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Gelfand-Kirillov dimensions and annihilator varieties of highest weight modules of exceptional Lie algebras

Zhanqiang Bai, Fan Gao, Yutong Wang, Xun Xie

TL;DR

This work delivers an efficient, uniform algorithm for computing the Gelfand–Kirillov dimension of simple highest weight modules over exceptional Lie algebras, by reducing to Lusztig’s a-function on a carefully chosen Weyl-subgroup element wλ and exploiting a structural decomposition Φ_{[λ]}≅Φ_{[λ]_1}×…×Φ_{[λ]_k}. Simultaneously, it determines the annihilator variety via Sommers duality, leveraging the Springer correspondence and PyCox for exceptional factors; this yields a direct path from a weight λ to both GKdim L(λ) and V(Ann L(λ)). The paper provides explicit GK-dim classifications and annihilator data for types G2, F4, and the E-series (E6–E8), including integral and nonintegral cases, with extensive tables in the appendix. The approach connects deep representation-theoretic invariants to nilpotent-orbit theory, enabling a practical online tool for computing these invariants and highlighting the role of pseudo-maximal root subsystems in the reduction process. Overall, the methods unify combinatorial Weyl-group data, dualities, and algorithmic computation to advance understanding of highest weight modules in exceptional types and their geometric annihilator structures.

Abstract

In this paper, we give an efficient algorithm to compute the Gelfand-Kirillov dimensions of simple highest weight modules of exceptional Lie algebras. By using the Sommers-Achar duality, we also determine the annihilator varieties of these highest weight modules.

Gelfand-Kirillov dimensions and annihilator varieties of highest weight modules of exceptional Lie algebras

TL;DR

This work delivers an efficient, uniform algorithm for computing the Gelfand–Kirillov dimension of simple highest weight modules over exceptional Lie algebras, by reducing to Lusztig’s a-function on a carefully chosen Weyl-subgroup element wλ and exploiting a structural decomposition Φ_{[λ]}≅Φ_{[λ]_1}×…×Φ_{[λ]_k}. Simultaneously, it determines the annihilator variety via Sommers duality, leveraging the Springer correspondence and PyCox for exceptional factors; this yields a direct path from a weight λ to both GKdim L(λ) and V(Ann L(λ)). The paper provides explicit GK-dim classifications and annihilator data for types G2, F4, and the E-series (E6–E8), including integral and nonintegral cases, with extensive tables in the appendix. The approach connects deep representation-theoretic invariants to nilpotent-orbit theory, enabling a practical online tool for computing these invariants and highlighting the role of pseudo-maximal root subsystems in the reduction process. Overall, the methods unify combinatorial Weyl-group data, dualities, and algorithmic computation to advance understanding of highest weight modules in exceptional types and their geometric annihilator structures.

Abstract

In this paper, we give an efficient algorithm to compute the Gelfand-Kirillov dimensions of simple highest weight modules of exceptional Lie algebras. By using the Sommers-Achar duality, we also determine the annihilator varieties of these highest weight modules.

Paper Structure

This paper contains 26 sections, 33 theorems, 132 equations, 5 figures, 36 tables.

Table of Contents

  1. Introduction
  2. Main result
  3. Preliminaries
  4. Hecke algebra and -functions
  5. GK dimensions and associated varieties
  6. Annihilator variety
  7. PyCox
  8. Pseudo-maximal root subsystem
  9. The computation of for exceptional types
  10. Integral highest weight modules
  11. Nonintegral highest weight modules
  12. Longest element of a Weyl group
  13. GK dimensions of highest weight modules of type
  14. GK dimensions of highest weight modules of type
  15. GK dimensions of highest weight modules of type
  16. ...and 11 more sections

Key Result

Theorem 1.1

For a given nonintegral weight $\lambda$, suppose that $\lambda=w_{\lambda}\mu$, where $\mu$ is antidominant and $w_{\lambda}$ is the unique minimal length element in the integral Weyl subgroup $W_{[\lambda]}$. Suppose $\Phi_{[\lambda]}\simeq \Phi_1\times \Phi_2\times \cdots\times \Phi_k$ is a direc In particular, when ${\Phi_i}$ is of classical type, we can determine the value of $\mathbf{a} (w_{

Figures (5)

  • Figure 1: Extended Dynkin diagram of $G_2$
  • Figure 2: Extended Dynkin diagram of $F_4$
  • Figure 3: Extended Dynkin diagram of $E_6$
  • Figure 4: Extended Dynkin diagram of $E_7$
  • Figure 5: Extended Dynkin Diagram of $E_8$

Theorems & Definitions (74)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7: Hum08
  • Proposition 2.8: BX
  • ...and 64 more