The terms “Expression” and “Representation” are frequently used in mathematics and everyday language, both carrying the broad meaning of “indicating or describing something.” However, their usage and nuances differ. In this article, we will clarify the distinctions between these two concepts and explain why the field of mathematics known as “表現論” in Japanese is called “Representation Theory” in English.
What is Expression?
General Meaning
Expression refers to the act or result of conveying thoughts or emotions to the outside world. Examples include words, facial expressions, gestures, or works of art. It is often used as a means to “directly convey feelings or ideas.”
In a Mathematical Context
In mathematics, an expression refers to the act of “concretely representing” something using formulas or symbols.
- Example: \( x^2 + y^2 = 1 \) is a formula (“expression”) that represents a circle.
- Key Feature: Expression emphasizes “presenting the object itself.”
What is Representation?
General Meaning
Representation refers to the act or form of symbolizing or reproducing something. It includes nuances of “acting on behalf of” or “reproducing” something.
In a Mathematical Context
In mathematics, representation refers to the process of describing abstract objects (such as groups or algebras) in concrete forms like matrices or linear transformations.
- Example: Representing the rotation group \( SO(3) \) as \( 3 \times 3 \) matrices.
- Key Feature: Representation emphasizes “translating and reproducing abstract structures into concrete forms.”
The Difference Between Expression and Representation
| Aspect | Expression | Representation |
| Focus | Demonstrating the object itself | Reproducing abstract objects concretely |
| Examples | Formulas, sentences, works of art | Matrices, linear transformations |
| Mathematical Use | Describing objects with formulas or symbols | Translating abstract structures into concrete forms |
Why is 表現論 Called “Representation Theory”?
What is Representation Theory?
Representation theory is a branch of abstract algebra focused on describing abstract structures, such as groups and algebras, in concrete forms, particularly as linear transformations or matrices. The field emphasizes the “concretization” or “reproduction” of objects.
Why “Representation” Was Chosen
- Focus on Concretization
Representation theory aims to “translate” abstract structures into manipulable forms. This makes “representation” an appropriate term. - Historical Context
The term “representation” was influenced by the French word “représentation.” French mathematicians were pioneers in this field, and the term was adopted in English. - Difference from Expression
While “expression” focuses on “showing something,” “representation” highlights the process of “translating and reproducing abstract objects into another form.”
Cases Where the Distinction is Ambiguous
Expression and representation sometimes overlap depending on the context. For instance, the equation \( x^2 + y^2 = 1 \), which represents a circle, can be viewed as both an “expression” and a “representation” of the circle.
However, in mathematics, “representation” generally refers to the process of “translating abstract structures into another form,” and the two terms are typically distinguished in usage.
Conclusion
Both expression and representation aim to “represent” something, but their approaches and purposes differ. In mathematics, “expression” focuses on “indicating the object itself,” while “representation” deals with “reproducing abstract structures in concrete forms.”
Representation theory is named as such because it seeks to concretize abstract objects, making them manipulable and understandable. This name aptly reflects the mathematical rigor and objectives of the field.