[Paper Review] Polynomial functors and categorifications of Fock space

Jiuzu Hong, Antoine Touzé+1 more|arXiv (Cornell University)|Nov 22, 2011

Algebraic structures and combinatorial models14 references4 citations

TL;DR

This paper constructs a $χ$-action on the category $π$ of strict polynomial functors over an infinite field of characteristic $p$, categorifying the Fock space representation of the Kac-Moody algebra $χ$ (isomorphic to $χ⁡_{\infty}$ if $p=0$ and $\hat{\mathfrak{sl}}_p$ if $p>0$). The categorification is realized via Chuang-Rouquier functors, establishing derived equivalences between blocks of the same $p$-weight and recovering the Misra-Miwa crystal via socle functors on simple objects.

ABSTRACT

Fix an infinite field $k$ of characteristic $p$, and let $\g$ be the Kac-Moody algebra $\mathfrak{sl}_{\infty}$ if $p=0$ and $\hat{\mathfrak{sl}}_p$ otherwise. Let $\PP$ denote the category of strict polynomial functors defined over $k$. We describe a $\g$-action on $\PP$ (in the sense of Chuang and Rouquier) categorifying the Fock space representation of $\g$.

Motivation & Objective

• To establish a $χ$-categorification of the Fock space representation using the category $π$ of strict polynomial functors.
• To provide a more canonical and intrinsic setting for the $χ$-categorification compared to prior constructions involving inverse limits of general linear group representations.
• To demonstrate that blocks of $π$ of the same $p$-weight are derived equivalent via the $χ$-categorification machinery.
• To recover the Misra-Miwa crystal structure on the set of simple objects in $π$ using Kashiwara operators defined via socle functors.

Proposed method

• Define exact endo-functors $E_i, F_i$ on $π$ corresponding to the Chevalley generators of $χ$, inducing the $χ$-action on the Grothendieck group.
• Construct the $χ$-action via functors $E_i, F_i$ satisfying the Serre relations, using natural transformations and tensor product structures on polynomial functors.
• Verify the defining relations of the $χ$-action by checking the commutation relations on the level of functors, particularly the Serre relation involving $[E_i, E_j]$ for $|i-j|=1$.
• Use the socle functor to define Kashiwara operators $\tilde{e}_i, \tilde{f}_i$ on the set of simple objects in $π$, thereby recovering the crystal structure.
• Leverage the $χ$-categorification to deduce derived equivalences between blocks of $π$ of equal $p$-weight via the Chuang-Rouquier machinery for $χ⁡_2$-categorifications.
• Apply the theory to show that the category $π$ admits a $χ$-categorification independent of prior constructions, simplifying and clarifying earlier results in [HY].

Experimental results

Research questions

• RQ1 Can the Fock space representation of the Kac-Moody algebra $χ$ be categorified via the category $π$ of strict polynomial functors?
• RQ2 Does the $χ$-categorification on $π$ lead to derived equivalences between blocks of the same $p$-weight?
• RQ3 Can the Misra-Miwa crystal structure on Fock space be reconstructed from the $χ$-categorification on $π$?
• RQ4 How does the $χ$-categorification on $π$ compare to earlier constructions involving inverse limits of general linear group representations?
• RQ5 What is the role of the socle functor in realizing Kashiwara operators on the set of simple objects in $π$?

Key findings

• The category $π$ of strict polynomial functors admits a $χ$-action in the sense of Chuang and Rouquier, categorifying the Fock space representation of $χ$.
• Blocks of $π$ of the same $p$-weight are derived equivalent, as a consequence of the $χ$-categorification and the Chuang-Rouquier machinery.
• The Misra-Miwa crystal structure on Fock space is recovered from the $χ$-categorification via Kashiwara operators defined as compositions of Chevalley functors with the socle functor on simple objects.
• The $χ$-categorification on $π$ provides a more canonical and intrinsic framework than earlier constructions based on inverse limits of general linear group representations.
• The action of $χ$ on $π$ is explicitly realized through functors $E_i, F_i$ satisfying the Serre relations, verified by direct computation on tensor products of vector spaces.
• The construction is valid for $p \neq 2$; the case $p=2$ is excluded only for expositional simplicity, though the results extend to $p=2$.