Double Schubert polynomials for the classical groups

Ikeda, Takeshi; Mihalcea, Leonardo C.; Naruse, Hiroshi

doi:10.1016/j.aim.2010.07.008

Cited by 54 publications

(145 citation statements)

References 34 publications

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“…Moreover, the corresponding left divided differences δ i on H n := H * Tn (IG(n − k, 2n)) from [IMN1,§2.5] are compatible with the geometrization map π n : C (k) [t] → H n . According to [IMN1,Prop. 2.3] (see also [T3,Eqn.…”

Section: Divided Difference Operators On Z[c T]mentioning

confidence: 88%

“…This last result is a special case of the main theorem of [BKT2]. Finally, a third proof of (33) is provided by Ikeda and Matsumura in [IM,§8.2], starting from the Pfaffian formula [IMN1,Thm. 1.2] for the equivariant Schubert class of a point on the complete symplectic flag variety Sp 2n /B.…”

Section: Divided Difference Operators On Z[c T]mentioning

confidence: 99%

“…We emphasize that the double theta polynomials Θ λ (c | t) defined here are new, and are not a special case of the type C double Schubert polynomials of [IMN1], which represent the equivariant Schubert classes on complete symplectic flag varieties. Recently, Hudson, Ikeda, Matsumura, and Naruse [HIMN] proved an analogue of our cohomological formula in connective K-theory, and Anderson and Fulton [AF] obtained related Chern class formulas for more general degeneracy loci.…”

Section: Introductionmentioning

confidence: 99%

“…The surjections (2) are induced from the natural inclusions W n ֒→ W n+1 of the Weyl groups of type C, as in [BH,§2], [IMN1,§10], and §3 of the present work. Moreover, the variables t i are identified with the characters of the maximal tori T n in a compatible fashion, as in loc.…”

Section: Introductionmentioning

confidence: 99%

“…1 In her dissertation, Wilson gave a direct proof that the Θ λ (c | t) satisfy the equivariant Chevalley rule in H T (IG k ), i.e., the combinatorial formula for the expansion of the products σ 1 · σ λ in the basis of stable equivariant Schubert classes. This fits into a program for proving Theorem 1, and its extension to equivariant 1 Wilson actually worked with the principal specialization Θ λ (x, z | t) of Θ λ (c | t) in the ring of type C double Schubert polynomials of [IMN1]. quantum cohomology, by applying Mihalcea's characterization theorem [Mi1,Cor.…”

Section: Introductionmentioning

confidence: 99%

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Double theta polynomials and equivariant Giambelli formulas

Tamvakis

Wilson

2015

Math. Proc. Camb. Phil. Soc.

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Abstract. We use Young's raising operators to introduce and study double theta polynomials, which specialize to both the theta polynomials of Buch, Kresch, and Tamvakis, and to double (or factorial) Schur S-polynomials and Q-polynomials. These double theta polynomials give Giambelli formulas which represent the equivariant Schubert classes in the torus-equivariant cohomology ring of symplectic Grassmannians, and we employ them to obtain a new presentation of this ring in terms of intrinsic generators and relations.

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Section: Divided Difference Operators On Z[c T]mentioning

confidence: 88%

Section: Divided Difference Operators On Z[c T]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Double theta polynomials and equivariant Giambelli formulas

Tamvakis

Wilson

2015

Math. Proc. Camb. Phil. Soc.

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k-Schur Functions and Affine Schubert Calculus

Lam

Lapointe

Morse

et al. 2014

Fields Institute Monographs

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Affine Schubert Calculus

Lam

Lapointe

Morse

et al. 2014

K-Schur Functions and Affine Schubert Calculus

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The cohomology of the affine flag varietyFl G of a complex reductive group G is a comodule over the cohomology of the affine Grassmannian Gr G . We give positive formulae for the coproduct of an affine Schubert class in terms of affine Stanley classes and finite Schubert classes, in (torus-equivariant) cohomology and K-theory. As an application, we deduce monomial positivity for the affine Schubert polynomials of the second author. THOMAS LAM, SEUNGJIN LEE, AND MARK SHIMOZONOThe class ξ v G/B is considered an element of H * T (Fl G ) via pullback under evaluation at the identity (see (3.5)). The same formulae hold in non-equivariant cohomology.In the majority of this article (Sections 2-4) we will work in torus-equivariant K-theory K * T (Fl) of the affine flag variety. The coproduct formula for holds in (torus-equivariant) K-theory with Demazure product replacing length-additive products (see Theorem 4.7). Our proof relies heavily on the action of the affine nilHecke ring on K * T (Fl). Let us note that there are a number of different geometric approaches [KK, KS, LSSa] for constructing Schubert classes in K * T (Fl), see [LLMS, Section 3] for a comparison. However, our results holds at the level of Grothendieck groups and the precise geometric model (thick affine flag variety, thin affine flag variety, or based loop group) is not crucial.In Section 5, the proofs for cohomology are indicated. In Section 6, we give examples of our formula in classical type. In particular, we prove (Theorem 6.1) that the affine Schubert polynomials [Lee] are monomial positive, and we explain how the Billey-Haiman formula [BH] for type C or D Schubert polynomials (see also [IMN]) is a consequence of our coproduct formula.By taking an appropriate limit, the coproduct formula for backstable (double) Schubert polynomials [LLS] can be deduced from Theorem 1.1. Whereas the proofs in [LLS] are essentially combinatorial, the present work relies heavily on equivariant localization and the nilHecke algebra. Affine nilHecke ring and the equivariant K-theory of the affine flag varietyThe proofs of our results for a complex reductive group easily reduces to that of a semisimple simply-connected group. To stay close to our main references [KK, LSSa], we work with the latter. Henceforth, we fix a complex semsimple simply-connected group G.The results of this section are due to Kostant and Kumar [KK]. Our notation agrees with that of [LSSa].2.1. Small-torus affine K-nilHecke ring. Let T ⊂ G be the maximal torus with character group, or weight lattice P . We have P = i∈I Zω i where ω i denotes a fundamental weight and I denotes the finite Dynkin diagram of G. LetP = Zδ ⊕ i∈Î ZΛ i be the affine weight lattice with fundamental weights Λ i for i in the affine Dynkin node set I = I ∪ {0}, and let δ denote the null root. The natural projection cl :P → P has kernel Zδ ⊕ ZΛ 0 and satisfies cl(Λ i ) = ω i for i ∈ I. Let af : P →P be the section of cl given by afThe finite Weyl group W acts naturally on P and on R(T ), where R(T ) ∼ = Z[P ] = λ∈P Ze λ is t...

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Double Schubert polynomials for the classical groups

Cited by 54 publications

References 34 publications

Double theta polynomials and equivariant Giambelli formulas

Double theta polynomials and equivariant Giambelli formulas

k-Schur Functions and Affine Schubert Calculus

Affine Schubert Calculus

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