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Schubert valuations on Grassmann varieties

Rocco Chirivì, Xin Fang, Peter Littelmann

Abstract

The goal of the paper is twofold: on one side it provides an order structure on the set of all maximal chains in the Bruhat poset of Schubert varieties in a Grassmann variety; on the other hand, using this order structure, it works out explicit formulae for the valuation and the Newton-Okounkov body associated to each maximal chain appearing in the framework of Seshadri stratification.

Schubert valuations on Grassmann varieties

Abstract

The goal of the paper is twofold: on one side it provides an order structure on the set of all maximal chains in the Bruhat poset of Schubert varieties in a Grassmann variety; on the other hand, using this order structure, it works out explicit formulae for the valuation and the Newton-Okounkov body associated to each maximal chain appearing in the framework of Seshadri stratification.
Paper Structure (18 sections, 23 theorems, 86 equations)

This paper contains 18 sections, 23 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. A poset structure on maximal chains
  3. Basic facts
  4. Maximal chains
  5. Root operators on standard rows
  6. Maximal chains and the symmetric group on r letters
  7. Root operators on maximal chains
  8. The image of the chains in the symmetric group
  9. Newton-Okounkov bodies
  10. Geometric valuation
  11. Change of sequence of parameters
  12. Schubert valuations on Grassmann varieties
  13. Valuations from birational sequences
  14. Birational sequences and valuations
  15. Explicit formulae for $\mathfrak{C}_{\mathrm{right}}$
  16. ...and 3 more sections

Key Result

Proposition 2.1

We have In particular $I(d, n-1)$ is the subset of $I(d,n)$ consisting of rows not containing $n$, and the set $I(d-1,n-1) * n$ is the subset of $I(d,n)$ consisting of those standard rows containing $n$.

Theorems & Definitions (66)

  • Proposition 2.1
  • proof
  • Remark 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Proposition 2.8
  • ...and 56 more