page_title: Glossary of representation theory
section_title: S
hierarchical_section_title: Glossary of representation theory / S
caption_attribution_description: English: Issai Schur.
context_page_description: This is a glossary of representation theory in mathematics.
  The term "module" is often used synonymously for a representation; for the module-theoretic terminology, see also glossary of module theory.
  See also Glossary of Lie groups and Lie algebras, list of representation theory topics and Category:Representation theory.
  Notations: We write. Thus, for example, a one-representation of a group G is of the form.
context_section_description: Schur
  1.  Issai Schur
  2.  Schur's lemma states that a G-linear map between irreducible representations must be either bijective or zero.
  3.  The Schur orthogonality relations on a compact group says the characters of non-isomorphic irreducible representations are orthogonal to each other.
  4.  The Schur functor  constructs representations such as symmetric powers or exterior powers according to a partition . The characters of  are Schur polynomials.
  5.  The Schur–Weyl duality computes the irreducible representations occurring in tensor powers of -modules.
  6.  A Schur polynomial is a symmetric function, of a type occurring in the Weyl character formula applied to unitary groups.
  7.  Schur index.
  8.  A Schur complex.
  semisimple
  A semisimple representation (also called a completely reducible representation) is a direct sum of simple representations.
  simple
  Another term for "irreducible".
  smooth
  1.  A smooth representation of a locally profinite group G is a complex representation such that, for each v in V, there is some compact open subgroup  K of G that fixes v; i.e.,  for every g in K.
  2.  A smooth vector in a representation space of a Lie group is a vector v such that  is a smooth function.
  Specht
  Specht module
  Steinberg
  Steinberg representation.
  subrepresentation
  A subrepresentation of a representation  of G is a vector subspace W of V such that  is well-defined for each g in G.
  Swan
  The Swan representation is used to define the Swan conductor.
  symmetric
  1.  A symmetric power of a representation V is a representation  with the group action induced by .
  2.  In particular, the symmetric square of a representation V is a representation  with the group action induced by .
  system of imprimitivity
  A concept in the Mackey theory. See system of imprimitivity.
caption: Issai Schur
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