A Generalization of the Gysin Homomorphism for Flag Bundles

Ilori, Samuel A.

doi:10.2307/2373843

Cited by 3 publications

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“…, x n the Chern roots of a complex vector bundle E. This is the fundamental formula given in [18, §0, p.2]. 35 From the above interpretation, we can obtain the generating function for S L k (x n ) (k ∈ Z). We argue as follows: Set…”

Section: 3mentioning

confidence: 87%

“…Various Gysin formulas. As mentioned in the introduction, various types of Gysin formulas related to the Gysin maps are known (see e.g., Akyildiz [2], Akyildiz-Carrell [4], Buch [14], Damon [16], [17], Darondeau-Pragacz [18], Fel'dman [22], Fulton [23], Fulton-Pragacz [25], Harris-Tu [27], Ilori [35], Jozefiak-Lascoux-Pragacz [37], Kajimoto [39], Quillen [62], Pragacz [58], [60], [61], Sugawara [65], Tu [68], [69]). In this subsection, we shall take up typical examples of these formulas.…”

Section: Brumfiel-madsen Formulamentioning

confidence: 99%

“…-Gysin formulas for flag bundles, Grassmann bundles, or projective bundles are described in Borel-Hirzebruch [7, §8], [8, §20], Buch [14, §7], Damon [16], [17], Darondeau-Pragacz [18], Fel'dman [22, §4], Fulton [23, §14.2], Fulton-Pragacz [25, Chapter IV, Appendices E, F], Harris-Tu [27, §2], Ilori [35], Kajimoto-Sugawara [40], Pragacz [58, §2], [60, §4], [61], Quillen [62], Sugawara [65], Tu [68], [69], Vishik [70, §5.7]. -For a connected complex (semi-simple) Lie group G C with a Borel subgroup B and a parabolic subgroup P containing B, Gysin formulas for the natural projection G C /B −→ G C /P are described in Akyildiz [2], Akyildiz-Carrell [3], [4], Fulton-Pragacz [25, Appendix E], Brion [12], Kajimoto [39].…”

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Universal Gysin formulas for the universal Hall-Littlewood functions

Nakagawa,

Naruse

2016

Preprint

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It is known that the usual Schur S-and P -polynomials can be described via the Gysin homomorphisms for flag bundles in the ordinary cohomology theory. Recently, P. Pragacz generalized these Gysin formulas to the Hall-Littlewood polynomials. In this paper, we introduce a universal analogue of the Hall-Littlewood polynomials, which we call the universal Hall-Littlewood functions, and give Gysin formulas for various flag bundles in the complex cobordism theory. Furthermore, we give two kinds of the universal analogue of the schur polynomials, and some Gysin formulas for these functions are established.

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Section: 3mentioning

confidence: 87%

Section: Brumfiel-madsen Formulamentioning

confidence: 99%

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confidence: 99%

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Universal Gysin formulas for the universal Hall-Littlewood functions

Nakagawa,

Naruse

2016

Preprint

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“…The reformulation (2) and the idea to iterate the obtained formula on chains (or "towers") of projective bundles originally appeared in the paper [5] by the first author. The idea of generalizing this formula to flag bundles was signaled by Bérczi, and it became clear that this suggestion was relevant as we recovered a formula for type A of Ilori from [12]. After the first version of the paper was completed, Manivel informed us that, independently, in their recent paper [14], Kaji and Terasoma prove a formula for type A (in the particular case of full flag bundles).…”

Section: Introductionmentioning

confidence: 98%

Universal Gysin formulas for flag bundles

Darondeau

Pragacz

2017

Int. J. Math.

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Abstract. -We give push-forward formulas for all flag bundles of types A, B, C, D. The formulas (and also the proofs) involve only Segre classes of the original vector bundles and characteristic classes of universal bundles. As an application, we provide new determinantal formulas. IntroductionA proper morphism F : Y → X of non-singular algebraic varieties over an algebraically closed field yields an additive mapX of Chow groups induced by push-forward cycles, called the Gysin map (see Fulton's book [7]; note that the theory developed in this book allows one to generalize the results of the present paper to singular varieties over a field and their Chow groups; moreover, for complex varieties, one can also use the cohomology rings with integral coefficients). We will alternatively denote F * by X Y . Push-forward formulas show how the classes of algebraic cycles on Y go via the Gysin map to classes of algebraic cycles on X. In the present paper, we are interested in push-forwards in flag bundles. We shall give formulas for the classical types A, B, C, D. These have a universal character in three aspects: -these involve characteristic classes of universal vector bundles; -these universally hold for any polynomial in such classes; -these use in a universal way only the Segre classes of the original vector bundles.The starting point of our argument is a reformulation of the classical formula for push-forward of powers of the hyperplane class in a projective bundle, that we recall. Let E → X be a vector bundle of rank n on a variety X. Let P(E) → X be the projective bundle of lines in E and let ξ ≔ c 1 (O P(E) (1)) be the hyperplane class. For any i, the ith Segre class of E isThis is a definition in [7] and a lemma in preceding intersection theory (see e.g. [13, Lemma 1]). Then, consider a polynomial f (ξ) = i α i ξ i with coefficients α i in the Chow ring of X (here, we identify A• X with a subring of A • P(E) and throughout the text, we will often omit pullback notation for vector bundles and algebraic cycles). Using the projection formula

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Universal Gysin formulas for the universal Hall-Littlewood functions

Nakagawa

Naruse

2018

An Alpine Bouquet of Algebraic Topology

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A Generalization of the Gysin Homomorphism for Flag Bundles

Cited by 3 publications

References 1 publication

Universal Gysin formulas for the universal Hall-Littlewood functions

Universal Gysin formulas for the universal Hall-Littlewood functions

Universal Gysin formulas for flag bundles

Universal Gysin formulas for the universal Hall-Littlewood functions

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