Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$t$-quantized Cartan matrix and R-matrices for cuspidal modules over quiver Hecke algebras

Masaki Kashiwara, Se-jin Oh

TL;DR

This paper shows that the $t$-quantized Cartan matrix ${oldsymbol{ extsf{C}}_t(t)=oldsymbol{ extsf{C}}(1,t)}$ and its AR-quiver framework encode the $Z$-invariants of R-matrices between cuspidal modules in quiver Hecke algebras of arbitrary finite type. By folding simply-laced AR-quivers along Dynkin diagram automorphisms, the authors relate non-simply-laced types to ADE-type data, deriving explicit degree polynomials $oldsymbol{ rak{d}}_{i,j}(t)$ and their folded counterparts $ ilde{oldsymbol{ rak{d}}}_{i,j}(t)$. They prove that the invariant $oldsymbol{ ext{d}}(S_Q(oldsymbol{eta}),S_Q(oldsymbol{eta'}))$ equals the coefficient of $t^{|p-s|-1}$ in $ ilde{oldsymbol{ extunderbar{B}}}_{i,j}(t)$ after mapping $(i,p)$ and $(j,s)$ through $oldsymbol{ extphi}_Q$, and that the composition length of the convolution product of cuspidal modules is governed by these degree polynomials. The work advances a unified combinatorial approach to R-matrix denominators and convolution lengths across all finite types, including BCFG, by leveraging AR-quiver statistics and Q-data folding. It strengthens the link between quiver Hecke algebras and quantum groups, providing tools to compute explicit invariants from Dynkin-quiver data.

Abstract

As every simple module of a quiver Hecke algebra appears as the image of the R-matrix defined on the convolution product of certain cuspidal modules, knowing the $\mathbb{Z}$-invariants of the R-matrices between cuspidal modules is quite significant. In this paper, we prove that the $(q,t)$-Cartan matrix specialized at $q=1$ of an arbitrary finite type, called the $t$-quantized Cartan matrix, informs us of the invariants of R-matrices. To prove this, we use combinatorial AR-quivers associated with Dynkin quivers and their properties as crucial ingredients.

$t$-quantized Cartan matrix and R-matrices for cuspidal modules over quiver Hecke algebras

TL;DR

This paper shows that the -quantized Cartan matrix and its AR-quiver framework encode the -invariants of R-matrices between cuspidal modules in quiver Hecke algebras of arbitrary finite type. By folding simply-laced AR-quivers along Dynkin diagram automorphisms, the authors relate non-simply-laced types to ADE-type data, deriving explicit degree polynomials and their folded counterparts . They prove that the invariant equals the coefficient of in after mapping and through , and that the composition length of the convolution product of cuspidal modules is governed by these degree polynomials. The work advances a unified combinatorial approach to R-matrix denominators and convolution lengths across all finite types, including BCFG, by leveraging AR-quiver statistics and Q-data folding. It strengthens the link between quiver Hecke algebras and quantum groups, providing tools to compute explicit invariants from Dynkin-quiver data.

Abstract

As every simple module of a quiver Hecke algebra appears as the image of the R-matrix defined on the convolution product of certain cuspidal modules, knowing the -invariants of the R-matrices between cuspidal modules is quite significant. In this paper, we prove that the -Cartan matrix specialized at of an arbitrary finite type, called the -quantized Cartan matrix, informs us of the invariants of R-matrices. To prove this, we use combinatorial AR-quivers associated with Dynkin quivers and their properties as crucial ingredients.
Paper Structure (37 sections, 70 theorems, 322 equations, 4 tables)

This paper contains 37 sections, 70 theorems, 322 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Backgrounds
  3. Finite Cartan data
  4. Convex orders
  5. Statistics
  6. Dynkin quivers
  7. $t$-quantized Cartan matrix
  8. Auslander-Reiten (AR) quivers
  9. Quivers
  10. Classical AR-quiver $\Gamma_Q$
  11. $D_{n+1}$-case
  12. $A_{2n-1}$-case
  13. Relation between $\mathfrak{d}_{i,j}(t)$ and $\widetilde{\mathfrak{d}}_{i,j}(t)$
  14. Labeling of AR-quivers in type BCFG
  15. $B_{n}$-case
  16. ...and 22 more sections

Key Result

Theorem 2.5

For each Dynkin quiver $Q=(\triangle,\xi)$, there exists a $Q$-adapted reduced expression ${\underline{w}}_0$ of $w_0 \in \mathsf{W}_\triangle$, and the set of all $Q$-adapted reduced expressions is a commutation class of $w_0$.

Theorems & Definitions (132)

  • Definition 2.1
  • Definition 2.2: cf. McNa15Oh18
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.5: OS19BKO22
  • Definition 2.6
  • Definition 2.7
  • Example 2.8
  • Lemma 2.9
  • Definition 3.1
  • ...and 122 more