from the notion of P -Borel in [8].…”

Section: 2mentioning

confidence: 93%

“…Given any weakly increasing sequence ℓ we can associate a terrace sequence ℓ ′ as follows. Consider (8) l a − a, l a+1 − a, l a+2 − (a + 1), · · · , l b − (b − 1).…”

Section: 1mentioning

confidence: 99%

“…The ideals L(J , P ) will then be non-standard polarizations of L p (J , P ). The notion of P -stable is not the same as the notion of P -Borel in [8]. Rather the notion of P -stable is more subtle.…”

Section: Introductionmentioning

confidence: 99%

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Poset Ideals of P-Partitions and Generalized Letterplace and Determinantal Ideals

Fløystad

2018

Acta Math Vietnam

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For any finite poset P we have the poset of isotone maps Hom(P, N), also called P op -partitions. To any poset ideal J in Hom(P, N), finite or infinite, we associate monomial ideals: the letterplace ideal L(J , P ) and the Alexander dual coletterplace ideal L(P, J ), and study them. We derive a class of monomial ideals in k[x p , p ∈ P ] called P -stable. When P is a chain we establish a duality on strongly stable ideals. We study the case when J is a principal poset ideal. When P is a chain we construct a new class of determinantal ideals which generalizes ideals of maximal minors and whose initial ideals are letterplace ideals of principal poset ideals.

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“…Only if there is a longest b-chain going through a we can say this. Our notion of P -stable is therefore quite distinct from the notion of P -Borel in [8].…”

Section: 2mentioning

confidence: 93%

“…Given any weakly increasing sequence ℓ we can associate a terrace sequence ℓ ′ as follows. Consider (8) l a − a, l a+1 − a, l a+2 − (a + 1), · · · , l b − (b − 1).…”

Section: 1mentioning

confidence: 99%

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Poset Ideals of P-Partitions and Generalized Letterplace and Determinantal Ideals

Fløystad

2018

Acta Math Vietnam

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“…Remark 3.2. We caution the reader that the Borel partial order is often given in the literature as the reverse of how we have presented it in Definition 3.2 (c.f [11,8]). We have made the choice in Definition 3.2 so that the Borel partial order is compatible with monomial orders.…”

Section: Borel Idealsmentioning

confidence: 99%

“…We now explain what we mean by principal L-Borel ideals and give the idea for which collections of principal L-Borel ideals have a Koszul multi-Rees algebra. We have taken the notation of an L-Borel ideal from [11], where Q-Borel ideals are introduced for a partially ordered set Q on the underlying variables of the polynomial ring. An ideal I ⊂ R = K[x 1 , .…”

Section: Introductionmentioning

confidence: 99%

Koszul multi-Rees algebras of principal $L$-Borel Ideals

DiPasquale¹,

Nezhad²

2020

Preprint

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Given a monomial m in a polynomial ring and a subset L of the variables of the polynomial ring, the principal L-Borel ideal generated by m is the ideal generated by all monomials which can be obtained from m by successively replacing variables of m by those which are in L and have smaller index. Given a collection I = {I 1 , . . . , Ir} where I i is L i -Borel for i = 1, . . . , r (where the subsets L 1 , . . . , Lr may be different for each ideal), we prove in essence that if the bipartite incidence graph among the subsets L 1 , . . . , Lr is chordal bipartite, then the defining equations of the multi-Rees algebra of I has a Gröbner basis of quadrics with squarefree lead terms under lexicographic order. Thus the multi-Rees algebra of such a collection of ideals is Koszul, Cohen-Macaulay, and normal. This significantly generalizes a theorem of Ohsugi and Hibi on Koszul bipartite graphs. As a corollary we obtain that the multi-Rees algebra of a collection of principal Borel ideals is Koszul. To prove our main result we use a fiber-wise Gröbner basis criterion for the kernel of a toric map and we introduce a modification of Sturmfels' sorting algorithm. xi xj m ∈ I. In this case we call xi xj m an L-Borel move.The ideal I is a principal L-Borel ideal if its generators are obtained by L-Borel moves from a single monomial, which we call the L-Borel generator of I. For example, if L is the linear ordering on {x 1 , . . . , x 5 } considered above, x 1 , x 3 , x 5 is a principal L-Borel ideal with L-Borel generator x 5 .

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