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Standard monomials and invariant theory of arc spaces II: Symplectic group

Andrew R. Linshaw, Bailin Song

Abstract

This is the second in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field $K$, we construct a standard monomial basis for the arc space of the Pfaffian variety over $K$. As an application, we prove the arc space analogue of the first and second fundamental theorems of invariant theory for the symplectic group.

Standard monomials and invariant theory of arc spaces II: Symplectic group

Abstract

This is the second in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field , we construct a standard monomial basis for the arc space of the Pfaffian variety over . As an application, we prove the arc space analogue of the first and second fundamental theorems of invariant theory for the symplectic group.

Paper Structure

This paper contains 24 sections, 31 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Invariant theory
  3. Standard monomials and Pfaffian varieties
  4. Arc spaces
  5. Standard monomials for arc spaces
  6. Application in invariant theory
  7. Standard monomials
  8. Generators
  9. Ordering
  10. Relations
  11. Standard monomials
  12. A canonical basis
  13. A ring homomorphism
  14. Tableaux
  15. Application
  16. ...and 9 more sections

Key Result

Theorem 1.2

(FFT and SFT for $Sp_h(K)$ and $W = K^{\oplus h}$)

Theorems & Definitions (56)

  • Theorem 1.2
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.9
  • Corollary 1.10
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.5
  • proof
  • Lemma 2.7
  • ...and 46 more