Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

GLS homogenization tilde map

Fayadh Kadhem

Abstract

In the construction of a cluster algebra on the homogeneous coordinate ring of a partial flag variety by Geiß, Leclerc and Schr{ö}er, they defined a special map denoted by ``tilde". This map lifts each element $f$ of the coordinate ring of a Schubert cell uniquely to an element $\widetilde{f}$ of the (multi-homogeneous) coordinate ring of the corresponding partial flag variety. The significance of this map appears from its essential role; it lifts the cluster algebra of the coordinate ring of a cell to a cluster algebra living in the coordinate ring of the corresponding partial flag variety. This paper takes a closer look at this map and gives an explicit algorithm to calculate it for the \textit{generalized minors}.

GLS homogenization tilde map

Abstract

In the construction of a cluster algebra on the homogeneous coordinate ring of a partial flag variety by Geiß, Leclerc and Schr{ö}er, they defined a special map denoted by ``tilde". This map lifts each element of the coordinate ring of a Schubert cell uniquely to an element of the (multi-homogeneous) coordinate ring of the corresponding partial flag variety. The significance of this map appears from its essential role; it lifts the cluster algebra of the coordinate ring of a cell to a cluster algebra living in the coordinate ring of the corresponding partial flag variety. This paper takes a closer look at this map and gives an explicit algorithm to calculate it for the \textit{generalized minors}.
Paper Structure (7 sections, 9 theorems, 81 equations)

This paper contains 7 sections, 9 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Cluster algebras
  3. Schubert cells and partial flag varieties
  4. Lift degree and explicit algorithm
  5. An application to cluster algebras
  6. Examples
  7. General theorems

Key Result

Lemma 3.14

For any element $f$ of the coordinate ring $\mathbb C[N_K]$ there is a unique homogeneous element $\widetilde{f}$ in $\mathbb C [G/P_{K}^{-}]$ whose projection to $\mathbb C[N_K]$ is $f$ and whose multi-degree is minimal with respect to the partial ordering $\preceq$ of weights.

Theorems & Definitions (48)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 3.1
  • Example 3.2
  • Definition 3.3
  • Example 3.4
  • ...and 38 more