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Manuscripta Mathematica

, Volume 144, Issue 3–4, pp 311–339 | Cite as

Nilpotency indices, degrees of iterations of affine triangular automorphisms, and Schubert calculus

  • Shu KawaguchiEmail author
Article
  • 101 Downloads

Abstract

Let f be a triangular automorphism of the affine N-space of degree d with Jacobian 1 over a \({\mathbb{Q}}\)-algebra R. We introduce a weighted nilpotency index ν(f) for f, and give a bound of deg(f n ) in terms of N, d and ν(f) for all \({n \in \mathbb{Z}}\). When N = 2, our formula, combined with computation of the Hilbert series of certain graded algebras, yields the estimate deg(f n ) ≤ d 2d + 1 for all \({n \in \mathbb{Z}}\). If n varies through all integers, this estimate turns out to be sharp and is related, somewhat unexpectedly, to the Schubert calculus on the Grassmannian G(d − 1, 2d − 2). Numerical computation for small degrees suggests that this estimate restricted to the inverse degree (i.e. n = −1) is also sharp if d ≥ 3.

Mathematics Subject Classification (2000)

08A35 13B25 14J50 14R10 

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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsKyoto UniversityKyotoJapan

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