Let G be a connected reductive linear algebraic group over C with an involution theta. Denote by K the subgroup of fixed points. In certain cases, the K-orbits in the flag variety G/B are indexed by the twisted identities t = {theta(omega(-1))omega | omega is an element of W} in the Weyl group W. Under this assumption, we establish a criterion for rational smoothness of orbit closures which generalises classical results of Carrell and Peterson for Schubert varieties. That is, whether an orbit closure is rationally smooth at a given point can be determined by examining the degrees in a "Bruhat graph" whose vertices form a subset of t. Moreover, an orbit closure is rationally smooth everywhere if and only if its corresponding interval in the Bruhat order on t is rank symmetric. less thanbrgreater than less thanbrgreater thanIn the special case K = Sp(2n) (C), G = SL(2n) (C), we strengthen our criterion by showing that only the degree of a single vertex, the "bottom one", needs to be examined. This generalises a result of Deodhar for type A Schubert varieties.

Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.

Inversion arrangements and Bruhat intervals2011In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 118, no 7, p. 1897-1906Article in journal (Refereed)

Abstract [en]

Let W be a finite Coxeter group. For a given w is an element of 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 is an element of W that satisfy (*). A couple of immediate consequences are derived: (1) The criterion only involves the order ideal of w as an abstract poser. 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 is an element of W is rationally smooth, then w satisfies (*).

Given a permutation statistic s : G(n) -greater than R, Ilk, define the mean statistic s as the class function giving the mean of a over. conjugacy classes. We describe a way to calculate the expected value of a on a product of t independently chosen elements from the uniform distribution on a union of conjugacy classes Gamma subset of G(n). In order to apply the formula, one needs to express the class function 3 as a linear combination of irreducible G(n)-characters. We provide such expressions for several commonly studied permutation statistics, including the exceedance number, inversion number, descent number, major index and k-cycle number. In particular, this leads to formulae for the expected values of said statistics.

A root ideal arrangement A_I is the set of reflecting hyperplanes corresponding to the roots in an order ideal I of the root poset on the positive roots of a finite crystallographic root system. A characterisation of supersolvable root ideal arrangements is obtained. Namely, A_I is supersolvable if and only if I is chain peelable, meaning that it is possible to reach the empty poset from I by in each step removing a maximal chain which is also an order filter. In particular, supersolvability is preserved under taking subideals. We identify the maximal ideals that correspond to non-supersolvable arrangements. There are essentially two such ideals, one in type D_4 and one in type F_4. By showing that A_I is not line-closed if I contains one of these, we deduce that the Orlik-Solomon algebra OS(A_I) has the Koszul property if and only if A_I is supersolvable.