🧩Representation Theory Unit 8 – Tensor Products of Representations
Tensor products are a crucial operation in linear algebra, combining vector spaces to create new ones. They're fundamental in representation theory, allowing us to construct new representations from existing ones and study how abstract algebraic structures can be represented as linear transformations.
Understanding tensor products is key to grasping advanced concepts in quantum mechanics, algebraic geometry, and more. They're used to decompose representations, define characters, and construct important algebras. Mastering tensor products opens doors to exploring complex mathematical structures and their applications.
Tensor products are a fundamental operation in linear algebra that combine two vector spaces to create a new vector space
Given two vector spaces V and W, their tensor product is denoted as V⊗W
Elements of V⊗W are linear combinations of simple tensors v⊗w, where v∈V and w∈W
Tensor products have a universal property that allows them to be defined uniquely up to isomorphism
Representation theory studies how abstract algebraic structures (groups, algebras) can be represented as linear transformations on vector spaces
Tensor products play a crucial role in constructing new representations from existing ones
The dimension of the tensor product space is the product of the dimensions of the individual spaces: dim(V⊗W)=dim(V)⋅dim(W)
Tensor products are associative: (U⊗V)⊗W≅U⊗(V⊗W)
The tensor product of two finite-dimensional representations is also a representation
Tensor Product Basics
The tensor product of two vectors v∈V and w∈W is denoted as v⊗w
This is a simple tensor, which forms a basis for the tensor product space V⊗W
Tensor products are bilinear, meaning they are linear in each argument separately
(av1+bv2)⊗w=a(v1⊗w)+b(v2⊗w)
v⊗(cw1+dw2)=c(v⊗w1)+d(v⊗w2)
The tensor product of two linear maps f:V→V′ and g:W→W′ is a linear map f⊗g:V⊗W→V′⊗W′
Defined by (f⊗g)(v⊗w)=f(v)⊗g(w)
Tensor products are distributive over direct sums: (V1⊕V2)⊗W≅(V1⊗W)⊕(V2⊗W)
The tensor product of a vector space with a field k is isomorphic to the original vector space: V⊗k≅V
Constructing Tensor Products
To construct the tensor product of two vector spaces V and W, start with the free vector space generated by the Cartesian product V×W
Elements of this space are formal linear combinations of pairs (v,w), where v∈V and w∈W
Impose bilinearity relations on this space to obtain the tensor product space V⊗W
(av1+bv2,w)=a(v1,w)+b(v2,w)
(v,cw1+dw2)=c(v,w1)+d(v,w2)
The resulting space V⊗W is the quotient of the free vector space by the subspace generated by these relations
Given bases {vi} for V and {wj} for W, the tensor products vi⊗wj form a basis for V⊗W
This basis is called the tensor product basis
The dimension of V⊗W is the product of the dimensions of V and W
Tensor products can be constructed for more than two vector spaces by iterating the construction
Properties of Tensor Products
Tensor products are associative: (U⊗V)⊗W≅U⊗(V⊗W)
This allows for unambiguous notation of multiple tensor products: U⊗V⊗W
Tensor products are distributive over direct sums: (V1⊕V2)⊗W≅(V1⊗W)⊕(V2⊗W)
The tensor product of a vector space with a field k is isomorphic to the original vector space: V⊗k≅V
The tensor product of two finite-dimensional representations is also a representation
If ρ:G→GL(V) and σ:G→GL(W) are representations, then ρ⊗σ:G→GL(V⊗W) is a representation
Tensor products are compatible with dual spaces: (V⊗W)∗≅V∗⊗W∗
The tensor product of two irreducible representations is generally reducible
Decomposing tensor products of irreducible representations is a central problem in representation theory
Applications in Representation Theory
Tensor products are used to construct new representations from existing ones
The tensor product of two representations is also a representation
Decomposing tensor products of irreducible representations into irreducible components is a fundamental problem
Clebsch-Gordan coefficients describe this decomposition for SU(2) representations
Tensor products are used to define the character of a representation
The character of a representation ρ:G→GL(V) is the trace of the linear map ρ(g) for each g∈G
Tensor products appear in the definition of the tensor algebra and exterior algebra of a vector space
These algebras have important applications in physics and geometry
Tensor products are used to construct invariant theory and study the invariants of group actions
Schur functors, which are certain functors involving tensor products, are used to construct irreducible representations of the general linear group GL(V)
Computational Techniques
Computing tensor products of large matrices can be computationally expensive due to the high dimension of the resulting space
Efficient algorithms for tensor product computations often exploit the structure of the input matrices
Kronecker product formula: (A⊗B)(C⊗D)=(AC)⊗(BD)
Tensor network methods represent high-dimensional tensors as networks of lower-dimensional tensors
This can reduce computational complexity and memory requirements
Singular value decomposition (SVD) can be used to compress and approximate tensor products
Higher-order SVD (HOSVD) generalizes this to tensors of arbitrary order
Tensor decomposition methods, such as CP decomposition and Tucker decomposition, express a tensor as a sum of simpler tensors
These methods can reveal underlying structure and reduce dimensionality
Symbolic computation software, such as Mathematica and SymPy, can perform tensor product computations and simplifications
Examples and Case Studies
The tensor product of two vector spaces Rn and Rm is isomorphic to the space of n×m matrices Rn×m
Matrix multiplication can be interpreted as a tensor product operation
In quantum mechanics, the state space of a composite system is the tensor product of the state spaces of the individual systems
Entanglement arises from the tensor product structure of the state space
The tensor product of two representations of the Lie algebra sl2(C) decomposes into a direct sum of irreducible representations
This decomposition is described by the Clebsch-Gordan formula
The tensor product of two irreducible representations of the symmetric group Sn corresponds to the induction product of the associated Young diagrams
In the representation theory of finite groups, the character table encodes information about tensor products of irreducible representations
The Kronecker product of two character tables gives the character table of the tensor product representation
Advanced Topics and Extensions
Tensor categories generalize the notion of tensor products to categories with a monoidal structure
Examples include the category of vector spaces, the category of representations of a group, and the category of modules over a ring
Braided tensor categories have a braiding isomorphism that allows for non-trivial commutativity of the tensor product
Braided tensor categories are important in the study of quantum groups and topological quantum field theories
Fusion categories are semisimple rigid tensor categories with finitely many simple objects
They appear in the study of conformal field theory and the classification of subfactors
Hopf algebras are algebraic structures that generalize both algebras and coalgebras, with a compatible tensor product operation
Representations of Hopf algebras, known as Hopf modules, form a tensor category
Tensor networks have applications beyond computational techniques, such as in the study of many-body quantum systems and the holographic principle in quantum gravity
Tensor rank and tensor decomposition methods are active areas of research, with connections to algebraic geometry, complexity theory, and machine learning.