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Characteristic polynomials for classical Lie algebras

Chenyue Feng, Shoumin Liu, Xumin Wang

Abstract

In this paper, we will compute the characteristic polynomials for finite dimensional representations of classical complex Lie algebras and the exceptional Lie algebra of type G2, which can be obtained through the orbits of integral weights under the action of their corresponding Weyl groups and the invariant polynomial theory of the Weyl groups. We show that the characteristic polynomials can be decomposed into products of irreducible orbital factors, each of which is invariant under the action of their corresponding Weyl groups.

Characteristic polynomials for classical Lie algebras

Abstract

In this paper, we will compute the characteristic polynomials for finite dimensional representations of classical complex Lie algebras and the exceptional Lie algebra of type G2, which can be obtained through the orbits of integral weights under the action of their corresponding Weyl groups and the invariant polynomial theory of the Weyl groups. We show that the characteristic polynomials can be decomposed into products of irreducible orbital factors, each of which is invariant under the action of their corresponding Weyl groups.

Paper Structure

This paper contains 9 sections, 19 theorems, 96 equations, 1 table.

Table of Contents

  1. Introduction
  2. Some preliminary results
  3. Characteristic polynomials for $\mathfrak{sl}(n, \mathbb C)$ and $\mathfrak{sl}(4, \mathbb C)$
  4. Cases for the Lie algebra $\mathfrak{sl}(3, \mathbb C)$
  5. Characteristic polynomials for $\mathfrak{sl}(n, \mathbb C)$
  6. Some results for $\mathfrak{sl}(4, \mathbb C)$
  7. Characteristic polynomials for ${\rm B}_{n}$, ${\rm C}_{n}$, ${\rm D}_{n}(n\geq 4)$ and ${\rm G}_2$
  8. Characteristic polynomials for ${\rm B}_{n}$, ${\rm C}_{n}$, ${\rm D}_{n}(n\geq 4)$
  9. Characteristic polynomials for ${\rm G}_{2}$

Key Result

Theorem 1.1

If $G = \{1, g_1, \cdots, g_n\}$ is a finite group, then Moreover, each $f_{\pi}(z)$ is an irreducible polynomial of degree $d_{\pi}$.

Theorems & Definitions (41)

  • Theorem 1.1
  • Conjecture 1.1
  • Definition 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 31 more