mathematics

Weakly symmetric space

Weakly symmetric space is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Weakly symmetric space rather than just read about it. In short: In mathematics, a weakly symmetric space is a notion introduced by the Norwegian mathematician Atle Selberg in the 1950s as a generalisation of symmetric space, due to Élie Cartan. Geometrically the spaces are defined as complete Riemannian manifolds such that any two points can be exchanged by an isometry, the symmetric case being when the isometry is required to have period two.

Key takeaways

Weakly symmetric space belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Weakly symmetric space to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Weakly symmetric space from memory before moving on to harder problems.

Reference excerpt

In mathematics, a weakly symmetric space is a notion introduced by the Norwegian mathematician Atle Selberg in the 1950s as a generalisation of symmetric space, due to Élie Cartan. Geometrically the spaces are defined as complete Riemannian manifolds such that any two points can be exchanged by an isometry, the symmetric case being when the isometry is required to have period two. The classification of weakly symmetric spaces relies on that of periodic automorphisms of complex semisimple Lie algebras. They provide examples of Gelfand pairs, although the corresponding theory of spherical functions in harmonic analysis, known for symmetric spaces, has not yet been developed.

References Akhiezer, D. N.; Vinberg, E. B. (1999), "Weakly symmetric spaces and spherical varieties", Transf. Groups, 4: 3–24, doi:10.1007/BF01236659, S2CID 124032062 Helgason, Sigurdur (1978), Differential geometry, Lie groups and symmetric spaces, Academic Press, ISBN 0-12-338460-5 Kac, V. G. (1990), Infinite dimensional Lie algebras (3rd ed.), Cambridge University Press, ISBN 0-521-46693-8 Kobayashi, Toshiyuki (2002). "Branching problems of unitary representations". Proceedings of the International Congress of Mathematicians, Vol. II. Beijing: Higher Ed. Press. pp. 615–627. Kobayashi, Toshiyuki (2004), "Geometry of multiplicity-free representations of GL(n), visible actions on flag varieties, and triunity", Acta Appl. Math., 81: 129–146, doi:10.1023/B:ACAP.0000024198.46928.0c, S2CID 14530010 Kobayashi, Toshiyuki (2007), "A generalized Cartan decomposition for the double coset space (U(n1)×U(n2)×U(n3))\U(n)/(U(p)×U(q))", J. Math. Soc. Jpn., 59: 669–691 Krämer, Manfred (1979), "Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen", Compositio Mathematica (in German), 38: 129–153 Matsuki, Toshihiko (1991), "Orbits on flag manifolds", Proceedings of the International Congress of Mathematicians, Vol. II, 1990 Kyoto, Math. Soc. Japan, pp. 807–813 Matsuki, Toshihiko (2013), "An example of orthogonal triple flag variety of finite type", J. Algebra, 375: 148–187, CiteSeerX 10.1.1.750.7197, doi:10.1016/j.jalgebra.2012.11.012, S2CID 119132477 {{citation}}: Cite uses deprecated parameter |citeseerx= (help) Mikityuk, I. V. (1987), "On the integrability of invariant Hamiltonian systems with homogeneous configuration spaces", Math. USSR Sbornik, 57 (2): 527–546, Bibcode:1987SbMat..57..527M, doi:10.1070/SM1987v057n02ABEH003084 Selberg, A. (1956), "Harmonic analysis and discontinuous groups in weakly symmetric riemannian spaces, with applications to Dirichlet series", J. Indian Math. Society, 20: 47–87 Stembridge, J. R. (2001), "Multiplicity-free products of Schur functions", Annals of Combinatorics, 5 (2): 113–121, doi:10.1007/s00026-001-8008-6, hdl:2027.42/41839, S2CID 18105235 Stembridge, J. R. (2003), "Multiplicity-free products and restrictions of Weyl characters", Representation Theory, 7 (18): 404–439, doi:10.1090/S1088-4165-03-00150-X Vinberg, E. B. (2001), "Commutative homogeneous spaces and co-isotropic symplectic actions", Russian Math. Surveys, 56 (1): 1–60, Bibcode:2001RuMaS..56....1V, doi:10.1070/RM2001v056n01ABEH000356, S2CID 250919435 Wolf, J. A.; Gray, A. (1968), "Homogeneous spaces defined by Lie group automorphisms. I, II", Journal of Differential Geometry, 2: 77–114, 115–159 Wolf, J. A. (2007), Harmonic Analysis on Commutative Spaces, American Mathematical Society, ISBN 978-0-8218-4289-8 Ziller, Wolfgang (1996), "Weakly symmetric spaces", Topics in geometry, Progr. Nonlinear Differential Equations Appl., vol. 20, Boston: Birkhäuser, pp. 355–368

Worked examples

Example 1 — a first encounter with Weakly symmetric space

Start with the simplest possible case. Write down what Weakly symmetric space claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Weakly symmetric space before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Weakly symmetric space ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Weakly symmetric space

In research: Weakly symmetric space appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Weakly symmetric space in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Weakly symmetric space is common in secondary-school and first-year university syllabi. It links to neighbouring topics Differential geometry, Differential geometry stubs, Harmonic analysis, so understanding it makes those chapters shorter.
In everyday life: Look for Weakly symmetric space outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Weakly symmetric space in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Weakly symmetric space means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Weakly symmetric space out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Weakly symmetric space in simple terms?

Why does Weakly symmetric space matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Weakly symmetric space?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Weakly symmetric space.