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See: Symmetric group representations, General linear group representations

表示论

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• 影片: A gentle introduction to group representation theory -Peter Buergisser
• 影片: Representation Theory Max Planck

读物

Overview

• Stack Exchange: Importance of Representation Theory Given a space X and the space of functions on it, F(X), the decomposition of F(X) is highly nontrivial and, given various Lie groups, leads to Fourier series X=S, Fourier transform X=R, spherical harmonics X=SO(3), modular forms X=SL2(R)/SL2(Z).
• Groups and Representation Theory by Kevin McGerty
• Google: representation theory
• Bill Casselman. Representations of finite groups

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