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Standard monomials and invariant theory for arc spaces I: general linear group

Andrew R. Linshaw, Bailin Song

Abstract

This is the first 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 determinantal variety over $K$. As an application, we prove the arc space analogue of the first and second fundamental theorems of invariant theory for the general linear group.

Standard monomials and invariant theory for arc spaces I: general linear group

Abstract

This is the first 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 determinantal variety over . As an application, we prove the arc space analogue of the first and second fundamental theorems of invariant theory for the general linear group.

Paper Structure

This paper contains 26 sections, 34 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Classical invariant theory
  3. Standard monomial theory
  4. Arc spaces
  5. Standard monomials for arc spaces
  6. Application in invariant theory
  7. Acknowledgment:
  8. Standard monomials
  9. Normalized derivatives
  10. Generators
  11. Ordering
  12. Relations
  13. Standard monomials
  14. A canonical basis
  15. A ring homomorphism
  16. ...and 11 more sections

Key Result

Theorem 1.2

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

Theorems & Definitions (64)

  • Theorem 1.2
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.9
  • Corollary 1.10
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.6
  • proof
  • Lemma 2.8
  • ...and 54 more