Introduction to lie algebras and representation theory

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Arrangements Of Hyperplanes

In this paper, we give new upper bounds on the regularity of edge ideals whose resolutions are k-step linear; surprisingly, the bounds are logarithmic in the number of variables. We also give various bounds for the projective dimension of such ideals, generalizing other recent results. By Alexander duality, our results also apply to unmixed square-free monomial ideals of codimension two. We also discuss and connect these results to more classical topics in commutative algebra.

/pdf/bounds-on-the-regularity-and-projective-dimension-of-ideals-4rgbsn5heh.pdf

Bounds on the regularity and projective dimension of ideals associated to graphs

Gamma-positivity is an elementary property that polynomials with symmetric coefficients may have, which directly implies their unimodality. The idea behind it stems from work of Foata, Sch\"utzenberger and Strehl on the Eulerian polynomials; it was revived independently by Br\"and\'en and Gal in the course of their study of poset Eulerian polynomials and face enumeration of flag simplicial spheres, respectively, and has found numerous applications since then. This paper surveys some of the main results and open problems on gamma-positivity, appearing in various combinatorial or geometric contexts, as well as some of the diverse methods that have been used to prove it.

Gamma-positivity in combinatorics and geometry

Socles of Buchsbaum modules, complexes and posets

We show that the Castelnuovo-Mumford regularity of the binomial edge ideal of a graph is bounded below by the length of its longest induced path and bounded above by the number of its vertices.

https://projecteuclid.org/download/pdf_1/euclid.jca/1372776630

Regularity bounds for binomial edge ideals

In 1971, McMullen and Walkup posed the following conjecture, which is called the generalized lower bound conjecture: If P is a simplicial d-polytope then its h-vector (h0, h1, …, hd) satisfies $ {h_0}\leq {h_1}\leq \ldots \leq {h_{{\left\lfloor {{d \left/ {2} \right.}} \right\rfloor }}} $. Moreover, if hr−1 = hr for some $ r\leq \frac{1}{2}d $ then P can be triangulated without introducing simplices of dimension ≤d − r.

/pdf/on-the-generalized-lower-bound-conjecture-for-polytopes-and-3v8cv06et6.pdf

On the generalized lower bound conjecture for polytopes and spheres

We study $h$-vectors of simplicial complexes which satisfy Serre's condition ($S_r$). We say that a simplicial complex $\Delta$ satisfies Serre's condition ($S_r$) if $\tilde H_i(\lk_\Delta(F);K)=0$ for all faces $F \in \Delta$ and for all $i < \min \{r-1,\dim \lk_\Delta(F)\}$, where $\lk_\Delta(F)$ is the link of $\Delta$ with respect to $F$ and where $\tilde H_i(\Delta;K)$ is the reduced homology groups of $\Delta$ over a field $K$. The main result of this paper is that if $\Delta$ satisfies Serre's condition ($S_r$) then (i) $h_k(\Delta)$ is non-negative for $k =0,1,...,r$ and (ii) $\sum_{k\geq r}h_k(\Delta)$ is non-negative.

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2009/0016/0006/MRL-2009-0016-0006-a010.pdf

H-vectors of simplicial complexes with Serre's conditions

In this paper, we study the conjecture of Kuhnel and Lutz, who state that a combinatorial triangulation of the product of two spheres $$\mathbb S^i \times \mathbb S^j$$
 with $$j \ge i$$
 is tight if and only if it has exactly $$i+2j+4$$
 vertices. To approach this conjecture, we use graded Betti numbers of Stanley–Reisner rings. By using recent results on graded Betti numbers, we prove that the only if part of the conjecture holds when $$j>2i$$
 and that the if part of the conjecture holds for triangulations all whose vertex links are simplicial polytopes. We also apply this algebraic approach to obtain lower bounds on the numbers of vertices and edges of triangulations of manifolds and pseudomanifolds.

Tight combinatorial manifolds and graded Betti numbers

Given a semisimple complex linear algebraic group $G$ and a lower ideal $I$ in positive roots of $G$, three objects arise: the ideal arrangement $\mathcal{A}_I$, the regular nilpotent Hessenberg variety $\mbox{Hess}(N,I)$, and the regular semisimple Hessenberg variety $\mbox{Hess}(S,I)$. We show that a certain graded ring derived from the logarithmic derivation module of $\mathcal{A}_I$ is isomorphic to $H^*(\mbox{Hess}(N,I))$ and $H^*(\mbox{Hess}(S,I))^W$, the invariants in $H^*(\mbox{Hess}(S,I))$ under an action of the Weyl group $W$ of $G$. This isomorphism is shown for general Lie type, and generalizes Borel's celebrated theorem showing that the coinvariant algebra of $W$ is isomorphic to the cohomology ring of the flag variety $G/B$. 
This surprising connection between Hessenberg varieties and hyperplane arrangements enables us to produce a number of interesting consequences. For instance, the surjectivity of the restriction map $H^*(G/B)\to H^*(\mbox{Hess}(N,I))$ announced by Dale Peterson and an affirmative answer to a conjecture of Sommers-Tymoczko are immediate consequences. We also give an explicit ring presentation of $H^*(\mbox{Hess}(N,I))$ in types $B$, $C$, and $G$. Such a presentation was already known in type $A$ or when $\mbox{Hess}(N,I)$ is the Peterson variety. Moreover, we find the volume polynomial of $\mbox{Hess}(N,I)$ and see that the hard Lefschetz property and the Hodge-Riemann relations hold for $\mbox{Hess}(N,I)$, despite the fact that it is a singular variety in general.

Satoshi Murai

Papers

Regularity bounds for binomial edge ideals

On the generalized lower bound conjecture for polytopes and spheres

H-vectors of simplicial complexes with Serre's conditions

Tight combinatorial manifolds and graded Betti numbers

Hessenberg varieties and hyperplane arrangements