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Andrius Kulikauskas

  • m a t h 4 w i s d o m - g m a i l
  • +370 607 27 665
  • My work is in the Public Domain for all to share freely.

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  • 读物 书 影片 维基百科

Introduction E9F5FC

Questions FFFFC0

Software

See: Symmetric group representations, General linear group representations

表示论


https://www.math4wisdom.com/wiki/Research/RepresentationTheory?action=edit

  • 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?
  • Think about commutativity and noncommutativity, the exterior algebra.
  • Think about (A-B)^n. This relates to the Euler characteristic.

Open problem: Calculating plethysm.


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读物

Overview

Observations

We have {$$\begin{pmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{pmatrix} \begin{pmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{pmatrix}= \begin{pmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{pmatrix}$$} However, the constant map from a group {$G$} to this matrix is not a representation. The reason is that this is not an invertible matrix. A representation takes us from {$G$} to {$GL(V)$} and the latter consists of invertible matrices.

Note that a subspace {$W$} of {$V$} typically does not have a unique complement. Consider, for example, one-dimensional {$W$} and two-dimensional {$V$}. There are many different lines that can be the complement of {$W$}. Together {$W$} and its complement span {$V$}.

Given a representation {$\theta$} and a scalar {$\alpha$}, the function {$\alpha\theta$} is not, in general, a representation because typically {$\alpha^2\neq\alpha$}.

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.

Geometry and logic

  • Permutation matrices act on a vector space. This can be thought of in terms of both logic-combinatorics and geometry. Logically, we have labels, and geometrically, they are directions-dimensions. In either case, the labels or directions are assured to be all different.

Compact group

Representation of a compact group is based on an integral measure that is translatable. With my orthogonal polynomials, the group is not compact, so the measure is not translatable.

Topics to Learn

From Amritanshu Prasad. Representation Theory: A Combinatorial Viewpoint

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
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This page was last changed on September 11, 2021, at 03:06 PM