Schubert Calculus

Kleiman, S. L.; Laksov, Dan

doi:10.2307/2317421

Cited by 112 publications

(119 citation statements)

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“…These results allow to prove [12,35,36] the following theorem of central importance for the whole development presented in the rest of this paper.…”

Section: From Grassmannians To Schubert Varietiesmentioning

confidence: 57%

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Kontsevich–Witten model from 2+1 gravity: new exact combinatorial solution

Kholodenko¹

2002

Journal of Geometry and Physics

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In previous publications ( J. Geom. Phys.38 (2001) 81-139 and references therein ) the partition function for 2+1 gravity was constructed for the fixed genus Riemann surface.With help of this function the dynamical transition from pseudo-Anosov to periodic (Seifert-fibered) regime was studied. In this paper the periodic regime is studied in some detail in order to recover major results of Kontsevich (Comm.Math.Phys. 147 (1992) 1-23 ) inspired by earlier work of Witten on topological two dimensional quantum gravity.To achieve this goal some results from enumerative combinatorics have been used. The logical developments are extensively illustrated using geometrically convincing figures. This feature is helpful for development of some non traditional applications (mentioned through the entire text) of obtained results to fields other than theoretical particle physics.

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“…These results allow to prove [12,35,36] the following theorem of central importance for the whole development presented in the rest of this paper.…”

Section: From Grassmannians To Schubert Varietiesmentioning

confidence: 57%

“…We shall call these planes as d-planes following [35] . d-planes in P n can be represented by the points in the projective space P N whose dimension N is given by…”

Section: From Chern Classes To Grassmanniansmentioning

confidence: 99%

Kontsevich–Witten model from 2+1 gravity: new exact combinatorial solution

Kholodenko¹

2002

Journal of Geometry and Physics

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“…. , λ m ) of kplanes may be computed using the algorithms in the Schubert calculus (see [14] or [5] or the Introduction to [9] …”

Section: Give a Partition λ And A Flagmentioning

confidence: 99%

Galois groups of Schubert problems via homotopy computation

Leykin

Sottile

2009

Math. Comp.

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Abstract. Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate the problem in pure mathematics of determining Galois groups in the Schubert calculus. For example, we show by direct computation that the Galois group of the Schubert problem of 3-planes in C 8 meeting 15 fixed 5-planes non-trivially is the full symmetric group S 6006 .

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“…are the Plücker coordinates of L. They define an injection G(1, 3) → IP 5 [12]. It is easy to see that this mapping is well defined: if L and K are two representations of the same line, and L = KA for a 2 × 2 matrix A, the Plücker coordinates of L are equal to the Plücker coordinates of K multiplied by the determinant of A.…”

Section: The Image Of a 3d Linementioning

confidence: 99%