See: Symmetric group representations, General linear group representations

Open problem: Calculating plethysm.

What do we know about matrices whose power to some N equals 1? Then the power symmetric function equals ... ? And the other symmetric functions of the eigenvalues...?

What can we say about the representation theory of Coxeter groups?

Videos

- A gentle introduction to group representation theory -Peter Buergisser
- Google: symmetric group general linear group representation theory
- Google: representation theory
- Google: representation theory symmetric group
- Google: representation theory general linear group
- Google: Schur Weyl duality

Texts

- Groups and Representation Theory by Kevin McGerty
- Slides: Group Representations and Symmetric Functions by Bruce Sagan
- Book: "The Symmetric Group: Representations, Combinatorial Algorithms and Symmetric Functions" by Bruce Sagan

Wikipedia

Think about commutativity and noncommutativity, the exterior algebra.

Think about (A-B)^n. This relates to the Euler characteristic.

Elements of a noncyclic group can't be mapped to corresponding roots of unity if there are elements whose orders are relatively prime. For then you could combine the two roots of unity to get a generator that is a product of the two orders.

**Representation Theory: A Combinatorial Viewpoint** Amritanshu Prasad

Basic Concepts of Representation Theory

- Representations and Modules
- Invariant Subspaces and Simplicity
- Complete Reducibility
- Maschke's Theorem
- Decomposing the Regular Module
- Tensor Products
- Characters
- Representations over Complex Numbers

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

Representations of General Linear Groups

- Polynomial Representations
- Schur Algebras
- Schur Algebras and Symmetric Groups
- Modules of a Commutant
- Characters of the Simple Representations
- Polynomial Representations of the Torus
- Weight Space Decompositions

Parsiųstas iš http://www.ms.lt/sodas/Book/RepresentationTheory

Puslapis paskutinį kartą pakeistas 2020 sausio 27 d., 21:41