Polynomial functors and categorifications of Fock space

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 $\g$.

💡 Research Summary

The paper establishes a categorical action of the Kac‑Moody algebra 𝔤 on the category 𝒫 of strict polynomial functors over an infinite field k of characteristic p, thereby providing a categorification of the Fock space representation of 𝔤. The algebra 𝔤 is taken to be 𝔰𝔩_∞ when p = 0 and the affine algebra \hat{𝔰𝔩}_p when p > 0. After recalling the standard Chevalley generators {e_i, f_i}, the root and weight lattices, and the Fock space B (the algebra of symmetric functions in infinitely many variables), the authors describe the classical action of 𝔤 on B: e_i adds an i‑colored box to a Young diagram, while f_i removes one. This yields the basic Fock space representation with basis given by Schur functions s_λ.

The central goal is to lift this representation to the categorical level. Using the framework of Chuang‑Rouquier, a 𝔤‑categorification consists of an adjoint pair of exact endofunctors (E, F) on a k‑linear additive category, together with natural transformations X∈End(E) and T∈End(E²), a weight decomposition, and an action of the degenerate affine Hecke algebra DH_n on End(Eⁿ). The authors construct such data on 𝒫. Objects of 𝒫 are functors M: V_k→V_k (V_k = finite‑dimensional k‑vector spaces) satisfying polynomiality conditions; each M(V) carries a polynomial representation of GL(V). The category decomposes as a direct sum of homogeneous subcategories 𝒫_d, each equivalent to the stable category of degree‑d polynomial GL_n‑modules via evaluation on a sufficiently large n‑dimensional space.

Key functors are the Schur functors S_λ and Weyl functors W_λ (the latter being the Kuhn dual of S_λ). They correspond to the classical Schur and Weyl modules and generate 𝒫_d. The authors define for each residue i∈ℤ/pℤ an “i‑box‑addition” functor E_i and an “i‑box‑removal” functor F_i by suitable compositions of induction, restriction, and tensor operations on polynomial functors; these functors shift the weight by the simple root α_i. The natural transformation X records the grading shift, while T encodes the braid‑type relations required for the Hecke algebra action. By verifying the relations of the degenerate affine Hecke algebra on End(Eⁿ), they ensure that the categorical relations lift the Lie algebra relations: