What Do You Know About Steinberg's Formula?

Steinberg's formula was introduced in 1961.

A closed-form expression is a mathematical expression that can be evaluated in a finite number of operations. It is a formula that directly gives the value of a function or a number, without the need for iterative calculations or approximations. It can be written using a finite number of standard mathematical operations, such as addition, subtraction, multiplication, division, exponentiation, and logarithms. Closed-form expressions are often preferred because they provide a concise and exact representation of a mathematical relationship or solution.

The Weyl character formula is a parameter of the Steinberg's formula that describes the characters of irreducible representations of compact Lie groups in terms of their highest weights. It provides a way to calculate the characters of these representations, which are important in the study of Lie groups and their representations. The formula relates the characters to the weights of the representations, allowing for a deeper understanding of the structure and properties of these groups.

In the context of a compact Lie group, the dimension of a maximal torus is referred to as the "rank." The rank represents the maximum number of linearly independent elements in the Cartan subalgebra of the Lie algebra associated with the Lie group. It is a fundamental concept in the theory of Lie groups and Lie algebras, and plays a crucial role in various aspects of their study and classification. Therefore, "rank" is the correct answer for the dimension of a maximal torus in a compact Lie group.

A Coxeter group is an abstract group that can be described in terms of reflections or kaleidoscopic mirrors. It is named after H.S.M. Coxeter, who extensively studied these groups. Coxeter groups have a set of generators and relations that define their structure, and they have many interesting properties and applications in various areas of mathematics, such as geometry and algebra.

The Weyl character formula was first proved in 1925.

The Peter-Weyl theorem states that any finite-dimensional unitary representation of a compact group can be decomposed into a direct sum of irreducible representations. This decomposition can be written as a direct sum of a finite number of parts, where each part corresponds to a different irreducible representation. Therefore, the correct answer is 3, indicating that the Peter-Weyl theorem has three parts in its decomposition.

A compact, connected, abelian Lie subgroup in a compact Lie group is referred to as a torus. This is because a torus is a type of compact, connected, abelian Lie subgroup that can be found in a compact Lie group.

A Dynkin diagram is a type of graph that has some edges doubled or tripled. It is a mathematical tool used in the study of Lie algebras and root systems. The diagram represents the structure of the algebra or system, with nodes representing the simple roots and edges representing the relationships between them. The doubling or tripling of edges indicates certain special properties or symmetries within the system.

The Peter-Weyl theorem states that any unitary representation of a compact Lie group can be decomposed into a direct sum of irreducible representations. This means that the representation can be completely reduced into its irreducible components. Therefore, the correct answer is "The complete reducibility of unitary representations of a compact Lie group."