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

Involution words are variations of reduced words for involutions in Coxeter groups, first studied under the name of "admissible sequences" by Richardson and Springer. They are maximal chains in Richardson and Springer's weak order on involutions. This article is the first in a series of papers on involution words, and focuses on their enumerative properties. We define involution analogues of several objects associated to permutations, including Rothe diagrams, the essential set, Schubert polynomials, and Stanley symmetric functions. These definitions have geometric interpretations for certain intervals in the weak order on involutions. In particular, our definition of "involution Schubert polynomials" can be viewed as a Billey-Jockusch-Stanley type formula for cohomology class representatives of O n -and Sp 2n -orbit closures in the flag variety, defined inductively in recent work of Wyser and Yong. As a special case of a more general theorem, we show that the involution Stanley symmetric function for the longest element of a finite symmetric group is a product of staircase-shaped Schur functions. This implies that the number of involution words for the longest element of a finite symmetric group is equal to the dimension of a certain irreducible representation of a Weyl group of type B.for z ∈ I(S 2n ).( 1.2) Elements of this set will be called fixed-point-free involution words, since v n is the involution of smallest possible length with no fixed points in S 2n . The setR FPF (z) is non-empty if and only if z ∈ I(S 2n ) has no fixed points, in which case every involution in the interval between v n and z in weak order will also be fixed-point-free. Fixed-point-free involution words are a special case of Rains and Vazirani's notion of "reduced expressions" for elements of quasiparabolic sets [42].is a reduced word for some element of W and the setR(y, z) is closed under the braid relations for (W, S). For y, z ∈ I and w ∈ A(y, z), we say that w is a relative atom from y to z.In the geometric cases, we define A(y) def = A(1, y) and A FPF (z) def = A(v n , z).Remark. The theorem-definition follows from results of Richardson and Springer [44]. A direct proof using our present notation appears in [20].The involution Rothe diagramsD(y) andD FPF (y) of y ∈ I(S n ) are defined in Section 3.2 as certain restrictions of the usual Rothe diagram D(y). The essential sets Ess(D(y)) and Ess(D FPF (y)) consist of southeast corners in the corresponding involution diagram. This closely mirrors the definition of Fulton's essential set Ess(D(w)) for w ∈ S n . In Proposition 3.16, we show that the involution essential sets determine a subset of the rank conditions sufficient to define Y K y when K = O n (C) or K = Sp n (C). The proof is largely a consequence of the analogous result for the B + -action, with some subtleties in the fixed-point-free case. These objects prove to be a key tool in our study of involution Schubert polynomials and involution Stanley symmetric functions. Sp(2n) w ] = 0, so as Υ FPF z ≡ [Y Sp(2n) w] it follo...

The involution Stanley symmetric functionsF y are the stable limits of the analogues of Schubert polynomials for the orbits of the orthogonal group in the flag variety. These symmetric functions are also generating functions for involution words, and are indexed by the involutions in the symmetric group. By construction eachF y is a sum of Stanley symmetric functions and therefore Schur positive. We prove the stronger fact that these power series are Schur P -positive. We give an algorithm to efficiently compute the decomposition ofF y into Schur Psummands, and prove that this decomposition is triangular with respect to the dominance order on partitions. As an application, we derive pattern avoidance conditions which characterize the involution Stanley symmetric functions which are equal to Schur P -functions. We deduce as a corollary that the involution Stanley symmetric function of the reverse permutation is a Schur P -function indexed by a shifted staircase shape. These results lead to alternate proofs of theorems of Ardila-Serrano and DeWitt on skew Schur functions which are Schur P -functions. We also prove new Pfaffian formulas for certain related involution Schubert polynomials.

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...