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For the group Z (the group of integers under addition), every irreducible representation is 1-dimensional. If V and W are 1-dimensional representations of Z, then Schur’s Lemma implies that any homomorphism between them is an isomorphism (unless the homomorphism is zero, which is not possible in this
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Consider the symmetric group S3. This group has irreducible representations of dimensions 1 and 2 over C. If ρ is an irreducible representation of S3 of dimension 1 (trivial representation), then Schur's Lemma tells us that any S3-homomorphism from this representation to any other representation
407:(including itself) is either an isomorphism or zero. In particular, if ρ is a 1-dimensional representation and σ is a 2-dimensional representation, any homomorphism from ρ to σ must be zero because these two representations are not isomorphic.
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in norm? A canonical example of this property, and commonly used to illustrate the Schur property, is the
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J. Schur, "Über lineare
Transformationen in der Theorie der unendlichen Reihen",
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of sequences whose series is absolutely convergent has the Schur property.
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This property was named after the early 20th century mathematician
509:(2015), Representations of Finite and Compact Groups. Springer.
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339:{\displaystyle \lim _{n\to \infty }\Vert x_{n}-x\Vert =0}
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145:{\displaystyle \ell _{1}}
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473:(1921) pp. 79-111
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278:{\displaystyle x}
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378:February 2021
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528:Issai Schur
426:Issai Schur
26:Issai Schur
18:mathematics
517:Categories
480:References
389:The space
167:, ||·||),
159:Definition
44:Motivation
507:Simon, B.
328:‖
322:−
309:‖
304:∞
301:→
134:ℓ
38:sequences
489:(1998),
436:See also
350:Examples
416:case).
497:
191:, and
454:Notes
495:ISBN
420:Name
228:has
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294:lim
232:if
36:of
16:In
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