Tensor product and quandle rings of connected quandles of prime order

Abstract

Let $\mathbb{C}$ be field of complex numbers and $X$ be a connected quandle of prime order. We study the regular representation of $X$ by describing the quandle ring $\mathbb{C}[X]$ as direct sum of right simple ideals. We provide description of tensor product of connected quandles of prime order. We further discuss multiplicity freeness of quandle ring decomposition for connected quandles of order $\leq 47$ and prove that $\mathbb{C}[X]$ decomposes multiplicity free for affine connected quandle $X$.