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Topological Noetherianity of the infinite half-spin representations

Christopher Chiu, Jan Draisma, Rob Eggermont, Tim Seynnaeve, Nafie Tairi

TL;DR

This work establishes that infinite half-spin representations admit a topological Noetherianity under the infinite spin group Spin(V_∞), and that half-spin varieties are determined by pulling back equations from a finite level, notably the isotropic Grassmannian arising at level 4. The authors develop a detailed framework connecting finite spin representations to the infinite setting via explicit contraction and multiplication maps, and they formulate half-spin varieties as Spin(V)-stable subschemes preserved by these maps; the main result shows stabilization of any descending chain of such varieties. A central construction is the inverse half-spin representation (and its direct spin counterpart) built as a projective limit with a Spin(V_∞)-action, and the proof uses a twist of GL(E_∞) actions, Eggermont22 results, and a degree-based induction on defining equations. They also prove a universality statement in dimension eight: the isotropic Grassmannian in spinor embedding is governed by the Cartan map from the level-4 cone, linking spinor and Plücker embeddings and enabling pullback-definability statements to hold uniformly in higher dimensions. Overall, the paper provides a topological-Noetherianity toolbox for infinite-dimensional spin representations with concrete geometric consequences for half-spin varieties and isotropic Grassmannians.

Abstract

We prove that the infinite half-spin representations are topologically Noetherian with respect to the infinite spin group. As a consequence we obtain that half-spin varieties, which we introduce, are defined by the pullback of equations at a finite level. The main example for such varieties is the infinite isotropic Grassmannian in its spinor embedding, for which we explicitly determine its defining equations.

Topological Noetherianity of the infinite half-spin representations

TL;DR

This work establishes that infinite half-spin representations admit a topological Noetherianity under the infinite spin group Spin(V_∞), and that half-spin varieties are determined by pulling back equations from a finite level, notably the isotropic Grassmannian arising at level 4. The authors develop a detailed framework connecting finite spin representations to the infinite setting via explicit contraction and multiplication maps, and they formulate half-spin varieties as Spin(V)-stable subschemes preserved by these maps; the main result shows stabilization of any descending chain of such varieties. A central construction is the inverse half-spin representation (and its direct spin counterpart) built as a projective limit with a Spin(V_∞)-action, and the proof uses a twist of GL(E_∞) actions, Eggermont22 results, and a degree-based induction on defining equations. They also prove a universality statement in dimension eight: the isotropic Grassmannian in spinor embedding is governed by the Cartan map from the level-4 cone, linking spinor and Plücker embeddings and enabling pullback-definability statements to hold uniformly in higher dimensions. Overall, the paper provides a topological-Noetherianity toolbox for infinite-dimensional spin representations with concrete geometric consequences for half-spin varieties and isotropic Grassmannians.

Abstract

We prove that the infinite half-spin representations are topologically Noetherian with respect to the infinite spin group. As a consequence we obtain that half-spin varieties, which we introduce, are defined by the pullback of equations at a finite level. The main example for such varieties is the infinite isotropic Grassmannian in its spinor embedding, for which we explicitly determine its defining equations.
Paper Structure (32 sections, 23 theorems, 121 equations)

This paper contains 32 sections, 23 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Purpose of this paper and main theorem
  3. Relations to the literature
  4. Organisation of this paper
  5. Finite spin representations and the spin group
  6. The Clifford algebra
  7. The Grassmann algebra as a Cl(V)-module
  8. Embedding so(V) into the Clifford algebra
  9. The half-spin representations
  10. Explicit formulas
  11. Highest weights of the half-spin representations
  12. The spin group
  13. Two actions of on
  14. The isotropic Grassmannian and infinite spin representations
  15. The isotropic Grassmannian in its spinor embedding
  16. ...and 17 more sections

Key Result

Theorem 1.1

The scheme $(\bigwedge\nolimits_\infty^+ E_\infty)^*$ is topologically $\mathop{\mathrm{Spin}}\nolimits(V_\infty)$-Noetherian. That is, every chain of $\mathop{\mathrm{Spin}}\nolimits(V_\infty)$-stable reduced closed subschemes stabilises.

Theorems & Definitions (56)

  • Theorem 1.1
  • Remark 2.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • proof
  • Proposition 3.4
  • proof : Proof of \ref{['prop:properties']}
  • ...and 46 more