k-Schur Functions and Affine Schubert Calculus

Lam, Thomas; Lapointe, Luc; Morse, Jennifer; Schilling, Anne; Shimozono, Mark; Zabrocki, Mike

doi:10.1007/978-1-4939-0682-6

Cited by 28 publications

(48 citation statements)

References 123 publications

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“…There are many conjecturally equivalent definitions of k-Schur functions in [15]. We follow the definition of k-Schur functions in [14] using strong strips.…”

Section: K-schur and Strong Schur Functionsmentioning

confidence: 99%

“…There is a coproduct structure on the nilHecke ring A. Kostant and Kumar showed that the structure constants of the coproduct in A are the same as the structure constants of the equivariant cohomology ring of F l. Details and proofs can be found in [8,15,19]. Let ∆ :…”

Section: 3mentioning

confidence: 99%

“…There are two definitions of Kac-Moody flag variety F l in [15], but we only recall F l as the Kac-Moody flag ind-variety in [8,10]. Let G af denote the Kac-Moody group of affine type associated with SL(n) and let B af denote its Borel subgroup.…”

Section: Affine Flag Varietiesmentioning

confidence: 99%

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Pieri rule for the affine flag variety

Lee

2017

Advances in Mathematics

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Abstract. We prove the affine Pieri rule for the cohomology of the affine flag variety conjectured by Lam, Lapointe, Morse and Shimozono. We study the cap operator on the affine nilHecke ring that is motivated by Kostant and Kumar's work on the equivariant cohomology of the affine flag variety. We show that the cap operators for Pieri elements are the same as Pieri operators defined by Berg, Saliola and Serrano. This establishes the affine Pieri rule.

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“…There are many conjecturally equivalent definitions of k-Schur functions in [15]. We follow the definition of k-Schur functions in [14] using strong strips.…”

Section: K-schur and Strong Schur Functionsmentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

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Pieri rule for the affine flag variety

Lee

2017

Advances in Mathematics

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“…There is a left action ψ → a · ψ ofK on K * T (Fl) given by the formulae (see [LLMSSZ,Chapter 4,Proposition 3.16] for the very similar cohomology case)…”

Section: Thinking Of Kmentioning

confidence: 99%

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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“…Let us say that we want to compute the coefficient of s (8) 65444433222 in s (8) 5444433222 s (8) 51 . Note that 65444433222 = 5444433222 {1,2,3,4,5,6} , so we are interested in LR arrays of shape 111111 and weight 51.…”

mentioning

confidence: 99%

The role of residue and quotient tables in the theory of k-Schur functions

Konvalinka

2015

Journal of Combinatorial Theory, Series A

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Recently, residue and quotient tables were defined by Fishel and the author, and were used to describe strong covers in the lattice of k-bounded partitions. In this paper, we prove (and, in some cases, conjecture) that residue and quotient tables can be used to describe many other results in the theory of k-bounded partitions and k-Schur functions, including k-conjugates, weak horizontal and vertical strips, and the Murnaghan-Nakayama rule. Evidence is presented for the claim that one of the most important open questions in the theory of k-Schur functions, a general rule that would describe their product, can be also concisely stated in terms of residue tables.

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k-Schur Functions and Affine Schubert Calculus

Cited by 28 publications

References 123 publications

Pieri rule for the affine flag variety

Pieri rule for the affine flag variety

Affine Schubert Calculus

The role of residue and quotient tables in the theory of k-Schur functions

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