[Paper Review] PBW--filtration over $\bz$ and compatible bases for $V_\bz(\la)$ in type ${ t A}_n$ and ${ t C}_n$

Evgeny Feigin, Ghislain Fourier+1 more|arXiv (Cornell University)|Apr 9, 2012

Algebraic structures and combinatorial models9 references3 citations

TL;DR

This paper establishes a PBW-filtration over the integers for highest weight representations $V_\mathbb{Z}(\lambda)$ in types $\mathbf{A}_n$ and $\mathbf{C}_n$, proving that the associated graded module $V^a_\mathbb{Z}(\lambda)$ is a torsion-free $S_\mathbb{Z}(\mathfrak{n}^{-,a})$-module with a compatible integer basis. The key result is a characteristic-free combinatorial graded character formula for $V^a_\mathbb{Z}(\lambda)$, valid over any field via integral realization of the PBW filtration.

ABSTRACT

We study the PBW-filtration on the highest weight representations $V(\la)$ of the Lie algebras of type ${ t A}_{n}$ and ${ t C}_{n}$. This filtration is induced by the standard degree filtration on $\U(\fn^-)$. In previous papers, the authors studied the filtration and the associated graded algebras and modules over the complex numbers. The aim of this paper is to present a proof of the results which holds over the integers and hence makes the whole construction available over any field.

Motivation & Objective

• To extend the PBW-filtration and associated graded structures on highest weight representations from the complex numbers to the integers, ensuring applicability over any field.
• To describe $V^a_\mathbb{Z}(\lambda)$ as a cyclic $S_\mathbb{Z}(\mathfrak{n}^{-,a})$-module by identifying the defining ideal $I_\mathbb{Z}(\lambda)$.
• To construct a compatible integer basis for $V^a_\mathbb{Z}(\lambda)$, proving it is torsion-free as a $\mathbb{Z}$-module.
• To derive a characteristic-free combinatorial graded character formula for $V^a_\mathbb{Z}(\lambda)$, valid over arbitrary fields.
• To establish a tensor product property for $V^a_\mathbb{Z}(\lambda + \mu)$ as a submodule of $V^a_\mathbb{Z}(\lambda) \otimes_\mathbb{Z} V^a_\mathbb{Z}(\mu)$, preserving the highest weight vector.

Proposed method

• Construct the PBW-filtration on $U(\mathfrak{n}^{-})$ over $\mathbb{Z}$ by lifting the standard degree filtration from the universal enveloping algebra to integral forms.
• Define the integral model $V_\mathbb{Z}(\lambda)$ as the image of $U_\mathbb{Z}(\mathfrak{n}^{-})$ acting on the highest weight vector $v_\lambda$, preserving integrality.
• Prove that the associated graded module $V^a_\mathbb{Z}(\lambda)$ is isomorphic to $S_\mathbb{Z}(\mathfrak{n}^{-,a}) / I_\mathbb{Z}(\lambda)$, with $I_\mathbb{Z}(\lambda)$ generated by relations over $\mathbb{Z}$.
• Use the diamond lemma and monomial orderings on the generators $f_{i,j}^{(s)}$ to show that the standard monomials form a $\mathbb{Z}$-basis for $V^a_\mathbb{Z}(\lambda)$, ensuring torsion-freeness.
• Establish the tensor product embedding $V^a_\mathbb{Z}(\lambda + \mu) \hookrightarrow V^a_\mathbb{Z}(\lambda) \otimes_\mathbb{Z} V^a_\mathbb{Z}(\mu)$ via the canonical map sending $v_{\lambda+\mu}$ to $v_\lambda \otimes v_\mu$.
• Derive the graded character formula by combining the integral PBW filtration with the known complex character formula, proving it holds over $\mathbb{Z}$ and hence over any field.

Experimental results

Research questions

• RQ1 Can the PBW-filtration on $V(\lambda)$ be defined over $\mathbb{Z}$, ensuring integrality of the filtration and associated graded module?
• RQ2 Is the associated graded module $V^a_\mathbb{Z}(\lambda)$ torsion-free as a $\mathbb{Z}$-module, and does it admit a compatible $\mathbb{Z}$-basis?
• RQ3 Can the structure of $V^a_\mathbb{Z}(\lambda)$ as a cyclic $S_\mathbb{Z}(\mathfrak{n}^{-,a})$-module be explicitly described via an integral ideal $I_\mathbb{Z}(\lambda)$?
• RQ4 Does the graded character of $V^a_\mathbb{Z}(\lambda)$ admit a combinatorial formula that is valid over any field, independent of the characteristic?
• RQ5 Is the tensor product property $V^a_\mathbb{Z}(\lambda + \mu) \hookrightarrow V^a_\mathbb{Z}(\lambda) \otimes_\mathbb{Z} V^a_\mathbb{Z}(\mu)$ preserved over $\mathbb{Z}$, with the highest weight vector mapping to the tensor product of highest weight vectors?

Key findings

• The PBW-filtration on $V(\lambda)$ over $\mathbb{C}$ lifts to an integral filtration on $V_\mathbb{Z}(\lambda)$, with the associated graded module $V^a_\mathbb{Z}(\lambda)$ isomorphic to $S_\mathbb{Z}(\mathfrak{n}^{-,a}) / I_\mathbb{Z}(\lambda)$ for an explicitly defined ideal $I_\mathbb{Z}(\lambda)$.
• The module $V^a_\mathbb{Z}(\lambda)$ is torsion-free over $\mathbb{Z}$, and admits a compatible $\mathbb{Z}$-basis consisting of standard monomials in the generators $f_{i,j}^{(s)}$.
• A characteristic-free combinatorial graded character formula for $V^a_\mathbb{Z}(\lambda)$ is established, generalizing the complex case to arbitrary fields.
• The tensor product property holds integrally: there exists a unique injective $S_\mathbb{Z}(\mathfrak{n}^{-,a})$-module homomorphism $V^a_\mathbb{Z}(\lambda + \mu) \hookrightarrow V^a_\mathbb{Z}(\lambda) \otimes_\mathbb{Z} V^a_\mathbb{Z}(\mu)$ sending $v_{\lambda+\mu}$ to $v_\lambda \otimes v_\mu$.
• The proof relies on the diamond lemma and monomial orderings to show that the standard monomials form a basis, and that all relations in the filtration are defined over $\mathbb{Z}$.
• The construction is valid for types $\mathbf{A}_n$ and $\mathbf{C}_n$, and the results are independent of the characteristic of the base field.