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Denominators of R-matrices, higher Dorey's rules and a generalization of T-systems for quantum affine algebras

Se-jin Oh, Travis Scrimshaw

TL;DR

The article develops a higher analogue of Dorey’s rule and a generalized T-system for quantum affine algebras, culminating in explicit denominator formulas $d_{M,N}(z)$ for all nonexceptional types (with mild ambiguities in $C_n^{(1)}$). It introduces higher Dorey rules, $i$-box combinatorics, and $ rak d$-invariants to control surjections between Kirillov–Reshetikhin modules and to classify simple tensor products, extending to twisted types via Schur–Weyl dualities. The results have broad implications for Schur positivity, quiver Hecke algebras, and monoidal categorification of cluster algebras, linking $R$-matrix denominators to categorical invariants and head/socle structures. The approach combines R-matrix theory, crystal combinatorics, and quiver representations to provide a unified framework for understanding tensor products and their heads, enabling explicit computation of denominators and deepened structural insight into KR modules and their interactions.

Abstract

We construct a higher level analogue of Dorey's rule, which describe certain surjective morphisms between Kirillov--Reshetikhin (KR) modules over quantum affine algebras. Building on this, we establish a generalized T-system of short exact sequences and prove the denominator formula between KR modules in all nonexceptional types, except with only mild ambiguities persisting in type $C_n^{(1)}$. As a consequence, we can completely classify when a tensor product of KR modules is simple. These results have further applications to Schur positivity statements, quiver Hecke algebras, and the recently introduced $\mathfrak{d}$-invariants in monoidal categories over quantum affine algebras and quiver Hecke algebras.

Denominators of R-matrices, higher Dorey's rules and a generalization of T-systems for quantum affine algebras

TL;DR

The article develops a higher analogue of Dorey’s rule and a generalized T-system for quantum affine algebras, culminating in explicit denominator formulas for all nonexceptional types (with mild ambiguities in ). It introduces higher Dorey rules, -box combinatorics, and -invariants to control surjections between Kirillov–Reshetikhin modules and to classify simple tensor products, extending to twisted types via Schur–Weyl dualities. The results have broad implications for Schur positivity, quiver Hecke algebras, and monoidal categorification of cluster algebras, linking -matrix denominators to categorical invariants and head/socle structures. The approach combines R-matrix theory, crystal combinatorics, and quiver representations to provide a unified framework for understanding tensor products and their heads, enabling explicit computation of denominators and deepened structural insight into KR modules and their interactions.

Abstract

We construct a higher level analogue of Dorey's rule, which describe certain surjective morphisms between Kirillov--Reshetikhin (KR) modules over quantum affine algebras. Building on this, we establish a generalized T-system of short exact sequences and prove the denominator formula between KR modules in all nonexceptional types, except with only mild ambiguities persisting in type . As a consequence, we can completely classify when a tensor product of KR modules is simple. These results have further applications to Schur positivity statements, quiver Hecke algebras, and the recently introduced -invariants in monoidal categories over quantum affine algebras and quiver Hecke algebras.

Paper Structure

This paper contains 49 sections, 148 theorems, 873 equations, 3 figures.

Table of Contents

  1. Background
  2. Quantum affine algebras
  3. Finite dimensional integrable modules
  4. Convex orders
  5. $\mathrm{Q}$-datum and quivers
  6. Subcategories
  7. R-matrices and Z-invariants
  8. R-matrices
  9. Z-invariants arising from the R-matrix
  10. Normal sequences and root modules
  11. KR modules, i-box combinatorics and Dorey's rule
  12. Kirillov--Reshetikhin modules
  13. i-boxes and T-systems
  14. Generalized Dorey's rule
  15. Kashiwara Crystals
  16. ...and 34 more sections

Key Result

Proposition 1.9

For $m,n\in \mathbb{Z}$ with $|m-n|>1$, if a simple module $M$ is contained in $\mathscr{C}_\mathcal{Q}[m] \cap \mathscr{C}_\mathcal{Q}[n]$, then $M \simeq \mathbf{1}$.

Figures (3)

  • Figure 1: Dynkin diagrams $\triangle$ for the exceptional affine types. The affine node is marked black.
  • Figure 2: The Dynkin diagram and automorphism $(\blacktriangle,\sigma)$ for non-simply-laced $\mathfrak{g}_0$.
  • Figure 3: The crystal corresponding to the natural representation $V(\Uplambda_1)$ for each of the Lie algebras of type ABCDG; here $n_- := n-1$.

Theorems & Definitions (275)

  • Remark 1.1
  • Definition 1.2: McN15Oh19
  • Definition 1.3
  • Example 1.4
  • Example 1.5
  • Example 1.6
  • Definition 1.7: HL15HL16KKKOIIIKO18OhS19
  • Remark 1.8
  • Proposition 1.9: KKOP23P
  • Example 1.10
  • ...and 265 more