Representation theory

1. Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. It explores how objects can be represented by simpler objects, such as matrices and linear transformations, and how these representations can provide insights into the structure and properties of the original objects. Representation theory has applications in various fields, including physics, computer science, and geometry, where it helps to understand complex structures by breaking them down into simpler components. Overall, representation theory plays a fundamental role in modern mathematics by providing powerful tools for studying and analyzing a wide range of mathematical structures.

What is a representation of a group?

A) An interpretation of group actions with graphs.
B) A text-based description of group operations.
C) A way to visually illustrate group elements.
D) A homomorphism from the group to the general linear group of a vector space.

2. What is an irreducible representation?

A) A representation with orthogonal basis vectors.
B) A representation with linearly independent elements.
C) A representation that has no non-trivial invariant subspaces.
D) A representation using complex numbers only.

3. In representation theory, what is the character of a representation?

A) The eigenvalues of the representation matrix.
B) The determinant of the matrix representing a group element.
C) The dimension of the vector space.
D) The trace of the matrix representing a group element.

4. What is the goal of studying representations of infinite-dimensional groups?

A) To analyze financial time series.
B) To develop geometric algorithms.
C) To solve partial differential equations.
D) To understand symmetry in quantum mechanics.

5. What is the concept of a unitary representation in representation theory?

A) A representation that preserves an inner product.
B) A representation using only unit vectors.
C) A representation with unity as a group element.
D) A representation with one element in each row and column.

6. What is the role of Schur functors in representation theory?

A) To describe geometric transformations.
B) To optimize matrices for numerical stability.
C) To analyze financial market data.
D) To classify representations of symmetric groups.

7. What is the center of a group in representation theory?

A) The set of elements that commute with all group elements.
B) The geometric center of a group representation.
C) The center of mass of all group elements.
D) The central point of a group element matrix.

8. What is meant by the term 'endomorphism' in representation theory?

A) A homomorphism of a group into itself.
B) A map between vector spaces.
C) A representation of a simple group.
D) A morphism from one group to another.

9. What is the adjoint representation of a Lie group?

A) The representation that corresponds to the group's Lie algebra.
B) A representation used in architectural design.
C) A representation involving adjacent matrices.
D) A representation with adjoint angles.

10. What is the relationship between representation theory and quantum mechanics?

A) Representation theory helps analyze symmetries and observables in quantum systems.
B) Representation theory creates quantum entanglement.
C) Representation theory measures quantum fluctuations.
D) Representation theory predicts quantum tunneling.

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