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Weyl elements in isotropic reductive groups

Egor Voronetsky

TL;DR

This work addresses Weyl elements in isotropic reductive group schemes over commutative rings, proposing an explicit square formula for $\alpha$-Weyl elements and exploring their existence and normalizers. By developing the relative root system framework, A1-gradings, and explicit constructions via Jordan and Albert algebras, it derives concrete descriptions of Weyl elements and their actions on the split torus and root subgroups, including explicit computations of $w^2$ across a wide range of types. The main contributions include a universal square formula for $^{w^2}u$ (with parity and root‑inclusion cases), a detailed treatment of $\alpha$-Weyl elements (existence locally and conjugation rules), and a classification of normalizers of long root subgroups in BC1-type indices, plus a canonical split torus with a root datum structure. The results unify Weyl-element theory for isotropic reductive groups over rings and provide tools with potential applications in structural theory and algebraic K-theory.

Abstract

We study Weyl elements in isotropic reductive groups over commutative rings. Our main result in an explicit formula for squares of such elements.

Weyl elements in isotropic reductive groups

TL;DR

This work addresses Weyl elements in isotropic reductive group schemes over commutative rings, proposing an explicit square formula for -Weyl elements and exploring their existence and normalizers. By developing the relative root system framework, A1-gradings, and explicit constructions via Jordan and Albert algebras, it derives concrete descriptions of Weyl elements and their actions on the split torus and root subgroups, including explicit computations of across a wide range of types. The main contributions include a universal square formula for (with parity and root‑inclusion cases), a detailed treatment of -Weyl elements (existence locally and conjugation rules), and a classification of normalizers of long root subgroups in BC1-type indices, plus a canonical split torus with a root datum structure. The results unify Weyl-element theory for isotropic reductive groups over rings and provide tools with potential applications in structural theory and algebraic K-theory.

Abstract

We study Weyl elements in isotropic reductive groups over commutative rings. Our main result in an explicit formula for squares of such elements.
Paper Structure (5 sections, 11 theorems, 53 equations, 5 tables)

This paper contains 5 sections, 11 theorems, 53 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Isotropic reductive groups
  3. Reductive groups graded by $\mathsf A_1$
  4. Normalizers of long root subgroups
  5. Weyl elements

Key Result

Lemma 1

Let $G$ be isotropic reductive group scheme over $K$. Then its automorphism group scheme $A = \mathbf{Aut}(G, L, U_\alpha)$ is an extension of a finite group scheme (a subgroup of the constant group sheaf $\mathbf{Out}(\widetilde{\Phi})$) by $L / \mathop{\mathrm{C}}\nolimits(G)$.

Theorems & Definitions (22)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 12 more