6.1 Definition and construction of induced representations
2 min read•july 25, 2024
Induced representations are a powerful tool in representation theory, allowing us to construct new representations of larger groups from existing ones of smaller subgroups. They bridge the gap between and larger group representations, revealing hidden connections.
The construction of induced representations involves tensor products or coset spaces, with both methods yielding results. The of an is calculated using a simple formula, reflecting the relative sizes of the original and larger groups.
Induced Representations: Definition and Construction
Role of induced representations
Top images from around the web for Role of induced representations
group theory - Lattice of subgroups - Mathematics Stack Exchange View original
Is this image relevant?
Representation Theory [The Physics Travel Guide] View original
Is this image relevant?
group theory - Lattice of subgroups - Mathematics Stack Exchange View original
Is this image relevant?
Representation Theory [The Physics Travel Guide] View original
Is this image relevant?
1 of 2
Top images from around the web for Role of induced representations
group theory - Lattice of subgroups - Mathematics Stack Exchange View original
Is this image relevant?
Representation Theory [The Physics Travel Guide] View original
Is this image relevant?
group theory - Lattice of subgroups - Mathematics Stack Exchange View original
Is this image relevant?
Representation Theory [The Physics Travel Guide] View original
Is this image relevant?
1 of 2
- Induced representations construct new representations from existing ones creating representations of larger groups from smaller subgroup representations
- Bridge representations of subgroups and larger groups analyzing structure of larger groups through subgroups (symmetric groups)
- Provide tool for studying relationships between different group representations revealing connections between seemingly unrelated representations
- Notation induces representation of subgroup to group facilitating clear communication in mathematical discussions
Construction of induced representations
- Tensor product construction forms starting with representation of subgroup
- represents group algebra of providing algebraic structure for induction
- Tensor product taken over ensures compatibility with subgroup action
- Coset space construction builds vector space with basis indexed by left cosets of in
- Choose representatives for left cosets of in
- Form vector space with basis indexed by these cosets
- Define action of on this space preserving coset structure
- Both methods yield isomorphic representations demonstrating consistency in construction approaches
Dimensions of induced representations
- Dimension formula calculates induced representation dimension
- denotes index of in counting number of left cosets
- represents dimension of original representation
- Factors affecting dimension include relative sizes of original and larger groups influencing complexity of induced representation
Examples of induced representations
- Permutation representations induce trivial representation of stabilizer subgroup resulting in action on cosets ( acting on -element subsets)
- Monomial representations induce one-dimensional representations of subgroups useful in theory (representations of dihedral groups)
- induces from trivial representation of trivial subgroup encompassing all irreducible representations
- constructs representations of symmetric groups via induction revealing combinatorial structure
- Parabolic induction in representation theory of reductive groups builds representations of Lie groups ()
Key Terms to Review (18)
Burnside's Lemma: Burnside's Lemma is a result in group theory that provides a way to count the number of distinct objects under the action of a group, using the concept of group orbits. It states that the number of distinct orbits of a set $X$ under a group $G$ is equal to the average number of points fixed by the elements of the group, which can be mathematically expressed as $$|X/G| = \frac{1}{|G|} \sum_{g \in G} |X^g|$$, where $|X^g|$ is the number of elements in $X$ fixed by the group element $g$. This lemma is particularly useful when studying representations and their induced representations.
Character: In representation theory, a character is a function that assigns to each group element the trace of its corresponding matrix representation. Characters provide deep insights into the structure of representations, revealing information about their irreducibility and symmetry properties.
Compact group: A compact group is a topological group that is both compact and Hausdorff, meaning it is closed and bounded in the context of its topology. Compact groups have significant implications in representation theory as they allow for the construction of induced representations and exhibit rich structures that simplify the analysis of their representations.
Dihedral group: The dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. It captures the essence of how geometric shapes can be manipulated while preserving their structure, making it a key example in understanding group actions and orbits, as well as representations of groups.
Dimension: Dimension in representation theory refers to the size of a vector space associated with a representation, specifically the number of basis vectors needed to span that space. This concept is crucial as it relates to understanding the structure of representations, particularly how they can be decomposed and analyzed, influencing topics such as irreducibility and induced representations.
