🧩Representation Theory Unit 8 – Tensor Products of Representations

Key Concepts and Definitions

• 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 $V \otimes W$
  • Elements of $V \otimes W$ are linear combinations of simple tensors $v \otimes w$, where $v \in V$ and $w \in W$
• 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: $\dim(V \otimes W) = \dim(V) \cdot \dim(W)$
• Tensor products are associative: $(U \otimes V) \otimes W \cong U \otimes (V \otimes W)$
• The tensor product of two finite-dimensional representations is also a representation

Tensor Product Basics

• The tensor product of two vectors $v \in V$ and $w \in W$ is denoted as $v \otimes w$
  • This is a simple tensor, which forms a basis for the tensor product space $V \otimes W$
• Tensor products are bilinear, meaning they are linear in each argument separately
  • $(av_1 + bv_2) \otimes w = a(v_1 \otimes w) + b(v_2 \otimes w)$
  • $v \otimes (cw_1 + dw_2) = c(v \otimes w_1) + d(v \otimes w_2)$
• The tensor product of two linear maps $f: V \to V'$ and $g: W \to W'$ is a linear map $f \otimes g: V \otimes W \to V' \otimes W'$
  • Defined by $(f \otimes g)(v \otimes w) = f(v) \otimes g(w)$
• Tensor products are distributive over direct sums: $(V_1 \oplus V_2) \otimes W \cong (V_1 \otimes W) \oplus (V_2 \otimes W)$
• The tensor product of a vector space with a field $k$ is isomorphic to the original vector space: $V \otimes k \cong V$

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 $V \times W$
  • Elements of this space are formal linear combinations of pairs $(v, w)$, where $v \in V$ and $w \in W$
• Impose bilinearity relations on this space to obtain the tensor product space $V \otimes W$
  • $(av_1 + bv_2, w) = a(v_1, w) + b(v_2, w)$
  • $(v, cw_1 + dw_2) = c(v, w_1) + d(v, w_2)$
• The resulting space $V \otimes W$ is the quotient of the free vector space by the subspace generated by these relations
• Given bases $\{v_i\}$ for $V$ and $\{w_j\}$ for $W$, the tensor products $v_i \otimes w_j$ form a basis for $V \otimes W$
  • This basis is called the tensor product basis
• The dimension of $V \otimes W$ 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: $(U \otimes V) \otimes W \cong U \otimes (V \otimes W)$
  • This allows for unambiguous notation of multiple tensor products: $U \otimes V \otimes W$
• Tensor products are distributive over direct sums: $(V_1 \oplus V_2) \otimes W \cong (V_1 \otimes W) \oplus (V_2 \otimes W)$
• The tensor product of a vector space with a field $k$ is isomorphic to the original vector space: $V \otimes k \cong V$
• The tensor product of two finite-dimensional representations is also a representation
  • If $\rho: G \to \text{GL}(V)$ and $\sigma: G \to \text{GL}(W)$ are representations, then $\rho \otimes \sigma: G \to \text{GL}(V \otimes W)$ is a representation
• Tensor products are compatible with dual spaces: $(V \otimes W)^* \cong V^* \otimes W^*$
• 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 $\text{SU}(2)$ representations
• Tensor products are used to define the character of a representation
  • The character of a representation $\rho: G \to \text{GL}(V)$ is the trace of the linear map $\rho(g)$ for each $g \in G$
• 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 $\text{GL}(V)$

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: $(A \otimes B)(C \otimes D) = (AC) \otimes (BD)$
• 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 $\mathbb{R}^n$ and $\mathbb{R}^m$ is isomorphic to the space of $n \times m$ matrices $\mathbb{R}^{n \times m}$
  • 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 $\mathfrak{sl}_2(\mathbb{C})$ 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 $S_n$ 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.