[Paper Review] The Khovanov-Lauda 2-category and categorifications of a level two quantum sl(n) representation

David Hill, Joshua Sussan|ArXiv.org|Oct 13, 2009

Algebraic structures and combinatorial models21 references4 citations

TL;DR

This paper constructs explicit 2-functors from the Khovanov-Lauda 2-category to two distinct categorifications of the irreducible quantum $σ\mathfrak{sl}_n$-representation of highest weight $2\omega_k$, namely the Huerfano-Khovanov diagrammatic category and a maximal parabolic subcategory of graded category $ϵ\mathcal{O}$ for $\mathfrak{gl}_{2k}$. The key contribution is proving the existence of these 2-functors by explicitly defining natural transformations that satisfy the defining relations of the Khovanov-Lauda 2-category, thereby realizing a 2-representation of $\mathcal{U}_q(\mathfrak{sl}_n)$ on these categories.

ABSTRACT

We construct 2-functors from a 2-category categorifying quantum sl(n) to 2-categories categorifying the irreducible representation of highest weight $ 2 ω_k. $

Motivation & Objective

• To construct a 2-functor from the Khovanov-Lauda 2-category to the Huerfano-Khovanov categorification of the irreducible representation $V_{2\omega_k}$.
• To establish a 2-representation of $\mathcal{U}_q(\mathfrak{sl}_n)$ on a maximal parabolic subcategory of graded category $\mathcal{O}$ for $\mathfrak{gl}_{2k}$.
• To verify that the natural transformations between functors in the categorification satisfy the defining relations of the Khovanov-Lauda 2-category.
• To demonstrate that the Grothendieck group of the category $\mathcal{P}_{k,n}$ is isomorphic to the irreducible representation $V_{2\omega_k}$.

Proposed method

• Explicitly define 2-functors $\Omega_{k,n}: \mathcal{KL} \to \mathcal{HK}_{k,n}$ and $\Pi_{k,n}: \mathcal{KL} \to \mathcal{P}_{k,n}$ from the Khovanov-Lauda 2-category to the diagrammatic and category $\mathcal{O}$ categorifications, respectively.
• Construct 1-morphisms as projective functors and 2-morphisms as natural transformations via the Soergel functor $\mathbb{V}$.
• Use the naturality of isomorphisms from [11, Section 6.2] to verify that the relations in the Khovanov-Lauda 2-category are satisfied.
• Leverage the Koszul grading on category $\mathcal{O}$ and cohomological calculations from [11] to define the 2-functor on the category $\mathcal{P}_{k,n}$.
• Establish a bijection between the set of projective-injective objects in $\mathcal{O}^{(k,k)}_{\overline{\lambda}}(\mathfrak{gl}_{2k})$ and column-decreasing, row-non-decreasing tableaux to compute weight space dimensions.
• Apply the Weyl character formula to show that the dimension of the $\lambda$-weight space in $[\mathcal{P}_{k,n}]_{\mathbb{Q}(q)}$ matches that of $V_{2\omega_k}$.

Experimental results

Research questions

• RQ1 Does the Khovanov-Lauda 2-category act on the Huerfano-Khovanov categorification of $V_{2\omega_k}$ via a 2-functor?
• RQ2 Can a 2-representation of $\mathcal{U}_q(\mathfrak{sl}_n)$ be realized on a maximal parabolic subcategory of graded category $\mathcal{O}$ for $\mathfrak{gl}_{2k}$?
• RQ3 Are the natural transformations in the Huerfano-Khovanov and category $\mathcal{O}$ categorifications isomorphic to the relations in the Khovanov-Lauda 2-category?
• RQ4 Is the Grothendieck group of the category $\mathcal{P}_{k,n}$ isomorphic to the irreducible representation $V_{2\omega_k}$?
• RQ5 Can the categorification of $V_{2\omega_k}$ be lifted from the classical limit at $q=1$ to the quantum setting using graded category $\mathcal{O}$?

Key findings

• There exists a 2-functor $\Omega_{k,n}: \mathcal{KL} \to \mathcal{HK}_{k,n}$ that maps the Khovanov-Lauda 2-category to the Huerfano-Khovanov diagrammatic categorification of $V_{2\omega_k}$.
• There exists a 2-functor $\Pi_{k,n}: \mathcal{KL} \to \mathcal{P}_{k,n}$ that realizes a 2-representation of $\mathcal{U}_q(\mathfrak{sl}_n)$ on the maximal parabolic subcategory of graded category $\mathcal{O}$ for $\mathfrak{gl}_{2k}$.
• The Grothendieck group $[\mathcal{P}_{k,n}]_{\mathbb{Q}(q)}$ is isomorphic to the irreducible representation $V_{2\omega_k}$ of $\mathcal{U}_q(\mathfrak{sl}_n)$.
• The dimension of the $\lambda$-weight space in $[\mathcal{P}_{k,n}]_{\mathbb{Q}(q)}$ is given by the Catalan number $\frac{1}{|S|+1}\binom{2|S|}{|S|}$, matching the dimension of the corresponding weight space in $V_{2\omega_k}$.
• The construction relies on the Soergel functor and cohomological isomorphisms from [11], ensuring naturality and satisfaction of Khovanov-Lauda relations.
• The categorification via category $\mathcal{O}$ is more explicit than previous approaches, utilizing the geometric Koszul grading and explicit cohomology calculations.