Finite group: A finite group is a set equipped with a binary operation that satisfies the group properties (closure, associativity, identity, and invertibility) and has a finite number of elements. This concept is crucial for understanding various topics in representation theory, as the structure and properties of finite groups significantly influence their representations and character theory.
Frobenius: Frobenius refers to a fundamental concept in representation theory that connects group representations and the way representations can be induced from a subgroup to a larger group. This principle not only establishes a method for constructing representations but also leads to deeper insights about the relationships between different groups and their actions. The Frobenius reciprocity theorem plays a crucial role in this framework, providing a connection between induced and restricted representations, making it essential for understanding linear representations and their applications.
Frobenius Reciprocity: Frobenius reciprocity is a fundamental concept in representation theory that describes a relationship between induced representations and restricted representations of groups. It states that there is a natural correspondence between homomorphisms from an induced representation to a representation and homomorphisms from the original representation to the restricted representation, facilitating the transition between different levels of group representations.
Group Action: A group action is a formal way of describing how a group interacts with a set by associating each group element with a transformation of that set. This concept is crucial as it helps in understanding how symmetries can be represented and analyzed, linking group theory to geometry and other mathematical structures. Group actions lead to important ideas like orbits and stabilizers, allowing us to study the structure of groups in more depth.
Induced representation: Induced representation is a way of constructing a representation of a group from a representation of one of its subgroups. This process is crucial in understanding how different representations can relate to each other, especially when dealing with larger groups built from smaller components. By inducing representations, we can analyze how characters behave under group actions and relate them to properties of the larger group through fundamental theorems.
Irreducibility: Irreducibility refers to a property of representations where a representation cannot be decomposed into smaller, non-trivial representations. This concept is essential in understanding how representations function within the framework of group theory and helps to establish the structure of representations through induction and restrictions. Recognizing whether a representation is irreducible has significant implications for the analysis of induced representations and their properties, as well as for understanding the Frobenius reciprocity theorem.
Isomorphic: In representation theory, isomorphic refers to the relationship between two representations that have a one-to-one correspondence in structure and behavior, meaning they are essentially the same in terms of their algebraic properties. This concept implies that if two representations are isomorphic, they can be transformed into each other by a linear transformation, preserving the group structure and operations. Isomorphic representations allow mathematicians to classify and understand different representations by studying their similarities.
Maschke's Theorem: Maschke's Theorem states that if a finite group is acting on a finite-dimensional vector space over a field whose characteristic does not divide the order of the group, then every representation of the group can be decomposed into a direct sum of irreducible representations. This theorem is fundamental in understanding the structure of representations, as it guarantees that every representation can be analyzed and simplified into simpler components, which is crucial for studying linear representations, matrix representations, and group algebras.
Principal Series Representations: Principal series representations are a class of representations associated with a reductive group, typically arising from the induced representations of a parabolic subgroup. These representations are crucial in understanding the structure and properties of the group, as they provide insight into its representation theory and harmonic analysis.
Regular Representation: The regular representation of a group is a specific type of linear representation where the group acts on itself by left multiplication. This construction allows one to view the group as a matrix representation, which is particularly useful for analyzing its structure and understanding its representations more generally.
Subgroup: A subgroup is a subset of a group that itself forms a group under the same operation. This means that a subgroup must contain the identity element, be closed under the group operation, and contain the inverse of each of its elements. Understanding subgroups is essential as they help to analyze the structure of groups, reveal symmetries, and facilitate the study of representations and their properties.
Symmetric group: The symmetric group, denoted as $$S_n$$, is the group of all permutations of a finite set of $$n$$ elements, capturing the essence of rearranging objects. This group is fundamental in understanding how groups act on sets, with its elements representing all possible ways to rearrange the members of the set, leading to various applications in algebra and combinatorics.
Young's Orthogonal Form: Young's Orthogonal Form is a specific representation of a group that organizes the irreducible representations of the symmetric group in a structured way. It is especially useful in the context of induced representations, as it helps to systematically relate these representations to simpler components, allowing for a clearer understanding of their structure and relationships within representation theory.