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Introduction E9F5FC

Understandable FFFFFF

Questions FFFFC0

Notes EEEEEE

Software

• 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

#### SymmetricFunctions

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 Puslapis paskutinį kartą pakeistas 2019 sausio 20 d., 21:16