[Paper Review] A brief review of abelian categorifications

Mikhail Khovanov, Volodymyr Mazorchuk+1 more|ArXiv.org|Feb 25, 2007

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TL;DR

This paper presents a framework for abelian categorification of semisimple representations of rings using abelian categories and exact functors, demonstrating how Grothendieck groups and functor actions lift algebraic structures with non-negative integer coefficients. Key results include categorifications of symmetric group and Hecke algebra representations via highest weight categories for $\mathfrak{sl}_n$, and explicit formulas for the determinant of the Cartan matrix in terms of combinatorial invariants.

ABSTRACT

This article contains a review of categorifications of semisimple representations of various rings via abelian categories and exact endofunctors on them. A simple definition of an abelian categorification is presented and illustrated with several examples, including categorifications of various representations of the symmetric group and its Hecke algebra via highest weight categories of modules over the Lie algebra sl(n). The review is intended to give non-experts in representation theory who are familiar with the topological aspects of categorification (lifting quantum link invariants to homology theories) an idea for the sort of categories that appear when link homology is extended to tangles.

Motivation & Objective

• To formalize a general framework for abelian categorification of $A$-modules with a basis having non-negative structure constants.
• To illustrate this framework through concrete examples, including categorifications of representations of the symmetric group and its Hecke algebra via $\mathfrak{sl}_n$ highest weight categories.
• To explore the categorical interpretation of algebraic invariants such as the determinant of the Cartan matrix in terms of Grothendieck groups and stable categories.
• To extend the framework to graded settings, relating the determinant of the graded Cartan matrix to Kazhdan-Lusztig polynomials and quantum integers.
• To conjecture that the determinant of the graded Cartan matrix is a product of quantum integers, supported by known cases and $q$-analogues of classical formulas.

Proposed method

• Define a weak abelian categorification as an abelian category $\mathcal{B}$ with an isomorphism $\varphi: K(\mathcal{B}) \xrightarrow{\sim} B$ and exact endofunctors $F_i$ such that $[F_i]$ lifts the action of basis elements $a_i$ on $B$.
• Use the Grothendieck group $K(\mathcal{B})$, generated by isomorphism classes of simple objects $[L_j]$, to realize the module $B$ with a distinguished basis.
• Ensure the composition $F_i F_j$ is isomorphic to $\bigoplus_k F_k^{c_{ij}^k}$, matching the multiplication structure constants $c_{ij}^k$ in the algebra $A$.
• Construct categorifications via direct sums of module categories over nilCoxeter algebras $R_n$, where $K(R_n\text{-mod}) \cong \mathbb{Z}$ and $[L_n]$ maps to $\frac{x^n}{n!}$ in the $\mathcal{A}_1$-module $B$.
• Compute the Cartan matrix of the category $\mathcal{C}^\mu$ of finite-dimensional modules over a symmetric algebra $A^\mu$, with entries $c_{a,b} = \dim \mathrm{Hom}(P_a, P_b)$ for indecomposable projectives $P_a$.
• Derive the determinant of the Cartan matrix using combinatorial formulas involving binomial coefficients and the Jantzen-Schaper formula, and extend to the graded case using quantum integers.

Experimental results

Research questions

• RQ1 How can one systematically construct abelian categorifications of $A$-modules with non-negative structure constants?
• RQ2 What is the categorical interpretation of the determinant of the Cartan matrix in terms of stable categories and Grothendieck groups?
• RQ3 Can the determinant of the graded Cartan matrix be expressed as a product of quantum integers for arbitrary partitions $\lambda$?
• RQ4 How do Kazhdan-Lusztig polynomials relate to the entries of the graded Cartan matrix in the categorification of $\mathfrak{sl}_n$ representations?
• RQ5 What is the role of the Shapovalov form in the categorification of irreducible $\mathfrak{sl}_n$-modules, and how does its determinant relate to the Cartan matrix?

Key findings

• The determinant of the Cartan matrix for the category $\mathcal{C}^\mu$ corresponding to $\lambda = (n)$ is 1, for $\lambda = (1^n)$ is $n!$, and for $\lambda = (n-1,1)$ is $n$.
• For $\lambda = (2^n)$, the determinant is $\prod_{i=1}^n (i+1)^{r_{n,i}}$, where $r_{n,i} = \binom{2n}{n-i} - 2\binom{2n}{n-i-1} + \binom{2n}{n-i-2}$, with $\binom{j}{s} = 0$ if $s < 0$.
• The determinant of the graded Cartan matrix for $\lambda = (2^n)$ is given by the same product, but with $i+1$ replaced by the quantum integer $[i+1] = 1 + q^2 + \cdots + q^{2i}$.
• The absolute value of the determinant of the Cartan matrix equals the cardinality of the Grothendieck group of the stable category $A^\mu\text{-\underline{mod}}$ when $\det(C) \neq 0$.
• The determinant of the graded Cartan matrix is algorithmically computable from Kazhdan-Lusztig polynomials of the symmetric group.
• The authors conjecture that for any partition $\lambda$, the determinant of the graded Cartan matrix is a product of quantum integers $[j]$ with appropriate multiplicities, supported by $q$-analogues of the Jantzen-Schaper formula.