Involution words II: braid relations and atomic structures

Abstract

Involution words are variations of reduced words for twisted involutions in Coxeter groups. They arise naturally in the study of the Bruhat order, of certain Iwahori

Schubert Polynomial Analogues for Degenerate Involutions

“…There are similar results for the symmetric subgroups Sp n (C) [8,14,17,25] and GL p (C) × GL q (C), p + q = n, [26,8] of GL n (C) . But there are symmetric subgroups associated to other reductive algebraic groups as well.…”

“…We now introduce a central combinatorial problem, to describe maximal chains of intervals in the weak order poset of involutions, using the language of [17]. Let τ, τ ′ ∈ I n and suppose that τ ≤ τ ′ .…”

“…Hamaker, Marberg and Pawlowski [14] gave the first detailed account of involution Schubert polynomials, creating a uniform combinatorial language for the study and connecting the combinatorics explicitly to the geometry of H-orbit closures in F . Hamaker, Marberg and Pawlowski [17] then developed the theory of involution words, describing the maximal chains for any interval in the weak order poset associated to H-orbits on F and providing evidence for their explicit conjecture that such chains are closely linked to ordinary Bruhat order on S n . Can, Wyser and the author [7] return to the study of the degenerate involution Schubert polynomial associated to the longest permutation viewed as a µ-involution and use geometric considerations to show that certain multiplicity-free sums of ordinary Schubert polynomials have very simple factorizations.…”

“…[17, Theorem 5.11] Let π, τ ∈ I n . Then A * (τ, τ ′ ) consists of all w ∈ S n such that for all (i, j), (k, l) ∈ Cyc(τ ′ ):…”

A symplectic refinement of shifted Hecke insertion

“…For example, A(321) = {231, 312} ⊂ {231, 312, 321} = B(321). Following [9], we refer to the elements of A(z) as atoms for z and to the elements of B(z) as Hecke atoms. Fix z ∈ I ∞ and suppose a 1 < a 2 < a 3 < .…”