Representation Theory

🧩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.

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 VV and WW, their tensor product is denoted as VWV \otimes W
    • Elements of VWV \otimes W are linear combinations of simple tensors vwv \otimes w, where vVv \in V and wWw \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(VW)=dim(V)dim(W)\dim(V \otimes W) = \dim(V) \cdot \dim(W)
  • Tensor products are associative: (UV)WU(VW)(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 vVv \in V and wWw \in W is denoted as vwv \otimes w
    • This is a simple tensor, which forms a basis for the tensor product space VWV \otimes W
  • Tensor products are bilinear, meaning they are linear in each argument separately
    • (av1+bv2)w=a(v1w)+b(v2w)(av_1 + bv_2) \otimes w = a(v_1 \otimes w) + b(v_2 \otimes w)
    • v(cw1+dw2)=c(vw1)+d(vw2)v \otimes (cw_1 + dw_2) = c(v \otimes w_1) + d(v \otimes w_2)
  • The tensor product of two linear maps f:VVf: V \to V' and g:WWg: W \to W' is a linear map fg:VWVWf \otimes g: V \otimes W \to V' \otimes W'
    • Defined by (fg)(vw)=f(v)g(w)(f \otimes g)(v \otimes w) = f(v) \otimes g(w)
  • Tensor products are distributive over direct sums: (V1V2)W(V1W)(V2W)(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 kk is isomorphic to the original vector space: VkVV \otimes k \cong V

Constructing Tensor Products

  • To construct the tensor product of two vector spaces VV and WW, start with the free vector space generated by the Cartesian product V×WV \times W
    • Elements of this space are formal linear combinations of pairs (v,w)(v, w), where vVv \in V and wWw \in W
  • Impose bilinearity relations on this space to obtain the tensor product space VWV \otimes W
    • (av1+bv2,w)=a(v1,w)+b(v2,w)(av_1 + bv_2, w) = a(v_1, w) + b(v_2, w)
    • (v,cw1+dw2)=c(v,w1)+d(v,w2)(v, cw_1 + dw_2) = c(v, w_1) + d(v, w_2)
  • The resulting space VWV \otimes W is the quotient of the free vector space by the subspace generated by these relations
  • Given bases {vi}\{v_i\} for VV and {wj}\{w_j\} for WW, the tensor products viwjv_i \otimes w_j form a basis for VWV \otimes W
    • This basis is called the tensor product basis
  • The dimension of VWV \otimes W is the product of the dimensions of VV and WW
  • Tensor products can be constructed for more than two vector spaces by iterating the construction

Properties of Tensor Products

  • Tensor products are associative: (UV)WU(VW)(U \otimes V) \otimes W \cong U \otimes (V \otimes W)
    • This allows for unambiguous notation of multiple tensor products: UVWU \otimes V \otimes W
  • Tensor products are distributive over direct sums: (V1V2)W(V1W)(V2W)(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 kk is isomorphic to the original vector space: VkVV \otimes k \cong V
  • The tensor product of two finite-dimensional representations is also a representation
    • If ρ:GGL(V)\rho: G \to \text{GL}(V) and σ:GGL(W)\sigma: G \to \text{GL}(W) are representations, then ρσ:GGL(VW)\rho \otimes \sigma: G \to \text{GL}(V \otimes W) is a representation
  • Tensor products are compatible with dual spaces: (VW)VW(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 SU(2)\text{SU}(2) representations
  • Tensor products are used to define the character of a representation
    • The character of a representation ρ:GGL(V)\rho: G \to \text{GL}(V) is the trace of the linear map ρ(g)\rho(g) for each gGg \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 GL(V)\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: (AB)(CD)=(AC)(BD)(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 Rn\mathbb{R}^n and Rm\mathbb{R}^m is isomorphic to the space of n×mn \times m matrices Rn×m\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 sl2(C)\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 SnS_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.


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© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.