Weak order on complete quadrics

Abstract

Using an action of the Richardson-Springer monoid on involutions, we study the weak order on the variety of complete quadrics. Maximal chains in the poset are explicitly determined. Applying results of Brion, our calculations describe certain cohomology classes in the complete flag variety.

Involution words II: braid relations and atomic structures

“…Let Γ(y) be the set formed by adding to Cyc(y) all pairs (a, b) with a = y(a) > b = y(b). To any γ, γ ′ ∈ Γ(y), we associate via (5.2) a certain involution σ(γ, γ ′ ) ∈ I(B 4 ) in the Weyl group of type B 4 . We then introduce a certain natural partial order ≺ on the finite set I(B 4 ); see (5.3).…”

“…Then w ∈ A(x, y) if and only if (a) wγ ∈ Γ(x) for all γ ∈ Cyc(y). As mentioned at the outset of this section, Theorem 5.10 gives a common generalization of [4,Theorem 3.7] and [5, Theorem 2.5 and Corollary 2.16]. We will use these results in the next section, so briefly indicate here how to recover them as special cases of our theorem.…”

“…4. The sets B(x) −1 = u −1 : u ∈ B(x) for x ∈ I(S n ) are the distinct equivalence classes in S n under ∼ B .…”

Bialgebras for Stanley symmetric functions

“…For integers p, q > 0, let N (p, q) = N (q, p) = P +Q P · r(p · · · 321) · r(q · · · 321) where P = p 2 and Q = q 2 . The following combines [7,Theorem 3.7] and [13, Theorem 1.4]: Proposition 1.2 (See [7,13]). If p = ⌊ n+1 2 ⌋ and q = ⌈ n+1 2 ⌉ then N (p, q) = w∈A(n) r(w).…”

Schubert Polynomial Analogues for Degenerate Involutions

“…One of their key results is that the inclusion order of H-orbit closures in F is given by the restriction of Bruhat order on S n to I n [22]. In a combinatorial framework, Can and the author [6] decomposed the (degenerate) involution Schubert polynomial associated to the longest permutation (which can be viewed as a µ-involution for any composition µ) as a sum of ordinary Schubert polynomials, by studying maximal chains in the associated weak order poset, using a result of Brion [4]. Can, Wyser and the author [8] then decomposed an arbitrary involution Schubert polynomial as a multiplicity-free sum of ordinary Schubert polynomials.…”