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Reflection length in the general linear and affine groups

Elise G. delMas, Joel Brewster Lewis

Abstract

We give an intrinsic criterion to tell whether a reflection factorization in the general linear group is reduced, and give a formula for computing reflection length in the general affine group.

Reflection length in the general linear and affine groups

Abstract

We give an intrinsic criterion to tell whether a reflection factorization in the general linear group is reduced, and give a formula for computing reflection length in the general affine group.

Paper Structure

This paper contains 13 sections, 17 theorems, 35 equations.

Table of Contents

  1. The general linear group
  2. The affine group
  3. Definitions
  4. Fundamental subspaces, tripartite classification
  5. The main theorem
  6. The case of the field of two elements
  7. Further remarks
  8. Precise statement of other affine results
  9. Other work on orthogonal linear groups
  10. Other work on general linear groups
  11. Affine version of the first main theorem?
  12. Classification of hyperbolic elements over $\mathbf{F}_2$
  13. Coincidences among small groups

Key Result

Theorem 1

If $G$ is and $R$ is the set of elements in $G$ that fix a hyperplane pointwise (the reflections), then the reflection length $\ell_R(g)$ of any element $g$ in $G$ is equal to $\dim \operatorname{im}(g - 1) = \operatorname{codim} \ker(g - 1)$.

Theorems & Definitions (41)

  • Theorem 1: Carter, Scherk, Dieudonne
  • Theorem 2: Carter, BradyWattEuclidean
  • Theorem : BradyMcCammond, LMPS
  • Proposition 3
  • proof
  • Remark 4
  • Definition 5
  • Proposition 6
  • proof
  • Theorem 7
  • ...and 31 more