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| Schur functor | |
|---|---|
| Name | Schur functor |
| Field | Representation theory; Algebraic geometry; Category theory |
| Introduced | Early 20th century |
| Named after | Issai Schur |
Schur functor is a construction associating to a vector space or locally free sheaf a new representation or sheaf built from polynomial operations and symmetry types. It organizes classical operations such as symmetric powers and exterior powers into a uniform family indexed by partitions or Young diagrams, and it plays a central role in the interactions among Issai Schur, Hermann Weyl, Alfred Young, Ferdinand Georg Frobenius, and later developments by Claude Chevalley, Jean-Pierre Serre, William Fulton, and Joe Harris. The concept appears in the literature on representations of GL_n, symmetric groups like S_n, and algebraic geometry over schemes studied by authors such as Grothendieck and Alexander Grothendieck.
A Schur functor assigns to each finite-dimensional vector space V over a field k a representation S^λ(V) indexed by a partition λ (or Young diagram), producing polynomial representations of GL(V) and commuting with base change along morphisms of schemes studied in the context of Grothendieck's theory. Fundamental properties relate to exactness and tensorial behavior underpinning results by Alexander Grothendieck, Michael Atiyah, Raoul Bott, and Jean-Louis Verdier: Schur functors are exact on sequences of locally free sheaves, respect direct sums via the Littlewood–Richardson rule developed by D. E. Littlewood and A. R. Richardson and interact with duals as in classical identities studied by Issai Schur and Hermann Weyl.
The classical construction uses Schur–Weyl duality between the actions of GL_n and S_r on tensor powers V^{⊗ r}, connecting foundational work of Issai Schur, Hermann Weyl, and Ferdinand Georg Frobenius. Young symmetrizers, coming from the combinatorial theory of Alfred Young and formalized by A. Kerber and Gordon James, project V^{⊗ r} onto isotypic components indexed by partitions λ via idempotents in the group algebra k[S_r] studied by Richard Brauer and Issai Schur. This approach connects to the theory of centralizer algebras including Schur algebra introduced by Issai Schur and later studied by J. A. Green, and to diagrammatic methods used by J. J. Graham and G. I. Lehrer.
Basic examples include symmetric powers Sym^r(V) linked to classical invariant theory associated with David Hilbert and Emmy Noether, exterior powers Λ^r(V) connected to de Rham theory examined by Élie Cartan and Henri Cartan, and divided powers Γ^r(V) relevant in positive characteristic contexts studied by Alexander Grothendieck and Jean-Pierre Serre. Hook and rectangular partitions produce Schur functors appearing in the representation theory of SL_n and in Bott's theorem attributed to Raoul Bott. The trivial partition yields the identity functor and the full-row or full-column partitions yield Sym^r and Λ^r, linking to classical constructions used by Bernhard Riemann and later by David Mumford in moduli problems.
Schur functors are polynomial functors in the sense of André Joyal and are compatible with natural transformations and monoidal structures studied in Category theory by Saunders Mac Lane and Max Kelly. They preserve direct sums and tensor products according to plethysm and the Littlewood–Richardson rule; plethysm problems investigated by I. G. Macdonald and Richard Stanley connect to combinatorial representation theory. Derived functors of Schur functors enter the realm of derived categories and homological algebra pioneered by Alexander Grothendieck and Jean-Louis Verdier, with spectral sequence techniques used by Jean-Pierre Serre and Henri Cartan in cohomological calculations.
In representation theory, Schur functors classify irreducible polynomial representations of GL_n and organize decomposition rules for tensor products in works by Weyl and Élie Cartan. They are indispensable in the study of syzygies and coordinate rings of varieties such as Grassmannian and Flag variety, central to the Schubert calculus developed by Hermann Schubert and modernized by William Fulton and Serge Lang. In algebraic geometry, Schur functors applied to the tautological bundle on moduli spaces studied by David Mumford and Igor Dolgachev yield vector bundles whose cohomology computes invariants used in geometric representation theory by George Lusztig and in the study of Beilinson–Bernstein localization by Alexander Beilinson and Joseph Bernstein.
The character theory of Schur functors is governed by Schur functions in the ring of symmetric functions developed by I. G. Macdonald, Alfred Young, and Ferdinand Georg Frobenius; this links to the Hall algebra studied by Philip Hall and to the theory of symmetric polynomials used by Gian-Carlo Rota and Richard Stanley. Over rings of positive characteristic or arbitrary commutative rings, divided powers and Steenrod operations investigated by Norman Steenrod and J. Peter May modify classical constructions, and the representation theory of Schur algebras over fields of characteristic p has been developed by J. A. Green and Andrew Mathas. Connections to quantum groups by Vladimir Drinfeld and Michio Jimbo appear via q-Schur algebras and categorification approaches advanced by Mikhail Khovanov and Jacob Rasmussen.