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• Amritanshu Prasad. Representation Theory: A Combinatorial Point of View
• Monica Vazirani. Representation Theory & Combinatorics of the Symmetry Group and Related Structures.
  • Lecture 1: Introduction
  • Lecture 2: Induction and Restriction
  • Lecture 3: Jucys-Murphy operators
• Math with MS. Representations of finite groups.
  • Lecture 22: Specht representations of symmetric groups

Literature

• Representations of the Symmetric Group Daphne Kao - my favorite
• GL_t for t is not an integer
• Irreducible Representations of the Symmetric Group and the General Linear Group Abhinav Verma
• Irreducible Representations of the Symmetric Group Redmond McNamara
• VGTU biblioteka: Young Tableaux: With Applications to Representation Theory and Geometry. William Fulton.
• Amritanshu Prasad
• Alain Lascoux. Young representations of the symmetric group.

Symmetric group, general linear group

• Andy Yu, Joshua Wiscons. Representation Theory of the Symmetric Group. Seminar Notes.
• Slides: Group Representations and Symmetric Functions by Bruce Sagan
• Book: "The Symmetric Group: Representations, Combinatorial Algorithms and Symmetric Functions" by Bruce Sagan
• Google: symmetric group general linear group representation theory
• Google: representation theory symmetric group
• Google: representation theory general linear group
• Google: Schur Weyl duality

Wikipedia

• Wikipedia: Representation theory of the symmetric group
• Wikipedia: Young symmetrizer
• Wikipedia: Specht module
• Schur functor
• Exterior algebra
• Symmetric power
• Plethysm
• Jucys–Murphy elements

Theory

{$S_{n}$} is the symmetric group on n letters. The irreducible representations of {$S_{n}$} are indexed by the conjugacy classes, which is to say, the partitions λ. Given a partition λ of the numbers 1,...,n, which is to say, a Young diagram filled with numbers, define {$R_{ \lambda }$} to be the permutations {$e_{\sigma}$} which preserve the numbers in each row, and {$C_{ \lambda }$} to be the permutations {$e_{\tau}$} which preserve the numbers in each column. Define:

{$$ s_{\lambda} = {\sum_{\sigma \in R_{ \lambda }}} e_{\sigma} \sum_{ \tau \in C_{ \lambda }} sgn(\tau) e_{\tau} $$}

Then the subspaces {$\mathbb{C}S_{n}\cdot s_{\lambda}$} are the irreducible representations indexed by λ.

For example, for the partition 21 filled [12][3] we have that

{$$s_{\lambda} = e_\imath + e_{13} - e_{12} + e_{132} $$}

Defining {$ A = e_\imath - e_{13}, B = e_{132} - e_{12}, C = e_{123} - e_{23} $} we have

{$ s_{\lambda} = A-B $}

{$ \imath \cdot s_{\lambda} = A-B $}

{$ e_{12} \cdot s_{\lambda} = A-B $}

{$ e_{13} \cdot s_{\lambda} = C-A $}

{$ e_{23} \cdot s_{\lambda} = B-C $}

{$ e_{123} \cdot s_{\lambda} = C-A $}

{$ e_{132} \cdot s_{\lambda} = B-C $}

We have a three-cycle that can be written with 2x2 matrices acting on basis vectors A-B and B-C. Study the three-cycle!

Discussion

There is a natural numbering of the cells in a partition based on their intepretation as paths in Pascal's triangle. In this numbering we assign numbers part by part. Within the part there may be a secondary numbering. The assigning of numbers part by part can go in either direction and is dual in that sense.

It's important here to conceive how sign comes to play in going to different parts.

In working with standard tableaux we use a very different numbering based on the innermost corner and building out in both directions. This numbering can be thought of as an internal point of view, in conditions, in context, not from the top of Pascal's triangle. I should investigate how these relative and absolute perspectives relate.

Topics to Learn

From Amritanshu Prasad. Representation Theory: A Combinatorial Viewpoint

Permutation Representations

• Group Actions and Permutation Representations
• Permutations
• Intertwining Permutation Representations
• Subset Representations
• Intertwining Partition Representations

The RSK Correspondence

• Semistandard Young Tableaux
• The RSK Correspondence
• Classification of Simple Representations of Sn

Character Twists

• Inversions and the Sign Character
• Twisting by a Multiplicative Character
• Conjugate of a Partition
• Twisting by the Sign Character
• The Dual RSK Correspondence
• Representations of Alternating Groups

Symmetric Functions

• The Ring of Symmetric Functions
• Other Bases for Homogeneous Symmetric Functions
• Specializations to m Variables
• Schur Functions and the Frobenius Character Formula
• Frobenius' Characteristic Function
• Branching Rules
• Littlewood-Richardson Coefficients
• The Hook-Length Formula
• The Involution s-lambda -> S-
• The Jacobi-Trudi Identities
• The Recursive Murnaghan-Nakayama Formula
• Character Values of Alternating Groups