Lie Groups & Representation Theory

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Abstractions about Lie groups, Lie algebras, lattices, roots, invariant functions, and their representations. They include exponential and infinitesimal transformations, discrete series, Eisenstein and theta representations, partition functions, Schur constructions, and representation rings.

23 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Abelian Lie group — A smooth Lie group whose multiplication is commutative, combining a finite-dimensional manifold with an abelian group structure and smooth operations.
• Covariant transformation — A component-transformation rule in which lower-index tensor components change with the inverse-coordinate or dual-basis matrix so the underlying geometric object and contractions remain invariant.
• Dedekind eta function — A holomorphic function on the upper half-plane defined by q^(1/24) times the infinite product of (1−q^n), with a weight-one-half modular transformation law.
• Dual lattice — The lattice of vectors pairing integrally with every vector of a full-rank lattice under a declared inner product.
• En (Lie algebra) — The Lie or Kac–Moody algebra family associated with the E-n branching Dynkin diagram, including exceptional finite cases and indefinite extensions.
• Exponential map (Lie theory) — The canonical smooth map sending a Lie-algebra element to the time-one point of its one-parameter subgroup in the Lie group.
• Frobenius–Schur indicator — An invariant distinguishing whether an irreducible complex representation is real, complex, or quaternionic in type.
• Gamas's theorem — A criterion stating that a tensor projected by an irreducible symmetric-group representation is nonzero exactly when its vectors can be partitioned into linearly independent subsets of sizes given by the partition’s columns.
• Infinitesimal transformation — A first-order generator describing the tangent direction of a continuous one-parameter family of transformations at the identity.
• Kostant partition function — A function counting the ways a weight can be expressed as a nonnegative integer combination of the positive roots of a root system.
• Partition algebra — An associative diagram algebra whose basis elements are set partitions and whose product concatenates diagrams while weighting closed middle components.
• Quaternionic discrete series representation — A discrete-series representation of a semisimple real Lie group whose symmetric space carries the quaternionic structure associated with an SU(2) factor in a maximal compact subgroup.
• Quaternionic representation — A complex group representation carrying an invariant antilinear equivariant operator whose square is minus the identity, equivalently a representation of quaternionic type.
• Real analytic Eisenstein series — A nonholomorphic automorphic series on the upper half-plane whose coprime-lattice sum is an eigenfunction of the hyperbolic Laplacian and admits meromorphic continuation.
• Real form (Lie theory) — A real Lie algebra or group whose scalar extension to the complex numbers recovers a specified complex Lie algebra or group.
• Representation ring — The Grothendieck ring of finite-dimensional group representations, with direct sum as addition and tensor product as multiplication.
• Restricted root system — The root system obtained by restricting a semisimple Lie algebra’s roots to a maximal abelian subspace in the noncompact part of a symmetric decomposition.
• Schur functor — A polynomial functor indexed by a partition that constructs an irreducible polynomial representation from tensor powers using prescribed row symmetries and column antisymmetries.
• SO(8) — The rank-four, 28-dimensional special orthogonal Lie group of orientation-preserving linear isometries of eight-dimensional Euclidean space, distinguished by triality in its Spin(8) cover.
• Steinberg formula — A Weyl-group and Kostant-partition-function formula for the multiplicity of an irreducible highest-weight representation inside a tensor product of two irreducible representations of a complex semisimple Lie algebra.
• Theta representation — A representation of the real Heisenberg group on a function space whose lattice-compatible vectors and transformation laws generate Jacobi theta functions.
• U-invariant — The supremum of dimensions of anisotropic quadratic forms over a field, equivalently the threshold above which every form is isotropic when finite.
• Verma module — A highest-weight module induced from a one-dimensional Borel representation, universal among highest-weight modules of the same weight.