Erdős–Ko–Rado theorems on the weak Bruhat lattice

Descents and the Weak Bruhat order

Let Dk be the set of permutations in Sn with k descents and Ak be the set of permutations with k ascents. For permutations of type A, which are the usual symmetric group elements, bijections σ : Dk → Ak satisfying σ(w) ≥ w in the weak Bruhat ordering are constructed for k = 1 and k = 2. Such a bijection is also described explicitly for k = 1 for permutations of type B. We discuss how this bijec...

Two theorems on lattice expansions

Computing the Size of Intervals in the Weak Bruhat Order

The weak Bruhat order on Sn is the partial order ≺ so that σ ≺ τ whenever the set of inversions of σ is a subset of the set of inversions of τ . We investigate the time complexity of computing the size of intervals with respect to ≺. Using relationships between two-dimensional posets and the weak Bruhat order, we show that the size of the interval [σ1, σ2] can be computed in polynomial time whe...

From the Weak Bruhat Order to Crystal Posets

We investigate the ways in which fundamental properties of the weak Bruhat order on a Weyl group can be lifted (or not) to a corresponding highest weight crystal graph, viewed as a partially ordered set; the latter projects to the weak order via the key map. First, a crystal theoretic analogue of the statement that any two reduced expressions for the same Coxeter group element are related by Co...

Weak Lefschetz Theorems

A new approach to Nori's weak Lefschetz theorem is described. The new approach, which involves the ¯ ∂-method, avoids moving arguments and gives much stronger results. In particular, it is proved that if X and Y are connected smooth projective varieties of positive dimension and f : Y − → X is a holomorphic immersion with ample normal bundle, then the image of π 1 (Y) in π 1 (X) is of finite in...