Transition formulas for involution Schubert polynomials

Abstract

The orbits of the orthogonal and symplectic groups on the flag variety are in bijection, respectively, with the involutions and fixed-point-free involutions in the symmetric group S n . Wyser and Yong have described polynomial representatives for the cohomology classes of the closures of these orbits, which we denote asŜ y (to be called involution Schubert polynomials) andŜ FPF y (to be called fixed-point-free involution Schubert polynomials). Our main results are explicit formulas decomposing the product ofŜ y (respectively,Ŝ FPF y ) with any y-invariant linear polynomial as a linear combination of other involution Schubert polynomials. These identities serve as analogues of Lascoux and Schützenberger's transition formula for Schubert polynomials, and lead to a self-contained algebraic proof of the nontrivial equivalence of several definitions ofŜ y andŜ FPF y appearing in the literature. Our formulas also imply combinatorial identities about involution words, certain variations of reduced words for involutions in S n . We construct operators on involution words based on the Little map to prove these identities bijectively. The proofs of our main theorems depend on some new technical results, extending work of Incitti, about covering relations in the Bruhat order of S n restricted to involutions.Let S Z denote the group of permutations of Z which fix all but finitely many points, and write S ∞ for the subgroup of elements in S Z with support contained in P = {1, 2, 3, . . . }. Define I ∞ (respectively, I Z ) as the subset of involutions in S ∞ (respectively, S Z ). We also write S n and I n for the subsets of S ∞ and I ∞ which fix all numbers outside [n] = {1, 2, . . . , n}, and F n ⊂ I n for the subset of fixed-point-free involutions. The Schubert polynomials are a family of homogeneous polynomials S w ∈ Z[x 1 , x 2 , . . .] indexed by w ∈ S ∞ . Write B for the subgroup of lower triangular matrices in GL n (C). It is well-known that the right B-orbits in the flag variety Fl(n) = B\GL n (C) are in bijection with S n , that the integral cohomology ring of Fl(n) is isomorphic to a quotient of Z[x 1 , x 2 , . . . , x n ], and that under this isomorphism, the Schubert polynomials {S w : w ∈ S n } correspond to the cohomology classes Poincaré dual to the closures of the aforementioned B-orbits; see [27] for details.The involution Schubert polynomials are homogeneous polynomialsŜ y indexed by y ∈ I ∞ serving a similar geometric purpose: the right orbits of O n (C) on Fl(n) are in bijection with I n , and the cohomology classes of their orbit closures are (up to a constant factor) represented by the involution Schubert polynomials {Ŝ y : y ∈ I n }. The family of fixed-point-free involution Schubert polynomials {Ŝ FPF z : z ∈ F n } plays an analogous role when n is even and O n (C) is replaced by Sp n (C). The precise definitions ofŜ y andŜ FPF z appear in Sections 3 and 4. We attribute the definitions of these polynomials to Wyser and Yong [32], although they occur as special cases of the cohomology representatives des...

On some actions of the 0-Hecke monoids of affine symmetric groups

“…Using this information, we prove that involutions inS n have what we call the Bruhat covering property:Theorem 1.1 (Bruhat covering property). If y ∈Ĩ n and t ∈S n is a reflection, then there exists at most one z ∈Ĩ n such that {wt : w ∈ A(y) and ℓ(wt) = ℓ(w) + 1} ∩ A(z) = ∅.The analogue of this result for involutions in S n was shown in [11], and served as a key lemma in proofs of "transition formulas" for certain involution Schubert polynomials. We conjecture that the same property holds for arbitrary Coxeter systems, in the following sense.Let (W, S) be a Coxeter system with length function ℓ : W → N and Demazure product • : W × W → W .…”

“…Can, Joyce, and Wyser's description of A(y) for y ∈ I n in [6] implies that w ∈ S n belongs to A(y) if and only if [w] E ∈ A([y] E ) for all subsets E ⊂ [n] which are invariant under y and contain at most two y-orbits; cf. [11,Corollary 3.19]. This "local" criterion for membership in A(y) was an important tool in the proofs of the main results in [11].…”

“…[11,Corollary 3.19]. This "local" criterion for membership in A(y) was an important tool in the proofs of the main results in [11]. Proof.…”

A symplectic refinement of shifted Hecke insertion

“…For z ∈ F ∞ , define z * ∈ F ∞ by conjugating z by n · · · 321 where n ∈ 2N is minimal such that z(i) = i − (−1) i for all integers i > n; the map w 1 w 2 · · · w l → (n − w 1 )(n − w 2 ) · · · (n − w l ) is then a bijection H Sp (z) → H Sp (z * ). [8,10,11,12]. Setting β = 0 in GP λ , alternatively, yields the well-known Schur P -function P λ .…”

Schubert Polynomial Analogues for Degenerate Involutions

“…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. Hamaker, Marberg and Pawlowski [16] have given a transition formula for involution Schubert polynomials, generalizing the transition formula for ordinary Schubert polynomials established by Lascoux and Schützenberger [19]. Hamaker, Marberg and Pawlowski [15] have also initiated a study of involution Stanley symmetric functions, a natural limit of involution Schubert polynomials, and shown that they can be expanded positively in the Schur P -basis; they have also established a similar result for fixed-point-free involution Stanley symmetric functions [13].…”