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

TL;DR

This paper introduces the $t$-quantized Cartan matrix for finite type quiver Hecke algebras, revealing its role in determining R-matrix invariants between cuspidal modules using combinatorial AR-quivers.

Contribution

It establishes a connection between the $t$-quantized Cartan matrix and R-matrix invariants, utilizing combinatorial AR-quivers for finite type algebras.

Findings

The $t$-quantized Cartan matrix encodes R-matrix invariants.

AR-quivers are key tools in the analysis.

The results apply to arbitrary finite type quiver Hecke algebras.

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.