Double eta polynomials and equivariant Giambelli formulas

Tamvakis, Harry

doi:10.1112/jlms/jdw032

Cited by 9 publications

(20 citation statements)

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“…This technique was profoundly refined by Buch, Kresch, and Tamvakis, who introduced and studied theta-and eta-polynomials as representatives for Schubert classes in the cohomology of isotropic Grassmannians [BKT1,BKT2]. "Double" versions of these polynomials were defined and shown to represent equivariant Schubert classes and degeneracy loci in [Wi,IM,TW,AF2,T1].…”

Section: Introductionmentioning

confidence: 99%

K-theoretic Chern class formulas for vexillary degeneracy loci

Anderson

2019

Advances in Mathematics

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Using raising operators and geometric arguments, we establish formulas for the K-theory classes of degeneracy loci in classical types. We also find new determinantal and Pfaffian expressions for classical cases considered by Giambelli: the loci where a generic matrix drops rank, and where a generic symmetric or skew-symmetric matrix drops rank.In an appendix, we construct a K-theoretic Euler class for evenrank vector bundles with quadratic form, refining the Chow-theoretic class introduced by Edidin and Graham. We also establish a relation between top Chern classes of maximal isotropic subbundles, which is used in proving the type D degeneracy locus formulas.Here p and q are the ranks of E and F , respectively, and c(F − E) is the K-theoretic Chern class. The leading term, where m = 0, is precisely Giambelli's determinantal formula in cohomology. 1 We will prove similar formulas for vexillary degeneracy loci, incorporating the stability properties of the cohomology formulas. 2 Our results generalize the formulas of [HIMN] for types A, B, and C. In general, the loci are described by conditions of the form dim(E p i ∩ F q i ) ≥ k i , and the results of [HIMN] are recovered as the special case where k i = i and p i = p for all i.

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Section: Introductionmentioning

confidence: 99%

K-theoretic Chern class formulas for vexillary degeneracy loci

Anderson

2019

Advances in Mathematics

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“…in Kazarian's paper [Ka]. The proof continues by using the arguments found in [TW,T7] to arrive at double theta and double eta polynomials, and then specializing to obtain their single versions in [BKT2,BKT3]. Finally, we establish more general versions of the Schubert splitting formulas of [T5] with the help of the double mixed Stanley functions and k-transition trees introduced in op.…”

Section: Introductionmentioning

confidence: 93%

“…Using raising operators, it is shown in [T7,Prop. 5] that if λ and µ are typed k-strict partitions such that |λ| = |µ| + 1 and w λ = s i w µ for some simple reflection s i ∈ W ∞ , then we have…”

Section: Iterating This Calculation Givesmentioning

confidence: 99%

“…In a recent preprint, Anderson and Fulton [AF2] use Young's raising operators and algebro-geometric arguments to define multi-theta and multi-eta polynomials, which extend the double theta and eta polynomials of [TW,T7,W] even further. The resulting degeneracy locus formulas and their proofs are an important contribution to the theory of theta and eta polynomials, but hold only for certain special elements w of the Weyl group of G. We leave the task of including them within the present algebraic and combinatorial framework to future research.…”

Section: Introductionmentioning

confidence: 99%

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Schubert polynomials and degeneracy locus formulas

Tamvakis

2018

EMS Series of Congress Reports

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In previous work [T6], we employed the approach to Schubert polynomials by Fomin, Stanley, and Kirillov to obtain simple, uniform proofs that the double Schubert polynomials of Lascoux and Schützenberger and Ikeda, Mihalcea, and Naruse represent degeneracy loci for the classical groups in the sense of Fulton. Using this as our starting point, and purely combinatorial methods, we obtain a new proof of the general formulas of [T5], which represent the degeneracy loci coming from any isotropic partial flag variety. Along the way, we also find several new formulas and elucidate the connections between some earlier ones. BKTY]. This answer was extended in a type uniform way to the symplectic and orthogonal Lie groups in [T5]. The formulas of [BKTY, T5] rely in part on a theory of Schubert polynomials, which express the classes of the degeneracy loci in terms of the Chern roots of the vector bundles E • and F • . The desired combinatorial theory of Schubert polynomials, together with its connection to geometry, was established in the papers [LS, L, Fu1] (for Lie type A) and [BH, T2, T3, IMN1, T5] (for Lie types B, C, and D).In previous work [T6, §7.3], we employed Fomin, Stanley, and Kirillov's nil-Coxeter algebra approach to Schubert polynomials [FS, FK] to give simple, uniform proofs that the double Schubert polynomials of Lascoux and Schützenberger [LS, L] and Ikeda, Mihalcea, and Naruse [IMN1] represent degeneracy loci of vector bundles, in the above sense. Our main goal in this paper is to begin with the same definition of Schubert polynomials from [T5, T6] and, by purely combinatorial methods, derive the splitting formulas for these polynomials found in [T5, §3 and §6]. The latter results then imply the general degeneracy locus formulas of [T5]. The proof of the corresponding type A splitting formula from [BKTY] is essentially combinatorial; this is clarified in [T5, §1.4] and §1 of the present paper. As in [T5], our arguments depend on two key results from [BKT2, BKT3], which state that the single Schubert polynomials indexed by Grassmannian elements of the Weyl groups are represented by (single) theta and eta polynomials. The original proofs of these theorems used the classical Pieri rules from [BKT1], which were derived geometrically in op. cit. by intersecting Schubert cells. More recent proofs by Ikeda and Matsumura [IM], the author and Wilson [TW, T7], and Anderson and Fulton [AF2] use localization in equivariant cohomology (following [Ar, KK]) or employ other geometric arguments stemming from Kazarian's work [Ka]. However, the statements of the aforementioned theorems from [BKT2, BKT3] are entirely combinatorial, and it is natural to seek proofs of them within the same framework. The corresponding result in type A is the elementary fact that the symmetric Schubert polynomials are equal to Schur polynomials. The approach we take here begins by extending Billey and Haiman's formula [BH, Prop. 4.15] for the single Schubert polynomials indexed by the longest element in the Weyl group of G to the double Schuber...

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“…The difficulty when working with sequences of divided differences applied to polynomials lies in choosing which path to follow in the weak Bruhat order, as the Leibnitz rule tends to destroy any nice formulas. The papers [TW,T4,T5] showed how divided differences can be used to obtain combinatorial proofs of the raising operator formulas for double theta and double eta polynomials, exploiting the fact that these polynomials behave well under the action of left divided differences. Therefore, as long as one remains among the Grassmannians elements, the choice of path through the left weak Bruhat order is immaterial.…”

Section: Introductionmentioning

confidence: 99%

Degeneracy locus formulas for amenable Weyl group elements

Tamvakis¹

2019

Preprint

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We define a class of amenable Weyl group elements in the Lie types B, C, and D, which we propose as the analogues of vexillary permutations in these Lie types. Our amenable signed permutations index flagged theta and eta polynomials, which generalize the double theta and eta polynomials of Wilson and the author. In geometry, we obtain corresponding formulas for the cohomology classes of symplectic and orthogonal degeneracy loci.

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Double eta polynomials and equivariant Giambelli formulas

Cited by 9 publications

References 15 publications

K-theoretic Chern class formulas for vexillary degeneracy loci

K-theoretic Chern class formulas for vexillary degeneracy loci

Schubert polynomials and degeneracy locus formulas

Degeneracy locus formulas for amenable Weyl group elements

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