Let (W,S) be a Coxeter system. We introduce the boolean com-plex of involutions ofWwhich is an analogue of the boolean complex ofWstudied by Ragnarsson and Tenner. By applying discrete Morse theory,we determine the homotopy type of the boolean complex of involutionsfor a large class of (W,S), including all finite Coxeter groups, finding thatthe homotopy type is that of a wedge of spheres of dimension |S|-1. In addition, we find simple recurrence formulas for the number of spheres inthe wedge

We prove that in finite, simply laced types, every Schubert variety indexed by an involution which is not the longest element of some standard parabolic subgroup is singular. 

Given an involutive automorphism theta of a Coxeter system (W, S), let J(theta) subset of W denote the set of twisted involutions. We provide a minimal set of moves that can be added to the braid moves, in order to connect all reduced S-expressions (also known as admissible sequences, reduced I-theta-expressions, or involution words) for any given w is an element of J(theta). This can be viewed as an analogue of the well-known word property for Coxeter groups. It improves upon a result of Hamaker, Marberg, and Pawlowski, and generalises similar statements valid in certain types due to Hu, Zhang, Wu, and Marberg. (C) 2018 Elsevier Inc. All rights reserved.

Special partial matchings (SPMs) are a generalisation of Brentis special matchings. Let a pircon be a poset in which every non-trivial principal order ideal is finite and admits an SPM. Thus pircons generalise Mariettis zircons. We prove that every open interval in a pircon is a PL ball or a PL sphere. It is then demonstrated that Bruhat orders on certain twisted identities and quasiparabolic W-sets constitute pircons. Together, these results extend a result of Can, Cherniaysky, and Twelbeck, prove a conjecture of Hultman, and confirm a claim of Rains and Vazirani.

When acts on the flag variety of , the orbits are in bijection with fixed point free involutions in the symmetric group . In this case, the associated Kazhdan-Lusztig-Vogan polynomials can be indexed by pairs of fixed point free involutions , where denotes the Bruhat order on . We prove that these polynomials are combinatorial invariants in the sense that if is a poset isomorphism of upper intervals in the Bruhat order on fixed point free involutions, then for all v amp;gt;= u.

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.

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.

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.

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 (*).