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On Mirković-Vilonen polytopes

Pierre Baumann

Abstract

Mirković-Vilonen polytopes encode in a geometrical way the numerical data present in the Kashiwara crystal $B(\infty)$ of a semisimple group $G$. We retrieve these polytopes from the coproduct of the Hopf algebra $\mathscr O(N)$ of regular functions on a maximal unipotent subgroup $N$ of $G$. We bring attention to a remarkable behavior that the classical bases (dual canonical, dual semicanonical, Mirković-Vilonen) of $\mathscr O(N)$ manifest with respect to the extremal points of these polytopes, which extends the crystal operations. This study leans on a notion of stability for graded bialgebras.

On Mirković-Vilonen polytopes

Abstract

Mirković-Vilonen polytopes encode in a geometrical way the numerical data present in the Kashiwara crystal of a semisimple group . We retrieve these polytopes from the coproduct of the Hopf algebra of regular functions on a maximal unipotent subgroup of . We bring attention to a remarkable behavior that the classical bases (dual canonical, dual semicanonical, Mirković-Vilonen) of manifest with respect to the extremal points of these polytopes, which extends the crystal operations. This study leans on a notion of stability for graded bialgebras.
Paper Structure (16 sections, 28 theorems, 20 equations)

This paper contains 16 sections, 28 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Stability for graded connected bialgebras
  3. Semistable elements
  4. Ordered Monomials
  5. The splitting isomorphisms
  6. Duality
  7. Polytopes and bases of $\mathscr O(N)$
  8. Notation
  9. Stability in $U(\mathfrak n)$ and $\mathscr O(N)$
  10. Polite bases
  11. The dual semicanonical basis
  12. Comparison with perfect bases
  13. Mirković and Vilonen's basis
  14. Recollection about the Geometric Satake Equivalence
  15. Cutting Mirković--Vilonen cycles
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $a\in A$ be a homogeneous element and write $\Delta(a)=\sum_{i=1}^nb_i\otimes c_i$ as above. Then $L(b_i)\subset L(a)$ and $R(c_i)\subset R(a)$ for all $i\in\llbracket1,n\rrbracket$.

Theorems & Definitions (28)

  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Proposition 2.9
  • Proposition 2.10
  • Proposition 2.11
  • ...and 18 more