Notes

Book

Math

Discovery

Andrius Kulikauskas

  • ms@ms.lt
  • +370 607 27 665
  • My work is in the Public Domain for all to share freely.

Lietuvių kalba

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.

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 2018 rugsėjo 12 d., 18:24
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