概论

数学

发现

Andrius Kulikauskas

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

用中文

  • 读物 书 影片 维基百科

Software


In what sense are there six natural bases for the symmetric functions?


  • How are the symmetric functions expressing the fundamentals of representation theory, linear algebra and matrix multiplication?
  • How are the symmetric functions expressing the fundamentals of group theory, the symmetric group and permutations?
  • How are the symmetric functions relating automorphisms of lists and automorphisms of sets?

Symmetric functions

  • 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?
  • Do the six natural bases of the symmetric functions correspond to the six transformations?
  • Understand the elementary symmetric functions in terms of the wedge product. And the homogeneous symmetric functions in terms of the inner product?
  • In category theory, where do symmetric functions come up? What are eigenvalues understood as? What would be symmetric functions of eigenvalues?

Symmetric functions of the eigenvalues of a matrix

  • Make sense combinatorially of the map between the homogeneous functions of eigenvalues in terms of words and in terms of products of Lyndon words.

Sources

Notes

  • Relate the monomial, forgotten, Schur symmetric functions of eigenvalues with the matrices {$(I-A)^{-1}$} and {$e^A$}.
  • Exercise: Get the eigenvalues for a generic matrix: 2x2, 3x3, etc.
  • Exercise: Look for a method to find the eigenvalues for a generic matrix. Express the solving of the equation as a way of relating the elementary functions. Are they related to the inverse Kostka matrix? And the impossibility of a combinatorial solution? And the nondeterminism issue, P vs NP?
  • Think again about the combinatorial intepretation of {$K^{-1}K=I$}.
  • What do the constraints on Lie groups and Lie algebras say about symmetric functions of eigenvalues.
  • Vandermonde determinant shows invertible - basis for finite Fourier transform
  • What is the connection between the universal grammar for games and the symmetric functions of the eigenvalues of a matrix?
  • How do symmetries of paths relate to symmetries of young diagrams

Partitions, duality, tableaux

Kostka matrix

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Šis puslapis paskutinį kartą keistas October 26, 2020, at 07:28 PM