Extremal Betti Numbers of Some Cohen–Macaulay Binomial Edge Ideals

Carla Mascia; Giancarlo Rinaldo

doi:10.1142/s1005386721000328

Extremal Betti Numbers of Some Cohen–Macaulay Binomial Edge Ideals

Algebra Colloquium ◽

10.1142/s1005386721000328 ◽

2021 ◽

Vol 28 (03) ◽

pp. 415-430

Author(s):

Carla Mascia ◽

Giancarlo Rinaldo

Keyword(s):

Betti Numbers ◽

Bipartite Graphs ◽

Poincaré Series ◽

Poincare Series ◽

Edge Ideals

We provide the regularity and the Cohen–Macaulay type of binomial edge ideals of Cohen–Macaulay cones, and we show the extremal Betti numbers of some classes of Cohen–Macaulay binomial edge ideals: Cohen–Macaulay bipartite and fan graphs. In addition, we compute the Hilbert–Poincaré series of the binomial edge ideals of some Cohen–Macaulay bipartite graphs.

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Poincaré series of Milnor algebras and free arrangements

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.2009.1146 ◽

2010 ◽

Vol 47 (4) ◽

pp. 513-521

Author(s):

Shahid Ahmad

Keyword(s):

Hyperplane Arrangement ◽

Betti Numbers ◽

Poincaré Series ◽

Poincare Series ◽

Complex Hyperplane

We show that for a free complex hyperplane arrangement \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{A}$$ \end{document}: f = 0, the Poincaré series of the graded Milnor algebra M(f) and the Betti numbers of the arrangement complement M(\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathcal{A}$$ \end{document}) determine each other. Examples show that this is false if we drop the freeness assumption.

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Poincaré Series of some Hypergraph Algebras

MATHEMATICA SCANDINAVICA ◽

10.7146/math.scand.a-15229 ◽

2013 ◽

Vol 112 (1) ◽

pp. 5 ◽

Cited By ~ 1

Author(s):

E. Emtander ◽

R. Fröberg ◽

F. Mohammadi ◽

S. Moradi

Keyword(s):

Betti Numbers ◽

Poincaré Series ◽

Poincare Series ◽

Graded Betti Numbers ◽

Graph Algebra ◽

Wheel Graph

A hypergraph $H=(V,E)$, where $V=\{x_1,\ldots,x_n\}$ and $E\subseteq 2^V$ defines a hypergraph algebra $R_H=k[x_1,\ldots, x_n]/(x_{i_1}\cdots x_{i_k}; \{i_1,\ldots,i_k\}\in E)$. All our hypergraphs are $d$-uniform, i.e., $|e_i|=d$ for all $e_i\in E$. We determine the Poincaré series $P_{R_H}(t)=\sum_{i=1}^\infty\dim_k\mathrm{Tor}_i^{R_H}(k,k)t^i$ for some hypergraphs generalizing lines, cycles, and stars. We finish by calculating the graded Betti numbers and the Poincaré series of the graph algebra of the wheel graph.

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Hilbert–Poincaré series of parity binomial edge ideals and permanental ideals of complete graphs

Collectanea mathematica ◽

10.1007/s13348-020-00294-2 ◽

2020 ◽

Author(s):

Trong Hoang Do ◽

Thomas Kahle

Keyword(s):

Poincaré Series ◽

Complete Graphs ◽

Poincare Series ◽

Edge Ideals

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Poincaré series, 3d gravity and averages of rational CFT

Journal of High Energy Physics ◽

10.1007/jhep04(2021)267 ◽

2021 ◽

Vol 2021 (4) ◽

Author(s):

Viraj Meruliya ◽

Sunil Mukhi ◽

Palash Singh

Keyword(s):

Poincaré Series ◽

Simons Theory ◽

Partition Functions ◽

Moody Algebra ◽

Modular Invariant ◽

Poincare Series ◽

Single Genus ◽

3D Gravity ◽

Chern Simons ◽

Chern Simons Theory

Abstract We investigate the Poincaré series approach to computing 3d gravity partition functions dual to Rational CFT. For a single genus-1 boundary, we show that for certain infinite sets of levels, the SU(2)k WZW models provide unitary examples for which the Poincaré series is a positive linear combination of two modular-invariant partition functions. This supports the interpretation that the bulk gravity theory (a topological Chern-Simons theory in this case) is dual to an average of distinct CFT’s sharing the same Kac-Moody algebra. We compute the weights of this average for all seed primaries and all relevant values of k. We then study other WZW models, notably SU(N)1 and SU(3)k, and find that each class presents rather different features. Finally we consider multiple genus-1 boundaries, where we find a class of seed functions for the Poincaré sum that reproduces both disconnected and connected contributions — the latter corresponding to analogues of 3-manifold “wormholes” — such that the expected average is correctly reproduced.

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