What are the representations of the symmetric group?

Sign representation and trivial representation

Trivial representation and geometric representation

Mesh representation and sign representation

Fibonacci representation and sign representation

How do you represent k=2 and n>1?

The space of two tensor decomposes into two parts

The space of two tensors decomposes into one part

The space of one tensor decomposes into halves

The space of two tensor remain unchanged

What is the result of the trivial representation of S2?

Both symmetric and asymmetric tensors

What Do You Know About Schur–weyl Duality?

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Quizzes Created: 129 | Total Attempts: 37,979

Questions: 10 | Attempts: 118 | Updated: Mar 20, 2023

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What Do You Know About Schurweyl Duality? - Quiz

In the representation theory branch of mathematics, Schur–Weyl duality relates unchangeable limited dimensional representations of the general symmetric and linear groups and gatherings.
The mathematical theorem shapes a prototype circumstance in representation hypothesis including two sorts of symmetry that decide each other. To know more about Schur–Weyl duality, take this short quiz.

Questions and Answers

1.

Which theorem is used to prove the Schur–Weyl duality?
- A.
  
  Double centralizer theorem
- B.
  
  Fenz theorem
- C.
  
  Optimum centralized theorem
- D.
  
  Colligated theorem
Correct Answer
A. Double centralizer theorem

Explanation
The Double Centralizer Theorem is used to prove the Schur-Weyl duality. This theorem states that for a pair of subalgebras, the centralizer of one subalgebra in the other is equal to the centralizer of the other subalgebra in the first one. In the context of Schur-Weyl duality, this theorem is used to show the relationship between the symmetric group and the general linear group, which are two important groups in representation theory.

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2.

How many symmetries are involved in the puzzle?
- A.
  
  2
- B.
  
  4
- C.
  
  6
- D.
  
  8
Correct Answer
A. 2

Explanation
The correct answer is 2 because symmetries involve reflections or rotations that result in the same pattern. In this case, there are only two possible symmetries - a reflection and a rotation.

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3.

Which of these does S(k) represent?
- A.
  
  Symmetry group
- B.
  
  Symbol representation
- C.
  
  Synchronization group
- D.
  
  Keptic group
Correct Answer
A. Symmetry group

Explanation
S(k) represents the Symmetry group. The other options, symbol representation, synchronization group, and skeptic group, do not accurately describe what S(k) represents. The Symmetry group refers to the collection of all symmetries of a given object or mathematical structure, and it is commonly denoted as S(k).

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4.

Which of these does GL(n) represent?
- A.
  
  Sequential number
- B.
  
  Geometrical linear number
- C.
  
  Geographical linear number
- D.
  
  General linear number
Correct Answer
D. General linear number

Explanation
GL(n) represents the general linear group of n-dimensional matrices. It is a mathematical concept used in linear algebra to denote the set of invertible matrices of size n by n. The term "general" implies that the matrices can have any real or complex entries, and "linear" refers to the linearity properties of matrix operations. Therefore, the correct answer is "General linear number."

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5.

Which kind of information is the puzzle used for?
- A.
  
  Scientific information
- B.
  
  Mechanics information
- C.
  
  Quantum information
- D.
  
  Earth-related study
- E.
  
  Energetics information
Correct Answer
C. Quantum information

Explanation
The puzzle is used for quantum information. Quantum information refers to the study and manipulation of information stored in quantum systems. This field combines principles from quantum mechanics and computer science to develop new methods of processing and transmitting information. The puzzle likely involves concepts and problems related to quantum information theory and quantum computing.

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6.

What are the representations of the symmetric group?
- A.
  
  Trivial representation and geometric representation
- B.
  
  Sign representation and trivial representation
- C.
  
  Mesh representation and sign representation
- D.
  
  Fibonacci representation and sign representation
Correct Answer
B. Sign representation and trivial representation

Explanation
The symmetric group is a group that consists of all possible permutations of a set. The sign representation of the symmetric group is a representation that assigns a sign to each permutation based on whether it is an even or odd permutation. The trivial representation is a representation that assigns the identity element of the group to every element of the set. Therefore, the correct answer is sign representation and trivial representation.

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7.

How do you represent k=2 and n>1?
- A.
  
  The space of two tensors decomposes into one part
- B.
  
  The space of two tensor decomposes into two parts
- C.
  
  The space of one tensor decomposes into halves
- D.
  
  The space of two tensor remain unchanged
Correct Answer
B. The space of two tensor decomposes into two parts

Explanation
When representing k=2 and n>1, the space of two tensors decomposes into two parts. This means that the original space is divided into two separate parts or subspaces. Each subspace represents a different aspect or component of the tensors, allowing for a more detailed analysis or representation.

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8.

What is the result of the trivial representation of S2?
- A.
  
  Symmetric tensors
- B.
  
  Asymmetric tensors
- C.
  
  Both symmetric and asymmetric tensors
- D.
  
  Tangential tensors
Correct Answer
A. Symmetric tensors

Explanation
The result of the trivial representation of S2 is symmetric tensors. This means that all elements in the representation are symmetric under permutation of indices.

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9.

Who discovered the puzzle?
- A.
  
  Hermann Schur and Issai Weyl
- B.
  
  Hermann Weyl and Issai Schur
- C.
  
  Mattias Weyl and Isaac Schur
- D.
  
  Matthew Weyl and John Schur
Correct Answer
B. Hermann Weyl and Issai Schur

Explanation
Hermann Weyl and Issai Schur discovered the puzzle.

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10.

What are the major groups used for determining the puzzle?
- A.
  
  Linear and Symmetric
- B.
  
  Geometric and Arithmetic
- C.
  
  Finite and infinite
- D.
  
  Inequality and Assymetric
Correct Answer
A. Linear and Symmetric

Explanation
The major groups used for determining the puzzle are linear and symmetric. Linear puzzles involve a logical sequence or progression, where each step leads to the next in a linear fashion. Symmetric puzzles involve patterns or arrangements that are balanced and mirror each other. Both linear and symmetric groups provide a framework for solving puzzles by following logical patterns and identifying symmetrical arrangements.

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