The Orbits of the Symplectic Group on the Flag Manifold

Bertiger, Anna

doi:10.48550/arxiv.1411.2302

Cited by 2 publications

(3 citation statements)

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“…We depict such matchings with the vertices on a horizontal axis, ordered from left to right, and edges shown as convex curves in the upper half plane. For example, (1,6)(2, 7)(3, 4)(5, 8) ∈ FPF 8 is represented as . .…”

Section: Fpf-involution Schubert Polynomialsmentioning

confidence: 99%

Fixed-point-free involutions and Schur $P$-positivity

Hamaker

Marberg

Pawlowski

2020

Journal of Combinatorics

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The orbits of the symplectic group acting on the type A flag variety are indexed by the fixed-point-free involutions in a finite symmetric group. The cohomology classes of the closures of these orbits have polynomial representativesŜ FPF z akin to Schubert polynomials. We show that the fixed-point-free involution Stanley symmetric functionsF FPF z , which are stable limits of the polynomialsŜ FPF z , are Schur P -positive. To do so, we construct an analogue of the Lascoux-Schützenberger tree, an algebraic recurrence that computes Schubert polynomials. As a byproduct of our proof, we obtain a Pfaffian formula of geometric interest forŜ FPF z when z is a fixed-point-free version of a Grassmannian permutation. We also classify the fixed-point-free involution Stanley symmetric functions that are single Schur P -functions, and show that the decomposition ofF FPF z into Schur P -functions is unitriangular with respect to dominance order on strict partitions. These results and proofs mirror previous work by the authors related to the orthogonal group action on the type A flag variety. * This author was partially supported by NSF grant 1148634. 5 Schur P -positivity 17 6 Triangularity 24 7 FPF-vexillary involutions 25 8 Pfaffian formulas 27A Index of symbols 31 minantal formula for S w when w is vexillary (2143-avoiding) or fully commutative (321-avoiding). When w is dominant (132-avoiding), S w is a monomial.Assume n is even and consider the symplectic group Sp n (C) acting on Fl(n). There are again finitely many orbits, now indexed by the fixed-point-free involutions in S n [22]. For a fixed-pointfree involution z ∈ S n , the cohomology class of the corresponding orbit closure Y z is represented by the fixed-point-free involution Schubert polynomialŜ FPF z introduced in [30] and described precisely by Definition 2.4. In [8], we gave a generating function-type description ofŜ FPF z in terms of reduced words and derived a simple product formula forŜ FPF z when z is a dominant fixed-point-free involution. In this paper, we continue to studyŜ FPF z and related combinatorics. Some of this combinatorics also appears in representation theory when studying the quasi-parabolic Iwahori-Hecke algebra modules defined by Rains and Vazirani [21].The groups O n (C) and GL p (C) × GL q (C) (with p + q = n) also act on Fl(n) with finitely many orbits. This paper is a continuation of the authors' previous work on the O n (C) case [11]. The GL p (C) × GL q (C) case has not yet been as thoroughly investigated, though there has been some recent progress in [4]; see also [5,31].The symmetric group S n of permutations of [n] = {1, 2, . . . , n} is a Coxeter group generated by the simple transpositions s i = (i, i + 1) for 1 ≤ i ≤ n − 1. For u ∈ S m and v ∈ S n , we write u × v for the permutation in S m+n that maps i → u(i) for i ∈ [m] and m + i → m + v(i) for i ∈ [n]. The Stanley symmetric function of w ∈ S n is then the stable limit F w def = lim m→∞ S 1m×wwhere 1 m denotes the identity element of S m . This is a well-defined homogeneous s...

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Section: Fpf-involution Schubert Polynomialsmentioning

confidence: 99%

Fixed-point-free involutions and Schur $P$-positivity

Hamaker

Marberg

Pawlowski

2020

Journal of Combinatorics

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“…The following basic properties are equivalent to [3, Lemma 2.2 and Corollary 2.3]. Lemma 4.8 (See [3]). If z ∈ F Z then lFPF (z) = 2n + c where n is the number of unordered pairs of nesting cycles of z, and c is the number of unordered pairs of crossing cycles of z.…”

Section: Bruhat Order On Fpf Involutionsmentioning

confidence: 99%

“…If z ∈ F Z then lFPF (z) = 2n + c where n is the number of unordered pairs of nesting cycles of z, and c is the number of unordered pairs of crossing cycles of z. Proposition 4.9 (See [3]). Let y ∈ F Z .…”

Section: Bruhat Order On Fpf Involutionsmentioning

confidence: 99%

Transition formulas for involution Schubert polynomials

Hamaker

Marberg

Pawlowski

2018

Sel. Math. New Ser.

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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...

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The Orbits of the Symplectic Group on the Flag Manifold

Cited by 2 publications

References 8 publications

Fixed-point-free involutions and Schur $P$-positivity

Fixed-point-free involutions and Schur $P$-positivity

Transition formulas for involution Schubert polynomials

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