Algebraic Structures in Representation Theory
Summary
Representation theory studies how algebraic objects such as groups, algebras and monoids act by linear transformations on vector spaces. Central to this field are the notions of modules over associative algebras and the decomposition of these modules into simpler components, often irreducible representations. Classical examples include the representation theory of finite groups, Lie algebras and quantum groups, where characters and structure constants such as Kronecker and Littlewood–Richardson coefficients encode deep combinatorial and geometric information. More recent advances blend category-theoretic frameworks, diagram algebras and homological methods to elucidate connections with algebraic geometry, quantum computing and mathematical physics. Through explicit constructions—such as partition algebras, quiver algebras and Hecke algebras—researchers bridge pure algebra with applications ranging from statistical mechanics to complexity theory.
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Technical terms
Algebra: A vector space equipped with a bilinear multiplication.
Module: A generalisation of a vector space on which an algebra acts linearly.
Irreducible representation: A nonzero module that contains no proper nontrivial submodules.
Kronecker coefficient: The multiplicity of an irreducible component in the tensor product of two symmetric‐group representations.
Plethysm coefficient: The multiplicity of an irreducible representation appearing in the composition of symmetric functions or Schur functors.
Partition algebra: A family of diagram algebras encoding centraliser algebras for symmetric‐group actions on tensor powers.
References
- Quantum Complexity of the Kronecker Coefficients. PRX Quantum (2024).
- The Computational Complexity of Plethysm Coefficients. computational complexity (2020).
- On representation theory of partition algebras for complex reflection groups. Algebraic Combinatorics (2020).
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