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Tensor product and quandle rings of connected quandles of prime order

Dilpreet Kaur, Pushpendra Singh

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 $X$.

Tensor product and quandle rings of connected quandles of prime order

Abstract

Let be field of complex numbers and be a connected quandle of prime order. We study the regular representation of by describing the quandle ring 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 and prove that decomposes multiplicity free for affine connected quandle .
Paper Structure (8 sections, 25 theorems, 37 equations, 1 table)

This paper contains 8 sections, 25 theorems, 37 equations, 1 table.

Table of Contents

  1. Introduction
  2. Review of quandles and quandle rings
  3. Inner automorphism groups of affine connected quandles
  4. Conjugacy classes of inner automorphism group
  5. Irreducible representations of $\mathbb{Z}_{p} \rtimes \mathbb{Z}_{n}$
  6. Decomposition of quandle ring of connected quandles of prime order
  7. Tensor product of connected quandles of prime order
  8. Multiplicity freeness of quandle ring decomposition of connected quandles

Key Result

Lemma 3.1

Let $(\mathcal{A}_{m},f)$ denote the affine connected quandle of order $m$. Then $Inn(\mathcal{A}_{m}) = \mathbb{Z}_{m} \rtimes \mathbb{Z}_{n}$.

Theorems & Definitions (60)

  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 50 more