Terminology issue regarding modules and representations

In summary, the conversation revolves around a question regarding the field F and its corresponding group algebra FS4. The representation X maps FS4 to 3x3 matrices over F and the FS4 module V is being discussed. The question is whether V is F3 or the free module of the representation of FS4 over itself. After further examination, it is determined that V is indeed F3.
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Homework Statement


Given a field F, FS4 is a group algebra... we have a representation X that maps FS4 to 3x3 matrices over (presumably) F. Let V denote the FS4 module corresponding to X... do stuff. My question is, what the heck is V supposed to be?

I assumed that V is F3, but that seemingly contradicts what the question wants me to do. So is V the free module of the representation of FS4 over itself (and then why bother having a representation)? I can't think of what else it could be

The full question can be found at

http://www.maths.ox.ac.uk/courses/2008/part-b/b2-algebra/b2a-introduction-representation-theory/materialsheet 5, question 3

EDIT: I was wrong... if V=F3 the question's conclusion is correct. For some reason I was associating the 4 in S4 with F instead, and thought it had characteristic 4... I noticed my error when I realized a) it has characteristic 0 b) 4 isn't the characteristic of a field. Good job by me wasting an hour working in a non-existent field
 
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Well the representation gives us a homomorphism from S_4 into GL(F^3). The induced action of the algebra FS_4 on F^3 induces an FS_4-module structure on F^3. So yes, V=F^3.
 

Related to Terminology issue regarding modules and representations

1. What is a module in the context of scientific terminology?

A module is a distinct and self-contained unit within a larger system or structure. In science, modules can refer to specific components of a biological organism, such as organs or organelles, or to discrete parts of a computer program or system.

2. How are modules and representations related?

In scientific terminology, modules and representations are closely related in that a representation is a way of describing or depicting a module. In other words, a representation is a visual or conceptual model of a module, which allows for better understanding and analysis of the module's function and properties.

3. What types of modules exist in science?

There are many types of modules in science, as they can pertain to various fields and disciplines. Some common examples include genetic modules, social modules, cognitive modules, and ecological modules. Each type of module serves a specific purpose and operates within a larger system.

4. How do scientists determine which representations are most accurate for a given module?

Determining the most accurate representation for a module often involves a combination of experimental data, mathematical models, and theoretical frameworks. Scientists may also use multiple representations to gain a more comprehensive understanding of a module and its interactions within a larger system.

5. What are the potential challenges with terminology issues surrounding modules and representations?

One potential challenge with terminology issues in this context is the potential for confusion or misinterpretation, as different fields of science may use different terminology for similar concepts. Additionally, the use of jargon and specialized language can make it difficult for non-experts to understand these concepts. It is important for scientists to clearly define and explain their terminology to ensure effective communication and understanding among colleagues and the general public.

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