10.3 Representation theory and characters

Representation theory dives into how algebraic groups and Lie algebras act on vector spaces. It's all about understanding these actions through homomorphisms, which map group elements to linear transformations.

Characters are the secret sauce of representation theory. These functions capture the essence of representations, making it easier to study their properties and decompose them into simpler parts.

Representations of Algebraic Groups and Lie Algebras

Defining Representations

• A representation of an algebraic group G is a homomorphism $\rho: G \to GL(V)$ from G to the general linear group of a vector space V over a field K
• A representation of a Lie algebra $\mathfrak{g}$ is a Lie algebra homomorphism $\rho: \mathfrak{g} \to gl(V)$ from $\mathfrak{g}$ to the Lie algebra of endomorphisms of a vector space V over a field K
• The dimension of the representation equals the dimension of the vector space V
• A subrepresentation of a representation $(\rho, V)$ is a subspace W of V that remains invariant under the action of G (or $\mathfrak{g}$)

Properties of Representations

• A representation is irreducible if it contains no proper non-zero subrepresentations
• The trivial representation is the one-dimensional representation where every group element (or Lie algebra element) acts as the identity
• Irreducible representations play a fundamental role in the study of algebraic groups and Lie algebras, as they serve as the building blocks for more complex representations
• The study of representations involves understanding how algebraic groups and Lie algebras act on vector spaces, which provides insight into their structure and properties

Character Functions of Representations

Defining Characters

• The character of a representation $(\rho, V)$ of a group G is the function $\chi: G \to K$ defined by $\chi(g) = tr(\rho(g))$, where tr denotes the trace
• Characters are class functions, meaning they remain constant on conjugacy classes of G
• For semisimple algebraic groups and Lie algebras, the character of a representation determines the representation up to isomorphism
• The character of a representation encodes important information about the representation, such as its dimension and decomposition into irreducible components

Properties of Characters

• The character of a direct sum of representations equals the sum of the characters of the individual representations
• The character of a tensor product of representations equals the product of the characters of the individual representations
• Schur's lemma states that a non-zero intertwining map between irreducible representations is an isomorphism, and the space of self-intertwining maps of an irreducible representation is one-dimensional
• Character theory provides a powerful tool for studying representations, as it allows one to reduce questions about representations to questions about functions on the group or Lie algebra

Decomposition of Representations

Decomposing Representations into Irreducibles

• The Weyl character formula expresses the character of an irreducible representation of a compact connected Lie group in terms of its highest weight
• Weyl's theorem states that every finite-dimensional representation of a compact connected Lie group or a semisimple complex Lie algebra is completely reducible, i.e., it decomposes as a direct sum of irreducible representations
• The multiplicity of an irreducible representation in a given representation can be computed using the inner product of characters
• Understanding the decomposition of representations into irreducibles is crucial for many applications, such as studying the structure of algebraic varieties with group actions

Tensor Products and Grothendieck Groups

• The tensor product of two irreducible representations decomposes into a direct sum of irreducible representations, with multiplicities given by the Littlewood-Richardson rule for $GL_n$ and analogous rules for other classical groups
• The Grothendieck group of a category of representations is the abelian group generated by isomorphism classes of representations, with relations coming from exact sequences
• The Grothendieck group encodes the decomposition of representations into irreducibles and provides a way to study representations using algebraic and combinatorial techniques
• Tensor products and Grothendieck groups are important tools for understanding the structure of representations and their interactions with each other

Representation Theory in Algebraic Geometry

Group Actions on Varieties

• The action of an algebraic group G on an algebraic variety X induces a representation of G on the coordinate ring $K[X]$ and other related vector spaces, such as the space of sections of a G-equivariant vector bundle
• Representations can be used to study the invariant theory of G-actions on varieties, i.e., the subrings of $K[X]$ fixed by the action of G
• The decomposition of the coordinate ring $K[X]$ into irreducible representations can provide information about the geometry of the variety X, such as its singularities and cohomology
• Studying group actions on varieties using representation theory allows one to understand the symmetries and structure of algebraic varieties

Applications to Moduli Spaces and Cohomology

• Character formulas and other representation-theoretic tools can be used to compute dimensions of invariant spaces, cohomology groups, and other geometric invariants
• Representation theory can be used to construct and study moduli spaces of geometric objects with symmetries, such as vector bundles or principal bundles on algebraic curves
• The Borel-Weil-Bott theorem relates the cohomology of line bundles on flag varieties to the representation theory of the corresponding algebraic group or Lie algebra
• Applying representation theory to problems in algebraic geometry provides a powerful set of tools for understanding the structure and properties of algebraic varieties and related objects