1.
Which theorem is used to prove the Schur–Weyl duality?
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.
2.
How many symmetries are involved in the puzzle?
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.
3.
Which of these does S(k) represent?
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).
4.
Which of these does GL(n) represent?
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."
5.
Which kind of information is the puzzle used for?
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.
6.
What are the representations of the symmetric group?
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.
7.
How do you represent k=2 and n>1?
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.
8.
What is the result of the trivial representation of S2?
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.
9.
Who discovered the puzzle?
Correct Answer
B. Hermann Weyl and Issai Schur
Explanation
Hermann Weyl and Issai Schur discovered the puzzle.
10.
What are the major groups used for determining the puzzle?
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.