Representation Theory

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17 domain-specific abstractions whose origin domain is Representation Theory.

• Capelli's identity — Correct the determinant identity det(AB)=det(A)det(B) for matrices of noncommuting multiplication and differentiation operators by adding an ordered diagonal shift, yielding a central invariant in gl_n representation theory.
• Category of representations — A category whose objects are representations of a fixed algebraic structure and whose morphisms are equivariant maps.
• Clifford theory — Representation-theoretic results describing how irreducible representations of a group restrict to a normal subgroup and how subgroup constituents extend or induce back to the group.
• Double affine braid group — A braid-like group associated with an affine root system that adds a second affine translation structure and whose group algebra leads to double affine Hecke algebras.
• Frobenius–Schur indicator — An invariant distinguishing whether an irreducible complex representation is real, complex, or quaternionic in type.
• 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.
• Maschke's theorem — Every finite-dimensional representation of a finite group over a field whose characteristic does not divide the group order decomposes as a direct sum of irreducible representations.
• Murnaghan–Nakayama rule — A signed rim-hook removal rule for computing irreducible character values of symmetric groups from partitions.
• Partition algebra — An associative diagram algebra whose basis elements are set partitions and whose product concatenates diagrams while weighting closed middle components.
• Projective representation — A homomorphism from a group to a projective linear group, equivalently linear operators whose multiplication respects the group law only up to nonzero scalar factors.
• 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.
• Representation ring — The Grothendieck ring of finite-dimensional group representations, with direct sum as addition and tensor product as multiplication.
• Restricted representation — The representation of a subgroup obtained by retaining the same vector space and limiting a group representation to subgroup elements.
• 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.
• Subrepresentation — An invariant subspace of a representation on which the original group, algebra or category action restricts to a representation in its own right.
• Theta representation — A representation of the real Heisenberg group on a function space whose lattice-compatible vectors and transformation laws generate Jacobi theta functions.