Group Representations and Young Tableaux

In summary, there are good resources available for understanding Young diagrams and tableaux for representations of the permutation groups Sn and the unitary groups U(n) of n x n unitary matrices. William Fulton's book "Young Tableaux" is a useful resource, as well as the book "Symmetry Groups" by Sagan. These resources explain how Young tableaux are used to figure out the irreducible representations of these groups and also provide a deeper understanding of why they work. Additionally, the book by Fulton also covers other related topics such as flag varieties and Schubert calculus.
  • #1
AKG
Science Advisor
Homework Helper
2,567
4
What are good resources on Young diagrams and tableaux for representations of the permutation groups Sn and the unitary groups U(n) of n x n unitary matrices?
 
Physics news on Phys.org
  • #2
William Fulton has an entire book on S_n (and matrix group) stuff. It's an LMS student text called something like 'Young Tableaux'
 
  • #3
Thanks, I'll look into that. My professor says that people use Young tableaux to figure out the irreducible representations of the symmetric groups and the unitary groups. However, he says that although people know the applications of Young tableaux, and have proven that they work, there's not much of an understanding as to why they work. As a project, he has asked me to try and figure it out, if possible. He recommended a book by an author named Sagan (forget the title, or the author's first name, but I have the book on hold). Do any resources come to mind that would help with this particular thing, i.e. figuring out why Young tableaux work? Does the Fulton book do this?
 
  • #4
I went looking for the text for a couple of courses that I took as a student and found it. This book was old when I used it, and is even older now. It looks much denser than I remember and may not be of much use to you. In any case, it is now in the http://www.ima.umn.edu/~miller/symmetrygroups.html" [Broken]. Chapters 4 and 9 may relevant.

Regards,
George
 
Last edited by a moderator:
  • #5
Hmm, I hate these 'why' questions. I'd say we understand rather well why Y.T. parametrize the reps of S_n, in fact we know a hell of a lot about them (they also parametrize modular reps and one can do many things involving hook lengths, p-cores, abacaus stuff..): they have a natural action on lablellings by S_n hence they are an S_n permutation module, they have an ordering respected by the action and induction and it all seems quite clear why to me, and yes Fulton explains all of this and far much more geometry besides (flag varieties and schubert calculus).
 

1. What are group representations and Young tableaux?

Group representations are mathematical tools used to study symmetries in physical systems. They involve representing elements of a group (such as rotations or reflections) as matrices, which can then be used to understand how the group acts on different objects. Young tableaux are graphical representations of these matrices, which can help visualize and analyze group representations.

2. How are group representations and Young tableaux related?

Young tableaux are a visual representation of group representations. They are constructed using a set of rules that correspond to the structure of a group, and they can provide insight into the properties and symmetries of a group representation. In some cases, Young tableaux can also be used to determine the irreducible representations of a group.

3. What is the significance of Young tableaux in physics?

Young tableaux are important in physics because they provide a way to understand the symmetries of physical systems. They are used in fields such as quantum mechanics, particle physics, and solid state physics to analyze and classify the behavior of particles and other physical objects under different symmetries.

4. How are Young tableaux used in solving problems?

Young tableaux are often used as a problem-solving tool in mathematics and physics. They can help to simplify complex matrix calculations and make it easier to identify patterns and symmetries in a given problem. Young tableaux can also be used to determine the eigenvalues and eigenvectors of a matrix, which can be useful in solving equations and understanding the behavior of physical systems.

5. Are there any real-world applications of group representations and Young tableaux?

Yes, there are many real-world applications of group representations and Young tableaux. They are used in various fields of physics, such as quantum mechanics, solid state physics, and particle physics, to study the symmetries of physical systems. They are also used in chemistry to understand the properties of molecules and in computer science for data compression and error correction. Additionally, group representations and Young tableaux have applications in other areas of mathematics, such as representation theory and algebraic geometry.

Similar threads

  • Linear and Abstract Algebra
Replies
1
Views
463
  • Linear and Abstract Algebra
Replies
1
Views
864
  • Linear and Abstract Algebra
Replies
3
Views
1K
Replies
13
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
2K
  • Linear and Abstract Algebra
Replies
3
Views
688
  • Linear and Abstract Algebra
Replies
12
Views
3K
  • Linear and Abstract Algebra
Replies
14
Views
2K
  • Linear and Abstract Algebra
Replies
2
Views
873
Back
Top