Reference Request: Product Of Representations And Characters

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Introduction

As a beginner in representation theory and character theory, it can be overwhelming to navigate the vast array of concepts and theorems that underlie these fields. When studying the representation theory of finite groups, one of the key topics that arises is the product of representations and characters. In this article, we will explore this topic in detail, providing a comprehensive overview of the relevant concepts and theorems.

Background

Representation theory is a branch of abstract algebra that studies the linear representations of groups. A representation of a group G is a homomorphism from G to the general linear group GL(V) of a vector space V. In other words, it is a way of assigning a linear transformation to each element of the group in a way that preserves the group operation.

Characters are a fundamental concept in representation theory, and they play a crucial role in the study of the product of representations. A character of a representation ρ of a group G is a function χρ: G → ℂ that assigns to each element g of G the trace of the linear transformation ρ(g). In other words, it is a function that assigns to each element of the group the sum of the diagonal entries of the matrix representing the linear transformation.

Product of Representations

The product of representations is a fundamental concept in representation theory, and it is defined as follows:

Let ρ1 and ρ2 be two representations of a group G on vector spaces V1 and V2, respectively. The product representation ρ1 ⊗ ρ2 is a representation of G on the tensor product space V1 ⊗ V2, defined by:

(ρ1 ⊗ ρ2)(g) = ρ1(g) ⊗ ρ2(g)

for all g in G.

Product of Characters

The product of characters is a fundamental concept in character theory, and it is defined as follows:

Let χ1 and χ2 be two characters of representations ρ1 and ρ2 of a group G, respectively. The product character χ1 ⊗ χ2 is a character of the product representation ρ1 ⊗ ρ2, defined by:

(χ1 ⊗ χ2)(g) = χ1(g)χ2(g)

for all g in G.

Properties of the Product of Representations

The product of representations has several important properties, which are listed below:

  • Associativity: The product of representations is associative, meaning that (ρ1 ⊗ ρ2) ⊗ ρ3 = ρ1 ⊗ (ρ2 ⊗ ρ3) for all representations ρ1, ρ2, and ρ3 of a group G.
  • Distributivity: The product of representations is distributive over addition, meaning that ρ1 ⊗ (ρ2 + ρ3) = ρ1 ⊗ ρ2 + ρ1 ⊗ ρ3 for all representations ρ1, ρ2, and ρ3 of a group G.
  • Unitarity: The product of representations is unitary, meaning that the character of the product representation is the product of the characters of the individual representations.

Properties of the Product of Characters

The product characters has several important properties, which are listed below:

  • Associativity: The product of characters is associative, meaning that (χ1 ⊗ χ2) ⊗ χ3 = χ1 ⊗ (χ2 ⊗ χ3) for all characters χ1, χ2, and χ3 of representations of a group G.
  • Distributivity: The product of characters is distributive over addition, meaning that χ1 ⊗ (χ2 + χ3) = χ1 ⊗ χ2 + χ1 ⊗ χ3 for all characters χ1, χ2, and χ3 of representations of a group G.
  • Unitarity: The product of characters is unitary, meaning that the character of the product representation is the product of the characters of the individual representations.

Applications of the Product of Representations

The product of representations has several important applications in representation theory and character theory, which are listed below:

  • Induced Representations: The product of representations is used to construct induced representations, which are representations of a group G that are induced from representations of a subgroup H.
  • Tensor Products: The product of representations is used to construct tensor products of representations, which are representations of a group G that are constructed from the tensor product of two representations.
  • Characters: The product of characters is used to construct characters of representations, which are functions that assign to each element of the group the sum of the diagonal entries of the matrix representing the linear transformation.

Conclusion

In conclusion, the product of representations and characters is a fundamental concept in representation theory and character theory. The product of representations is a way of constructing new representations from existing ones, and it has several important properties, including associativity, distributivity, and unitarity. The product of characters is a way of constructing new characters from existing ones, and it has several important properties, including associativity, distributivity, and unitarity. The product of representations and characters has several important applications in representation theory and character theory, including induced representations, tensor products, and characters.

References

  • Steinberg, B. (2012). Representation Theory of Finite Groups: An Introductory Approach. World Scientific Publishing.
  • Serre, J. P. (1977). Linear Representations of Finite Groups. Springer-Verlag.
  • Fulton, W. (1987). Young Tableaux: With Applications to Representation Theory and Geometry. Cambridge University Press.

