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Highest weight modules with respect to non-standard Gelfand-Tsetlin subalgebras

Juan Camilo Arias, Oscar Morales, Luis Enrique Ramirez

Abstract

In this paper we study realizations of highest weight modules for the complex Lie algebra $\mathfrak{gl}_n$ with respect to non-standard Gelfand-Tsetlin subalgebras. We also provide sufficient conditions for such subalgebras to have a diagonalizable action on these realizations.

Highest weight modules with respect to non-standard Gelfand-Tsetlin subalgebras

Abstract

In this paper we study realizations of highest weight modules for the complex Lie algebra with respect to non-standard Gelfand-Tsetlin subalgebras. We also provide sufficient conditions for such subalgebras to have a diagonalizable action on these realizations.

Paper Structure

This paper contains 16 sections, 29 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Root datum
  4. Partitions and quase-partitions of Delta
  5. Relation Gelfand-Tsetlin modules
  6. Relation modules
  7. Action on relation graphs
  8. An $S_n$-action on relation graphs
  9. Tableaux associated with graphs in the $S_n$-orbit of $G_{\mathfrak{h}}$
  10. Highest weight modules with respect to different Borel subalgebras
  11. Main results and applications
  12. sigma-relation modules and weights
  13. Applications
  14. Localization of sigma-relation modules
  15. Proof of Lemma \ref{['Lemma: sk and tableaux']}
  16. ...and 1 more sections

Key Result

Theorem 3.8

An ordered, non-critical, cross-less graph $G$ is a relation graph if and only if every connected component of $G$ satisfies the $\Diamond$-condition.

Theorems & Definitions (76)

  • Definition 3.1
  • Definition 3.2
  • Example 3.3
  • Definition 3.4
  • Definition 3.5
  • Remark 3.6
  • Definition 3.7
  • Theorem 3.8
  • proof
  • Definition 3.9
  • ...and 66 more