[Paper Review] Standard monomials and invariant theory for arc spaces III: special linear group

Andrew R. Linshaw, Bailin Song|arXiv (Cornell University)|Aug 20, 2021

Algebraic structures and combinatorial models29 references4 citations

TL;DR

This paper establishes the arc space analogue of the first and second fundamental theorems of invariant theory for the special linear group over an algebraically closed field. Using standard monomial theory, it proves that the invariant ring of the arc space of the affine quotient is generated by certain standard monomials, even when the arc space is nonreduced, resolving a key subtlety in the SL_h case compared to GL_h and Sp_h.

ABSTRACT

This is the third in a series of papers on standard monomial theory and invariant theory of arc spaces. For any algebraically closed field $K$, we prove the arc space analogue of the first and second fundamental theorems of invariant theory for the special linear group. This is more subtle than the results for the general linear and symplectic groups obtained in the first two papers because the arc space of the corresponding affine quotients can be nonreduced.

Motivation & Objective

• To extend standard monomial theory to arc spaces for the special linear group SL_h.
• To establish the first and second fundamental theorems of invariant theory in the arc space setting.
• To address the challenge that arc spaces of affine quotients can be nonreduced, unlike in the GL_h and Sp_h cases.
• To provide a standard monomial basis for the invariant ring of the arc space of the SL_h representation variety.
• To generalize classical invariant theory results to the infinite-dimensional setting of arc spaces.

Proposed method

• Constructs the arc space J_∞(V) as the inverse limit of jet schemes J_n(V), using the functor of points over K[[t]].
• Adapts standard monomial theory to the arc space by defining a partial order on generators: X^{(k)}_{u;v}, Y^{(k)}_u, Z^{(k)}_v.
• Uses normalized derivatives ∂̄^i to generate differential relations and define standard monomials in the arc space.
• Applies the theory of differential ideals and differentially finitely generated ideals to control relations in the invariant ring.
• Employs recursive reduction techniques and weight arguments (wt, R(E,·)) to prove nonstandard monomials vanish.
• Leverages known classical results (FFT and SFT for SL_h) as base cases and lifts them to the arc space via jet filtration.

Experimental results

Research questions

• RQ1 How can the first fundamental theorem of invariant theory for SL_h be lifted to the arc space setting?
• RQ2 What is the structure of the invariant ring K[V]^{SL_h} in the arc space J_∞(V), especially when the quotient is nonreduced?
• RQ3 Can standard monomial theory be extended to arc spaces of SL_h-representations to produce a basis for the invariant ring?
• RQ4 What are the defining relations (SFT) for the arc space invariant ring, and how do they differ from the classical case?
• RQ5 How do differential operators and jet structures affect the standard monomial basis in the arc space?

Key findings

• The arc space analogue of the first fundamental theorem holds: the invariant ring K[J_∞(V)]^{SL_h} is generated by the standard monomials X^{(k)}_{u;v}, Y^{(k)}_u, Z^{(k)}_v.
• The arc space analogue of the second fundamental theorem holds: the ideal of relations is generated by the same relations (1.2)–(1.4) lifted to the arc space via jet derivatives.
• The standard monomial basis for K[J_∞(V)]^{SL_h} is formed by ordered products of generators under the defined partial order, even when the arc space is nonreduced.
• The proof shows that nonstandard monomials vanish due to cancellation in differential relations, using weight and rank arguments on ∂̄-lists.
• The method successfully handles the nonreduced nature of the arc space quotient, a key difficulty absent in GL_h and Sp_h cases.
• The result generalizes classical invariant theory to infinite jet spaces, providing a complete algebraic description of SL_h invariants in the arc space.