[Paper Review] Integral Basis Theorem of cyclotomic Khovanov-Lauda-Rouquier algebras of Type A

Li Ge|arXiv (Cornell University)|Dec 11, 2014

Algebraic structures and combinatorial models15 references10 citations

TL;DR

This paper proves that cyclotomic Khovanov-Lauda-Rouquier (KLR) algebras of type A are free over the integers, establishing a graded cellular basis over ℤ. Using a multistage induction on n and multipartition lexicographic order, the authors show that the homogeneous basis elements ψst span a ℤ-free subalgebra closed under multiplication by generators, thereby classifying all graded irreducible representations via Kleshchev multipartitions and extending the classification to the full KLR algebra Rn.

ABSTRACT

In this paper we prove that the cyclotomic Khovanov-Lauda-Rouquier algebras in type A, $\mathscr R_n^Λ$, are $\mathbb{Z}$-free. We then extend the graded cellular basis of $\mathscr R_n^Λ$ constructed by Hu and Mathas to $\mathscr R_n$ and use this basis to give a classification of all irreducible $\mathscr R_n$-modules.

Motivation & Objective

• To prove that cyclotomic Khovanov-Lauda-Rouquier algebras RΛn are free over the integers, resolving a question posed by Hu and Mathas.
• To extend the graded cellular basis of RΛn to the full KLR algebra Rn, enabling classification of its irreducible modules.
• To provide a complete, explicit classification of all graded simple Rn-modules using a new labeling compatible with cyclotomic quotients.
• To establish that the cellular basis elements ψst form a ℤ-basis closed under multiplication by the algebra generators, ensuring integrality and freeness.

Proposed method

• Use of a multistage induction on n and lexicographic order of multipartitions to prove closure of the ψst basis under multiplication by generators.
• Reduction to the case e ≠ 2 via [9, Theorem 5.14], simplifying the quiver structure to a simply laced A-type quiver.
• Construction of a homogeneous basis {ψst} over ℤ, showing it spans a ℤ-free submodule of RΛn.
• Verification that the identity element lies in the span of {ψst}, ensuring the basis generates the full algebra.
• Extension of the cellular basis from RΛn to Rn using a sequence of weights Λ∞, defining a graded cellular structure on Rn.
• Application of Graham-Lehrer-style arguments and the theory of affine cellular algebras to classify irreducible modules.

Experimental results

Research questions

• RQ1 Is the cyclotomic Khovanov-Lauda-Rouquier algebra RΛn free over the integers for all dominant weights Λ and all e?
• RQ2 Can the graded cellular basis of RΛn be extended to the full KLR algebra Rn, and does it yield a complete classification of its irreducible modules?
• RQ3 Do the irreducible Rn-modules correspond to Kleshchev multipartitions under the grading, and is this labeling compatible with the cyclotomic quotients?
• RQ4 Is the cellular basis {ψst} closed under multiplication by the generators yr, ψs, and e(i), ensuring integrality and freeness of RΛn over ℤ?

Key findings

• The cyclotomic KLR algebra RΛn(Z) is a graded cellular algebra over ℤ with respect to the dominance order, and is free of rank ℓnn!.
• The homogeneous basis {ψst} of RΛn constructed by Hu and Mathas spans a ℤ-free submodule closed under multiplication by all generators, proving RΛn(Z) is ℤ-free.
• The basis extends to Rn, making Rn a graded cellular algebra, and enabling a complete classification of its graded irreducible modules.
• All graded simple Rn-modules are indexed by affine Kleshchev multipartitions ˆλ ∈ Pκ₀ and integer shifts k ∈ ℤ, forming the set {Dˆλ⟨k⟩}.
• The classification is compatible with the cyclotomic quotients: Dˆλ⟨k⟩ is isomorphic to Dλ⟨k⟩ when restricted to RΛn, and non-isomorphic modules correspond to distinct ˆλ.
• The result confirms that RΛn(O) is free over any integral domain O, as RΛn(O) ≅ RΛn(Z) ⊗Z O.