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Minimal Polynomials of Generalized Heisenberg Groups

Tillmann Jentsch

TL;DR

The paper analyzes minimal polynomials for homogeneous C0-spaces, focusing on H-type (generalized Heisenberg) groups with canonical left-invariant metrics. It develops a blueprint algorithm to compute the minimal polynomial from the symmetrized curvature data and derives a pointwise factorization of P_{ ext{min}} in terms of eigenvalue branches of -K^2, including global branches at 0 and 1. It then provides a detailed classification of eigenvalue branches for H-type groups, giving explicit formulas for low z-dimensions and a general multiplicity pattern k = 3n+1 (n even) or k = 3n (n odd) for dim z q 10, along with a Singer-invariant bound and Killing-tensor structure of the coefficients. The results yield concrete, invariant polynomials that capture the minimal-polynomial data and illuminate geometric properties (geodesics, curvature) of these nontrivial nilpotent groups, with implications for their C0-structure and automorphism groups.

Abstract

Every homogeneous Riemannian C_0-space (N,g) is associated with its minimal polynomial. To provide explicit examples, we compute the minimal polynomials for generalized Heisenberg groups equipped with their canonical left-invariant metrics.

Minimal Polynomials of Generalized Heisenberg Groups

TL;DR

The paper analyzes minimal polynomials for homogeneous C0-spaces, focusing on H-type (generalized Heisenberg) groups with canonical left-invariant metrics. It develops a blueprint algorithm to compute the minimal polynomial from the symmetrized curvature data and derives a pointwise factorization of P_{ ext{min}} in terms of eigenvalue branches of -K^2, including global branches at 0 and 1. It then provides a detailed classification of eigenvalue branches for H-type groups, giving explicit formulas for low z-dimensions and a general multiplicity pattern k = 3n+1 (n even) or k = 3n (n odd) for dim z q 10, along with a Singer-invariant bound and Killing-tensor structure of the coefficients. The results yield concrete, invariant polynomials that capture the minimal-polynomial data and illuminate geometric properties (geodesics, curvature) of these nontrivial nilpotent groups, with implications for their C0-structure and automorphism groups.

Abstract

Every homogeneous Riemannian C_0-space (N,g) is associated with its minimal polynomial. To provide explicit examples, we compute the minimal polynomials for generalized Heisenberg groups equipped with their canonical left-invariant metrics.
Paper Structure (36 sections, 48 theorems, 597 equations)

This paper contains 36 sections, 48 theorems, 597 equations.

Table of Contents

  1. Preliminaries
  2. The minimal polynomial of a homogeneous C0-space
  3. Generalized Heisenberg groups
  4. A blueprint algorithm to find the minimal polynomial of a homogeneous C0-space
  5. The main result
  6. Overview of the classification of the eigenvalue branches of -K^2
  7. Explicit formulas for some eigenvalue branches of -K^2
  8. Concluding remarks on Theorem \ref{['th:main_2']}
  9. Geometric properties of H-type groups
  10. Geodesics of $N$
  11. The symmetrized curvature tensor
  12. The characteristic family of linear operators
  13. Direct sums of Clifford modules
  14. Tensor products of Clifford modules
  15. The gradients of the eigenvalue branches of -K^2
  16. ...and 21 more sections

Key Result

Theorem 1.3

Let $(N,g)$ be a homogeneous Riemannian $\mathfrak{C}_0$-space with nonvanishing Ricci tensor. Then:

Theorems & Definitions (99)

  • Definition 1.1
  • Definition 1.2: BTV
  • Theorem 1.3
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • Lemma 3.3
  • ...and 89 more