Axel Hultman

The combinatorics of twisted involutions in coxeter groups

The open intervals in the Bruhat order on twisted involutions in a Coxeter group are shown to be PL spheres. This implies results conjectured by F. Incitti and sharpens the known fact that these posets are Gorenstein* over Z(2). We also introduce a Boolean cell complex which is an analogue for twisted involutions of the Coxeter complex. Several classical Coxeter complex properties are shared by our complex. When the group is finite, it is a shellable sphere, shelling orders being given by the linear extensions of the weak order on twisted involutions. Furthermore, the h-polynomial of the complex coincides with the polynomial counting twisted involutions by descents. In particular, this gives a type-independent proof that the latter is symmetric.

Bruhat intervals of length 4 in Weyl groups

We determine all isomorphism classes of intervals of length 4 in the Bruhat order on the Weyl groups A4, B4, D4 and F4. It turns out that there are 24 of them (some of which are dual to each other). Work of Dyer allows us to conclude that these are the only intervals of length 4 that can occur in the Bruhat order on any Weyl group. We also determine the intervals that arise already in the smaller classes of simply laced Weyl groups and symmetric groups.Our method combines theoretical arguments and computer calculations. We also present an independent, completely computerized, approach.

Inversion arrangements and Bruhat intervals

Let W be a finite Coxeter group. For a given w@?W, the following assertion may or may not be satisfied:(@?)The principal Bruhat order ideal of w contains as many elements as there are regions in the inversion hyperplane arrangement of w. We present a type independent combinatorial criterion which characterises the elements w@?W that satisfy (@?). A couple of immediate consequences are derived:(1)The criterion only involves the order ideal of w as an abstract poset. In this sense, (@?) is a poset-theoretic property. (2)For W of type A, another characterisation of (@?), in terms of pattern avoidance, was previously given in collaboration with Linusson, Shareshian and Sjostrand. We obtain a short and simple proof of that result. (3)If W is a Weyl group and the Schubert variety indexed by w@?W is rationally smooth, then w satisfies (@?).

Quotient Complexes and Lexicographic Shellability

AbstractLet Πn,k,k and Πn,k,h, h < k, denote the intersection lattices of the k-equal subspace arrangement of type

Fixed Points of Zircon Automorphisms

\mathcal{D}

Estimating the expected reversal distance after a fixed number of reversals

n and the k,h-equal subspace arrangement of type

Link complexes of subspace arrangements

\mathcal{B}