# On the Structure of Finite Groupoids and Their Representations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Schwinger’s Algebra of Measurements and the Groupoid Picture Of Quantum Mechanics

#### 1.2. A Simple Example: The Qubit

## 2. Finite Groupoids: A Structure Theorem

#### 2.1. Some Notations and Generalities on Groupoids, Subgroupoids and Connected Groupoids

#### 2.2. Quotient Spaces and Normal Subgroupoids

**Definition**

**1.**

**Proposition**

**1.**

**Proof.**

#### 2.3. The Structure of Finite Groupoids

**Proposition**

**2.**

**Proof.**

## 3. Representations of Finite Groupoids and Associative Algebras

#### 3.1. Linear Representations of Groupoids

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Theorem**

**1.**

**Lemma**

**1.**

**Proof.**

**Proposition**

**3.**

**Proof.**

#### 3.2. The Groupoid Algebra and Linear Representations of Groupoids

#### 3.3. The Fundamental Representation of a Finite Groupoid

**Proposition**

**4.**

**Proof.**

#### 3.4. Semisimplicity

**Theorem**

**2.**

- i.
- A is semisimple.
- ii.
- Any finite dimensional representation of A is completely reducible.
- iii.
- Any A-submodule U of an A-module V has supplementary factor, that is, there is an A-submodule W such that $V=U\oplus W$.

**Theorem**

**3.**

**Proof.**

**Corollary**

**1.**

## 4. The Structure of the Regular Representation of a Groupoid

#### 4.1. Characters

#### 4.2. Orthogonality of Characters

**Proposition**

**5.**

**Proof.**

**Proposition**

**6.**

**Theorem**

**4.**

**Proof.**

#### 4.3. The Left and Right Regular Representations of a Finite Groupoid

**Proposition**

**7.**

**Proof.**

**Theorem**

**5.**

## 5. Some Examples and Applications

#### 5.1. The Action Groupoid

#### 5.2. Loyd’s Puzzles

#### 5.2.1. The Groupoid ${\mathfrak{L}}_{2}$: The Four Squares Loyd’s Puzzle

#### 5.2.2. Representations of the Loyd’s Group ${\mathfrak{L}}_{2}$

#### 5.2.3. The Regular Representation

## 6. Conclusions and Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**The four states in ${\mathfrak{L}}_{2}$ and their configurations. Each state has three configurations (the order of the alternating group ${\mathcal{A}}_{3}$). There are 12 configurations, (the order of the alternating group ${\mathcal{A}}_{4}$). Thus, for instance, the first row depicts the configurations corresponding to the state ${s}_{1}$.

**Figure 5.**A graphical representation of the quotient groupoid ${\mathfrak{L}}_{2}/{\mathbf{G}}_{0}$ of the Loyd’s groupoid ${\mathfrak{L}}_{2}$ over the normal subgroupoid ${\mathbf{G}}_{0}$ (i.e., the groupoid of pairs of four elements).

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Ibort, A.; Rodríguez, M.A.
On the Structure of Finite Groupoids and Their Representations. *Symmetry* **2019**, *11*, 414.
https://doi.org/10.3390/sym11030414

**AMA Style**

Ibort A, Rodríguez MA.
On the Structure of Finite Groupoids and Their Representations. *Symmetry*. 2019; 11(3):414.
https://doi.org/10.3390/sym11030414

**Chicago/Turabian Style**

Ibort, Alberto, and Miguel A. Rodríguez.
2019. "On the Structure of Finite Groupoids and Their Representations" *Symmetry* 11, no. 3: 414.
https://doi.org/10.3390/sym11030414