𝘵-quantized Cartan matrix and R-matrices for cuspidal modules over 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 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 any finite type, called the 𝘵-quantized Cartan matrix, inform 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.