Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$J$-invariant of linear algebraic groups of outer type

Nikita Geldhauser, Maksim Zhykhovich

TL;DR

This work extends the $J$-invariant from inner to outer type semisimple groups by developing normed Chow rings and motives for generically quasi-split twisted flag varieties. It introduces a coherent framework (normed Chow rings, normed motives, and the normed upper motive) that yields a motivic decomposition analogous to the inner-type theory, and proves a Solomon-type structure for outer type via the ring $C^*$ and associated $J$-invariants. The paper provides explicit computations and tables for the outer-type cases, including unitary and orthogonal families, and links invariants to Rost invariants and Tits algebras, with special attention to degree-1 and degree-2 entries. This yields concrete, computable invariants for outer-type groups and deepens understanding of how motivic decompositions behave under generic quasi-splitting, with potential applications to classification problems and algebraic cycle theory.

Abstract

We extend the notion of the $J$-invariant to arbitrary semisimple linear algebraic groups and provide complete decompositions for the normed Chow motives of all generically quasi-split twisted flag varieties. Besides, we establish some combinatorial patterns for normed Chow groups and motives and provide some explicit formulae for values of the $J$-invariant.

$J$-invariant of linear algebraic groups of outer type

TL;DR

This work extends the -invariant from inner to outer type semisimple groups by developing normed Chow rings and motives for generically quasi-split twisted flag varieties. It introduces a coherent framework (normed Chow rings, normed motives, and the normed upper motive) that yields a motivic decomposition analogous to the inner-type theory, and proves a Solomon-type structure for outer type via the ring and associated -invariants. The paper provides explicit computations and tables for the outer-type cases, including unitary and orthogonal families, and links invariants to Rost invariants and Tits algebras, with special attention to degree-1 and degree-2 entries. This yields concrete, computable invariants for outer-type groups and deepens understanding of how motivic decompositions behave under generic quasi-splitting, with potential applications to classification problems and algebraic cycle theory.

Abstract

We extend the notion of the -invariant to arbitrary semisimple linear algebraic groups and provide complete decompositions for the normed Chow motives of all generically quasi-split twisted flag varieties. Besides, we establish some combinatorial patterns for normed Chow groups and motives and provide some explicit formulae for values of the -invariant.
Paper Structure (8 sections, 16 theorems, 64 equations)

This paper contains 8 sections, 16 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on linear algebraic groups of outer type
  3. Preliminaries on the category of Chow motives
  4. Background on the $J$-invariant for groups of inner type
  5. Normed Chow rings and motives
  6. Some numerology of algebraic groups
  7. Motives of generically quasi-split varieties
  8. $J$-invariant for groups of outer type

Key Result

Proposition 6.1

Let $G$ be a simple quasi-split group over $F$, let $B$ be a Borel subgroup of $G$ defined over $F$ and let $K$ be a finite Galois field extension such that the group $G_K$ is of inner type. a) If $G$ is a simple algebraic group of inner type over $F$, then the Poincaré polynomial $P(\mathop{\mathrm where $e_i$ are degrees of the fundamental polynomial invariants from Table tab3. b) If $G$ is an a

Theorems & Definitions (40)

  • Example 3.1
  • Definition 4.6: PSZ08
  • Remark 5.1
  • Proposition 6.1: Extended Solomon theorem
  • proof
  • Example 6.4
  • Lemma 7.1
  • proof
  • Lemma 7.2
  • proof
  • ...and 30 more