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Register a new userA generalized lower bound theorem for balanced manifolds
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A simplicial complex of dimension $d-1$ is said to be balanced if its graph is $d$-colorable. Juhnke-Kubitzke and Murai proved an analogue of the generalized lower bound theorem for balanced simplicial polytopes. We establish a generalization of their result to balanced triangulations of closed homology manifolds and balanced triangulations of orientable homology manifolds with boundary under an additional assumption that all proper links of these triangulations have the weak Lefschetz property. As a corollary, we show that if $Delta$ is an arbitrary balanced triangulation of any closed homology manifold of dimension $d-1 geq 3$, then $2h_2(Delta) - (d-1)h_1(Delta) geq 4{d choose 2}(tilde{beta}_1(Delta)-tilde{beta}_0(Delta))$, thus verifying a conjecture by Klee and Novik. To prove these results we develop the theory of flag $h$-vectors.
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A remarkable and important property of face numbers of simplicial polytopes is the generalized lower bound inequality, which says that the $h$-numbers of any simplicial polytope are unimodal. Recently, for balanced simplicial $d$-polytopes, that is simplicial $d$-polytopes whose underlying graphs are $d$-colorable, Klee and Novik proposed a balanced analogue of this inequality, that is stronger than just unimodality. The aim of this article is to prove this conjecture of Klee and Novik. For this, we also show a Lefschetz property for rank-selected subcomplexes of balanced simplicial polytopes and thereby obtain new inequalities for their $h$-numbers.
In this paper, we study Lefschetz properties of Artinian reductions of Stanley-Reisner rings of balanced simplicial $3$-polytopes. A $(d-1)$-dimensional simplicial complex is said to be balanced if its graph is $d$-colorable. If a simplicial complex is balanced, then its Stanley-Reisner ring has a special system of parameters induced by the coloring. We prove that the Artinian reduction of the Stanley-Reisner ring of a balanced simplicial $3$-polytope with respect to this special system of parameters has the strong Lefschetz property if the characteristic of the base field is not two or three. Moreover, we characterize $(2,1)$-balanced simplicial polytopes, i.e., polytopes with exactly one red vertex and two blue vertices in each facet, such that an analogous property holds. In fact, we show that this is the case if and only if the induced graph on the blue vertices satisfies a Laman-type combinatorial condition.
In 1967, Grunbaum conjectured that any $d$-dimensional polytope with $d+sleq 2d$ vertices has at least [phi_k(d+s,d) = {d+1 choose k+1 }+{d choose k+1 }-{d+1-s choose k+1 } ] $k$-faces. We prove this conjecture and also characterize the cases in which equality holds.
Computing the reversal distances of signed permutations is an important topic in Bioinformatics. Recently, a new lower bound for the reversal distance was obtained via the plane permutation framework. This lower bound appears different from the existing lower bound obtained by Bafna and Pevzner through breakpoint graphs. In this paper, we prove that the two lower bounds are equal. Moreover, we confirm a related conjecture on skew-symmetric plane permutations, which can be restated as follows: let $p=(0,-1,-2,ldots -n,n,n-1,ldots 1)$ and let $$ tilde{s}=(0,a_1,a_2,ldots a_n,-a_n,-a_{n-1},ldots -a_1) $$ be any long cycle on the set ${-n,-n+1,ldots 0,1,ldots n}$. Then, $n$ and $a_n$ are always in the same cycle of the product $ptilde{s}$. Furthermore, we show the new lower bound via plane permutations can be interpreted as the topological genera of orientable surfaces associated to signed permutations.
Let $A$ be a noetherian Koszul Artin-Schelter regular algebra, and let $fin A_2$ be a central regular element of $A$. The quotient algebra $A/(f)$ is usually called a (noncommutative) quadric hypersurface. In this paper, we use the Clifford deformation to study the quadric hypersurfaces obtained from the tensor products. We introduce a notion of simple graded isolated singularity and proved that, if $B/(g)$ is a simple graded isolated singularity of 0-type, then there is an equivalence of triangulated categories $underline{text{mcm}},A/(f)congunderline{text{mcm}},(Aotimes B)/(f+g)$ of the stable categories of maximal Cohen-Macaulay modules. This result may be viewed as a generalization of Kn{o}rrers periodicity theorem. As an application, we study the double branch cover $(A/(f))^#=A[x]/(f+x^2)$ of a noncommutative conic $A/(f)$.
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