- Intuitively understand the Schur functions as the characters of the irreducible representations of the general linear groups.
- Discover a more natural way to express the Schur functions than the rim hook tableaux. Do this for both symmetric function and for the functions of the eigenvalues.
- Understand the relations of the Schur functions to the binomial theorem.
- Consider the Schur functions in terms of the Jacobi-Trudi formulas - and what it means for the eigenvalue case.
- Understand how the role of the Schur functions and rim hook tableaux depends on the characteristic k of the field.
- Understand the combinatorics underlying the map between elementary (< decreasing slack) and homogeneous (<= increasing slack) bases, especially as it works in taking the power (=) basis to define the Schur (< x <=) bases. Consider this also in the case of the eigenvalues of a matrix. We also have a foursome, perhaps: Schur - monomial - forgotten? - power. The human bases - monomial and forgotten - map to elementary and homogeneous?
- Make sense combinatorially of the map between the homogeneous functions of eigenvalues in terms of words and in terms of products of Lyndon words.
- Consider what can be said of a postive definite matrix of eigenvalues of a matrix.
My Ph.D. thesis: Symmetric Functions of the Eigenvalues of a Matrix
Sources
Representation theory of the symmetric group
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