Alexander duals of multipermutohedron ideals

Kumar, Ajay; Kumar, C. Sathish

doi:10.1007/s12044-014-0164-9

Cited by 5 publications

(4 citation statements)

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“…In the case when x has distinct parts (x 1 > x 2 > • • • > x n ), the ideals I x are known as permutohedron ideals, and their minimal free resolution is constructed explicitly as a cellular resolution (see [MS05,Section 4.3.3] or [BS98]). When x has repeated parts, the cellular resolution is no longer minimal, but the Betti numbers of I x can still be determined [KK13]. One can then also derive (1.5) from the explicit knowledge of the (non-)vanishing behavior of the Betti numbers of I x .…”

Section: Introductionmentioning

confidence: 99%

Regularity of S_n-invariant monomial ideals

Raicu¹

2019

Preprint

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For a polynomial ring S in n variables, we consider the natural action of the symmetric group Sn on S by permuting the variables. For an Sn-invariant monomial ideal I ⊆ S and j ≥ 0, we give an explicit recipe for computing the modules Ext j S (S/I, S), and use this to describe the projective dimension and regularity of I. We classify the Sn-invariant monomial ideals I that have a linear free resolution, and also characterize those which are Cohen-Macaulay. We then consider two settings for analyzing the asymptotic behavior of regularity: one where we look at powers of a fixed ideal I, and another where we vary the dimension of the ambient polynomial ring and examine the invariant monomial ideals induced by I. In the first case we determine the asymptotic regularity for those ideals I that are generated by the Sn-orbit of a single monomial by solving an integer linear optimization problem. In the second case we describe the behavior of regularity for any I, recovering a recent result of Murai.

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Section: Introductionmentioning

confidence: 99%

Regularity of S_n-invariant monomial ideals

Raicu¹

2019

Preprint

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“…K n+1 in degree b and Supp(b) = {j : b j > 0} (see Theorem 5.11 of [11]). Since M (k) K n+1 = I(u(m)) [n] , a combinatorial description of all multidegrees b such that β i−1,b (M (k) K n+1 ) = 0 is given in terms of dual m-isolated subsets (see Definition 3.1 and Theorem 3.2 of [8]). For the particular case of m = (1, .…”

Section: K-skeleton Ideals Of Complete Graphsmentioning

confidence: 99%

mentioning

confidence: 99%

“…The multigraded Betti numbers of multipermutohedron ideals are described in [7]. Also, a combinatorial description of multigraded Betti numbers of Alexander duals of multipermutohedron ideals is given in [8]. Now from the identification of M (k) K n+1 with an Alexander dual of some multipermutohedron ideal, we obtain a combinatorial expression for the (i − 1) th Betti number β i−1 M (k) K n+1 (Theorem 12).…”

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confidence: 99%

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

Kumar

Lather

Sonica³

2021

Electron. J. Combin.

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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.

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Certain variants of multipermutohedron ideals

Kumar

Kumar²

2016

Proc Math Sci

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Alexander duals of multipermutohedron ideals

Cited by 5 publications

References 4 publications

Regularity of S_n-invariant monomial ideals

Regularity of S_n-invariant monomial ideals

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

Certain variants of multipermutohedron ideals

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