Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Highest Weight Varieties and Narayana Numbers

Boming Jia

Abstract

We compute the Hilbert series of the coordinate ring of some highest weight varieties. We also explain why Narayana numbers (and their generalizations) appear naturally in the numerator of the Hilbert series of the homogeneous coordinate ring of the Grassmannian $Gr(d,n+d+1)$ and of the minimal nilpotent adjoint orbit in $\mathfrak{sl}_\mathrm{n+1}(\mathbb{C})$.

Highest Weight Varieties and Narayana Numbers

Abstract

We compute the Hilbert series of the coordinate ring of some highest weight varieties. We also explain why Narayana numbers (and their generalizations) appear naturally in the numerator of the Hilbert series of the homogeneous coordinate ring of the Grassmannian and of the minimal nilpotent adjoint orbit in .
Paper Structure (4 sections, 10 theorems, 51 equations)

This paper contains 4 sections, 10 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Highest Weight Varieties.
  3. The Hilbert Series of $\mathbb{C}_{\mathrm{hom}}[Gr(d,\mathbb{C}^{n+1+d})]$.
  4. The Hilbert Series of $\mathbb{C}[\overline{\mathcal{O}}_\textrm{min}^{A_n}]$.

Key Result

Proposition 2.3

The ideal $I\subset\mathrm{Sym}(V_\lambda^*)$ satisfies

Theorems & Definitions (27)

  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3: Proposition III.1 in Garfinkle
  • proof
  • Definition 2.4
  • Corollary 2.5
  • proof
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3
  • ...and 17 more