In a seminal 1994 paper Lusztig (1994)
26, Lusztig extended the theory of total positivity by introducing the totally non-negative part
(
G
/
P
)
⩾
0
of an arbitrary (generalized, partial) flag ...variety
G
/
P
. He referred to this space as a “remarkable polyhedral subspace”, and conjectured a decomposition into cells, which was subsequently proven by the first author Rietsch (1998)
33. In Williams (2007)
40 the second author made the concrete conjecture that this cell decomposed space is the next best thing to a polyhedron, by conjecturing it to be a
regular CW complex that is homeomorphic to a closed ball. In this article we use discrete Morse theory to prove this conjecture up to homotopy-equivalence. Explicitly, we prove that the boundaries of the cells are homotopic to spheres, and the closures of cells are contractible. The latter part generalizes a result of Lusztig's (1998)
28, that
(
G
/
P
)
⩾
0
– the closure of the top-dimensional cell – is contractible. Concerning our result on the boundaries of cells, even the special case that the boundary of the top-dimensional cell
(
G
/
P
)
>
0
is homotopic to a sphere, is new for all
G
/
P
other than projective space.
Ideals with linear quotients Jahan, Ali Soleyman; Zheng, Xinxian
Journal of combinatorial theory. Series A,
2010, 2010-01-00, Volume:
117, Issue:
1
Journal Article
Peer reviewed
Open access
We study basic properties of monomial ideals with linear quotients. It is shown that if the monomial ideal
I has linear quotients, then the squarefree part of
I and each component of
I as well as
m
I
...have linear quotients, where
m
is the graded maximal ideal of the polynomial ring. As an analogy to the Rearrangement Lemma of Björner and Wachs we also show that for a monomial ideal with linear quotients the admissible order of the generators can be chosen degree increasingly.
Recently, methods have been proposed to reconstruct a 2-manifold surface from a sparse cloud of points estimated from an image sequence. Once a 3D Delaunay triangulation is computed from the points, ...the surface is searched by growing a set of tetrahedra whose boundary is maintained 2-manifold. Shelling is a step that adds one tetrahedron at once to the growing set. This paper surveys properties that helps to understand the shelling performances: shelling provides most tetrahedra enclosed by the final surface, but it can “get stuck” or block in unexpected cases.
Geometric lattices are characterized in this paper as those finite, atomic lattices such that every atom ordering induces a lexicographic shelling given by an edge labeling known as a minimal ...labeling. Equivalently, geometric lattices are shown to be exactly those finite lattices such that every ordering on the join-irreducibles induces a lexicographic shelling. This new characterization fits into a similar paradigm as McNamaraʼs characterization of supersolvable lattices as those lattices admitting a different type of lexicographic shelling, namely one in which each maximal chain is labeled with a permutation of {1,…,n}.
If a d-dimensional pure simplicial complex C has a shelling, which is a specific total order of all facets of C, C is said to be shellable. We consider the problem of deciding whether C is shellable ...or not. This problem is solved in linear time of m, the number of all facets of C, if d=1 or C is a pseudomanifold in d=2. Otherwise it is unknown at this point whether the decision of shellability can be solved in polynomial time of m. Thus, for the latter case, we had no choice but to apply a brute force method to the decision problem; namely checking up to the m! ways to see if one can arrange all the m facets of C into a shelling. In this paper, we introduce a new concept, called h-assignment, to C and propose a practical method using h-assignments to decide whether C is shellable or not. Our method can make the decision of shellability of C by smaller sized computation than the brute force method.
Let In be the set of involutions in the symmetric group Sn, and for A⊆{0,1,...,n}, letFnA={σ∈In|σ has a fixed points for some a∈A}. We give a complete characterisation of the sets A for which FnA, ...with the order induced by the Bruhat order on Sn, is a graded poset. In particular, we prove that Fn{1} (i.e., the set of involutions with exactly one fixed point) is graded, which settles a conjecture of Hultman in the affirmative. When FnA is graded, we give its rank function. We also give a short new proof of the EL-shellability of Fn{0} (i.e., the set of fixed point-free involutions), which was recently proved by Can, Cherniavsky, and Twelbeck.
The authors compute the（rational） Betti number of real toric varieties associated to Weyl chambers of type B, and furthermore show that their integral cohomology is p-torsion free for all odd primes ...p.
The order complex of inclusion poset $PF_n$ of Borel orbit closures in skew-symmetric matrices is investigated. It is shown that $PF_n$ is an EL-shellable poset, and furthermore, its order complex ...triangulates a ball. The rank-generating function of $PF_n$ is computed and the resulting polynomial is contrasted with the Hasse-Weil zeta function of the variety of skew-symmetric matrices over finite fields.
Monomial Ideals of Forest Type Jahan, Ali Soleyman; Zheng, Xinxian
Communications in algebra,
08/2012, Volume:
40, Issue:
8
Journal Article
Peer reviewed
As a generalization of the facet ideal of a forest, we define monomial ideal of forest type and show that monomial ideals of forest type are pretty clean. As a consequence, we show that if I is a ...monomial ideal of forest type in the polynomial ring S, then Stanley's decomposition conjecture holds for S/I. The other main result of this article shows that a clutter is totally balanced if and only if it has the free vertex property, and which is also equivalent to say that its edge ideal is a monomial ideal of forest type or is generated by an M sequence.
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