Further Reading

For further reading on the product of representations and characters, we recommend the following resources:

  • Representation Theory of Finite Groups: This is a comprehensive textbook on the representation theory of finite groups, written by Benjamin Steinberg.
  • Linear Representations of Finite Groups: This is a comprehensive textbook on the linear representations of finite groups, written by Jean-Pierre Serre.
  • Young Tableaux: This is a comprehensive textbook on Young tableaux, written by William Fulton.

Introduction

In our previous article, we explored the concept of the product of representations and characters in representation theory and character theory. In this article, we will answer some frequently asked questions about this topic, providing a comprehensive overview of the key concepts and theorems.

Q: What is the product of representations?

A: The product of representations is a way of constructing new representations from existing ones. Given two representations ρ1 and ρ2 of a group G on vector spaces V1 and V2, respectively, the product representation ρ1 ⊗ ρ2 is a representation of G on the tensor product space V1 ⊗ V2, defined by:

(ρ1 ⊗ ρ2)(g) = ρ1(g) ⊗ ρ2(g)

for all g in G.

Q: What is the product of characters?

A: The product of characters is a way of constructing new characters from existing ones. Given two characters χ1 and χ2 of representations ρ1 and ρ2 of a group G, respectively, the product character χ1 ⊗ χ2 is a character of the product representation ρ1 ⊗ ρ2, defined by:

(χ1 ⊗ χ2)(g) = χ1(g)χ2(g)

for all g in G.

Q: What are the properties of the product of representations?

A: The product of representations has several important properties, including:

  • Associativity: The product of representations is associative, meaning that (ρ1 ⊗ ρ2) ⊗ ρ3 = ρ1 ⊗ (ρ2 ⊗ ρ3) for all representations ρ1, ρ2, and ρ3 of a group G.
  • Distributivity: The product of representations is distributive over addition, meaning that ρ1 ⊗ (ρ2 + ρ3) = ρ1 ⊗ ρ2 + ρ1 ⊗ ρ3 for all representations ρ1, ρ2, and ρ3 of a group G.
  • Unitarity: The product of representations is unitary, meaning that the character of the product representation is the product of the characters of the individual representations.

Q: What are the properties of the product of characters?

A: The product of characters has several important properties, including:

  • Associativity: The product of characters is associative, meaning that (χ1 ⊗ χ2) ⊗ χ3 = χ1 ⊗ (χ2 ⊗ χ3) for all characters χ1, χ2, and χ3 of representations of a group G.
  • Distributivity: The product of characters is distributive over addition, meaning that χ1 ⊗ (χ2 + χ3) = χ1 ⊗ χ2 + χ1 ⊗ χ3 for all characters χ1, χ2, and χ3 of representations of a group G.
  • Unitarity: The product of characters is unitary, meaning that the character of the product representation is the product of the characters of the individual representations.

Q: What are the applications of the product of representations?

A: The product of representations has important applications in representation theory and character theory, including:

  • Induced Representations: The product of representations is used to construct induced representations, which are representations of a group G that are induced from representations of a subgroup H.
  • Tensor Products: The product of representations is used to construct tensor products of representations, which are representations of a group G that are constructed from the tensor product of two representations.
  • Characters: The product of characters is used to construct characters of representations, which are functions that assign to each element of the group the sum of the diagonal entries of the matrix representing the linear transformation.

Q: How do I use the product of representations in my research?

A: The product of representations is a powerful tool in representation theory and character theory, and it can be used in a variety of ways in your research. Here are a few examples:

  • Constructing new representations: The product of representations can be used to construct new representations from existing ones.
  • Analyzing characters: The product of characters can be used to analyze the characters of representations, which can be useful in a variety of applications.
  • Inducing representations: The product of representations can be used to induce representations from subgroups, which can be useful in a variety of applications.

Conclusion

In conclusion, the product of representations and characters is a fundamental concept in representation theory and character theory, and it has several important properties and applications. We hope that this article has been helpful in answering your questions about this topic, and that it has provided a comprehensive overview of the key concepts and theorems.

References

  • Steinberg, B. (2012). Representation Theory of Finite Groups: An Introductory Approach. World Scientific Publishing.
  • Serre, J. P. (1977). Linear Representations of Finite Groups. Springer-Verlag.
  • Fulton, W. (1987). Young Tableaux: With Applications to Representation Theory and Geometry. Cambridge University Press.

Further Reading

For further reading on the product of representations and characters, we recommend the following resources:

  • Representation Theory of Finite Groups: This is a comprehensive textbook on the representation theory of finite groups, written by Benjamin Steinberg.
  • Linear Representations of Finite Groups: This is a comprehensive textbook on the linear representations of finite groups, written by Jean-Pierre Serre.
  • Young Tableaux: This is a comprehensive textbook on Young tableaux, written by William Fulton.