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 as \({\hat{\mathfrak S}}_y\) (to be called involution Schubert polynomials) and \(\hat{\mathfrak S}^\mathtt{{FPF}}_y\) (to be called fixed-point-free involution Schubert polynomials). Our main results are explicit formulas decomposing the product of \({\hat{\mathfrak S}}_y\) (respectively, \(\hat{\mathfrak S}^\mathtt{{FPF}}_y\)) 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 of \({\hat{\mathfrak S}}_y\) and \( \hat{\mathfrak S}^\mathtt{{FPF}}_y\) 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.
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References
Abe, H., Billey, S.: Consequences of the Lakshmibai–Sandhya theorem: the ubiquity of permutation patterns in Schubert calculus and related geometry. Adv. Stud. Pure Math. 71, 1–52 (2016)
Bergeron, N., Billey, S.: RC-graphs and Schubert polynomials. Exp. Math. 2, 257–269 (1993)
Bertiger, A.: The orbits of the symplectic group on the flag manifold. Preprint, arXiv:1411.2302 (2014)
Braden, T., Billey, S.: Lower bounds for Kazhdan–Lusztig polynomials from patterns. Transform. Groups 8(4), 321–332 (2003)
Billey, S., Haiman, M.: Schubert polynomials for the classical groups. J. AMS 8, 443–482 (1995)
Billey, S.C., Jockusch, W., Stanley, R.P.: Some combinatorial properties of Schubert polynomials. J. Algebraic Comb. 2, 345–374 (1993)
Brion, M.: The behaviour at infinity of the Bruhat decomposition. Comment. Math. Helv. 73(1), 137–174 (1998)
Can, M.B., Joyce, M., Wyser, B.: Chains in weak order posets associated to involutions. J. Comb. Theory Ser. A 137, 207–225 (2016)
Can, M. B., Joyce, M., Wyser, B.: Wonderful symmetric varieties and schubert polynomials. Preprint arXiv:1509.03292 (2015)
Hamaker, Z., Marberg, E., Pawlowski, B.: Involution words: counting problems and connections to Schubert calculus for symmetric orbit closures. Preprint arXiv:1508.01823 (2015)
Hamaker, Z., Marberg, E., Pawlowski, B.: Involution words II: braid relations and atomic structures. J. Algebraic Comb. 45, 701–743 (2017)
Hamaker, Z., Marberg, E., Pawlowski, B.: Schur \(P\)-positivity and involution Stanley symmetric functions. In: IMRN, rnx274 (2017)
Hamaker, Z., Marberg, E., Pawlowski, B.: Fixed-point-free involutions and Schur \(P\)-positivity. Preprint arXiv:1706.06665 (2017)
Hu, J., Zhang, J.: On involutions in symmetric groups and a conjecture of Lusztig. Adv. Math. 287, 1–30 (2016)
Hultman, A.: Fixed points of involutive automorphisms of the Bruhat order. Adv. Math. 195, 283–296 (2005)
Hultman, A.: The combinatorics of twisted involutions in Coxeter groups. Trans. Am. Math. Soc. 359, 2787–2798 (2007)
Hultman, A.: Twisted identities in Coxeter groups. J. Algebraic Comb. 28, 313–332 (2008)
Hultman, A., Vorwerk, K.: Pattern avoidance and Boolean elements in the Bruhat order on involutions. J. Algebraic Comb. 30, 87–102 (2009)
Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge (1990)
Incitti, F.: The Bruhat Order on the Involutions of the Symmetric Group. J. Algebraic Comb. 20, 243–261 (2004)
Knutson, A.: Schubert polynomials and symmetric functions, notes for the Lisbon Combinatorics Summer School. http://www.math.cornell.edu/allenk/ (2012)
Knutson, A., Miller, E.: Subword complexes in Coxeter groups. Adv. Math. 184(1), 161–176 (2004)
Lam, T., Shimozono, M.: A Little bijection for affine Stanley symmetric functions. Seminaire Lotharingien de Combinatoire 54A, B54Ai (2006)
Lascoux, A., Schützenberger, M.-P.: Schubert polynomials and the Littlewood–Richardson rule. Lett. Math. Phys. 10(2), 111–124 (1985)
Little, D.P.: Combinatorial aspects of the Lascoux–Schützenberger tree. Adv. Math. 174(2), 236–253 (2003)
Macdonald, I.G.: Notes on Schubert Polynomials, Laboratoire de combinatoire et d’informatique mathématique (LACIM). Universite du Québec a Montréal, Montreal (1991)
Manivel, L.: Symmetric Functions, Schubert Polynomials, and Degeneracy Loci. American Mathematical Society, Providence (2001)
Monk, D.: The geometry of flag manifolds. Proc. Lond. Math. Soc. 9, 253–286 (1959)
Rains, E.M., Vazirani, M.J.: Deformations of permutation representations of Coxeter groups. J. Algebraic Comb. 37, 455–502 (2013)
Richardson, R.W., Springer, T.A.: The Bruhat order on symmetric varieties. Geom. Dedicata 35, 389–436 (1990)
Richardson, R.W., Springer, T.A.: Complements to: The Bruhat order on symmetric varieties. Geom. Dedicata 49, 231–238 (1994)
Wyser, B.J., Yong, A.: Polynomials for symmetric orbit closures in the flag variety. Transform. Groups 22, 267–290 (2017)
Wyser, B.J., Yong, A.: Polynomials for \(\text{ GL }_p\times \text{ GL }_q\) orbit closures in the flag variety. Sel. Math. 20(4), 1083–1110 (2014)
Acknowledgements
We thank Dan Bump, Michael Joyce, Vic Reiner, Alex Woo, Ben Wyser, and Alex Yong for many helpful conversations during the development of this paper. We also thank the anonymous referees for their useful comments and suggestions.
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Zachary Hamaker was supported by the IMA with funds provided by the National Science Foundation. Eric Marberg was supported through a fellowship from the National Science Foundation. Brendan Pawlowski was partially supported by NSF Grant 1148634.
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Hamaker, Z., Marberg, E. & Pawlowski, B. Transition formulas for involution Schubert polynomials. Sel. Math. New Ser. 24, 2991–3025 (2018). https://doi.org/10.1007/s00029-018-0423-1
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DOI: https://doi.org/10.1007/s00029-018-0423-1