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Morita equivalences between cyclotomic KLR algebras in types $\mathtt{C}_\infty$ and $\mathtt{A}_\infty$

Chris Bowman, Robert Muth, Liron Speyer, Louise Sutton

TL;DR

The paper proves a graded Morita equivalence between level-one cyclotomic KLR algebras of type $C_ olinebreak_ olinebreakinfty$ and level-two cyclotomic KLR algebras of type $A_ olinebreak_ olinebreakinfty$ by constructing an explicit isomorphism between a corner of the former and a tensor product with the latter. This equivalence preserves the graded cellular structures, enabling explicit, characteristic-free identifications of simple and Specht modules across the two types and yielding parallel graded decomposition data. The work reveals a folding phenomenon linking $C$-type and $A$-type algebras in the level-one setting and provides a framework to extract full Ext-quiver and submodule structure information for level-one type $C_ olinebreak_ olinebreakinfty$ algebras. Potential extensions to higher levels and other affine types are discussed, with expectations of substantial additional challenges. Overall, the results bridge representation theories across infinite-type settings and offer concrete combinatorial and categorical tools for understanding cyclotomic KLR algebras.

Abstract

We prove that level one cyclotomic KLR algebras in type $\mathtt{C}_\infty$ are graded Morita equivalent to level two cyclotomic KLR algebras in type $\mathtt{A}_\infty$. We hence deduce the graded decomposition numbers and full submodule structures of all level one cyclotomic KLR algebras in type $\mathtt{C}_\infty$.

Morita equivalences between cyclotomic KLR algebras in types $\mathtt{C}_\infty$ and $\mathtt{A}_\infty$

TL;DR

The paper proves a graded Morita equivalence between level-one cyclotomic KLR algebras of type and level-two cyclotomic KLR algebras of type by constructing an explicit isomorphism between a corner of the former and a tensor product with the latter. This equivalence preserves the graded cellular structures, enabling explicit, characteristic-free identifications of simple and Specht modules across the two types and yielding parallel graded decomposition data. The work reveals a folding phenomenon linking -type and -type algebras in the level-one setting and provides a framework to extract full Ext-quiver and submodule structure information for level-one type algebras. Potential extensions to higher levels and other affine types are discussed, with expectations of substantial additional challenges. Overall, the results bridge representation theories across infinite-type settings and offer concrete combinatorial and categorical tools for understanding cyclotomic KLR algebras.

Abstract

We prove that level one cyclotomic KLR algebras in type are graded Morita equivalent to level two cyclotomic KLR algebras in type . We hence deduce the graded decomposition numbers and full submodule structures of all level one cyclotomic KLR algebras in type .

Paper Structure

This paper contains 8 sections, 15 theorems, 73 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Cyclotomic Khovanov--Lauda--Rouquier algebras
  4. $\ell$-partitions and residues
  5. Standard tableaux and Specht modules
  6. Semistandard tableaux
  7. The isomorphism theorem
  8. The cellular basis under the isomorphism

Key Result

Theorem 2.11

Let $\mathfrak{g} = \mathfrak{sl}_\infty$ or $\mathfrak{sp}_\infty$. Then $\mathscr{R}_\beta^\Lambda(\mathfrak{g})$ is a graded cellular algebra with cellular basis

Figures (12)

  • Figure 1: An arbitrary partition $\nu$ labelling a Specht module of the level 1 type $\mathtt{C}_\infty$ KLR algebra. In grey we highlight the rectangular subpartition $\rho$ and in pink and blue we highlight the partitions ${\color{magenta}\lambda}$ and ${\color{cyan}\mu}$ which label a Specht module $S({\color{magenta}\lambda},{\color{cyan}\mu}')$ of the level 2 type $\mathtt{A}_\infty$ KLR algebra. The charges for the Specht modules are $\kappa_\mathtt{C} \in \mathbb{Z}_{\geqslant 0}$ and $({\color{magenta}\kappa_1},{\color{cyan}\kappa_2}) \in \mathbb{Z}_{>0}^2$, respectively.
  • Figure 2: The Dynkin diagrams of types $\mathtt{A}_\infty$ (above) and $\mathtt{C}_\infty$ (below).
  • Figure 3: The element $\psi^{\mathtt{T}_{\rho +{\color{magenta}\lambda}}}_{\mathtt{T}^{\rho +{\color{magenta}\lambda}}} y_{\mathtt{T}^{\rho + {\color{magenta}\lambda}}}$ corresponding to the tableaux in \ref{['bigeg']}. We have $\mathtt{T}_{\rho + {\color{magenta}\lambda}} = s_4 (s_3s_5)(s_2s_4s_6) \mathtt{T}^{\rho + {\color{magenta}\lambda}}$, where the bracketing indicates commuting elements (which can be reordered within the brackets at will to produce 12 distinct reduced words).
  • Figure 4: For $\kappa_\mathtt{C}=0$, $\rho = (4^4)$, and ${\color{magenta}\lambda} = {\color{magenta}(3,2,1)}$, we depict examples of the $\tt C$-residues of $\rho + {\color{magenta}\lambda}$, the tableau $\mathtt{T}^{\rho + {\color{magenta}\lambda}}$, and the tableau $\mathtt{T}_{\rho + {\color{magenta}\lambda}}$ respectively.
  • Figure 5: The leftmost tableau is of maximal degree, $3$ (note that every orange tile has degree $+1$ and every green has degree 0). The next three tableaux are all possible tableaux of degree $1$; in each case there is a unique pair of orange/green nodes of degree $0$/$-1$; these are $18$/$19$, $10$/$27$, and $2$/$35$ respectively.
  • ...and 7 more figures

Theorems & Definitions (35)

  • Definition 2.6
  • Definition 2.7
  • Definition 2.9
  • Definition 2.10
  • Theorem 2.11: mathas22
  • Theorem 2.12
  • Remark 2.14
  • Proposition 3.1
  • proof
  • Definition 3.2
  • ...and 25 more