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161:
The same function may be an invariant for one group of symmetries and equivariant for a different group of symmetries. For instance, under similarity transformations instead of congruences the area and perimeter are no longer invariant: scaling a triangle also changes its area and perimeter. However,
150:(a combination of a translation and rotation) to a triangle, and then constructing its center, produces the same point as constructing the center first, and then applying the same congruence to the center. More generally, all triangle centers are also equivariant under
200:
of the real numbers, so for instance it is unaffected by the choice of units used to represent the numbers. By contrast, the mean is not equivariant with respect to nonlinear transformations such as exponentials.
176:. In this way, the function mapping each triangle to its area or perimeter can be seen as equivariant for a multiplicative group action of the scaling transformations on the positive real numbers.
55:
with the action of the group. That is, applying a symmetry transformation and then computing the function produces the same result as computing the function and then applying the transformation.
693:
647:
62:, functions whose value is unchanged by a symmetry transformation of their argument. The value of an equivariant map is often (imprecisely) called an invariant.
667:
257:. That is, an intertwiner is just an equivariant linear map between two representations. Alternatively, an intertwiner for representations of a group
1033:
The centroid is the only affine equivariant center of a triangle, but more general convex bodies can have other affine equivariant centers; see e.g.
146:
are not invariant, because moving a triangle will also cause its centers to move. Instead, these centers are equivariant: applying any
Euclidean
353:). As a consequence, in important cases the construction of an intertwiner is enough to show the representations are effectively the same.
715:
1071:
242:
236:
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1179:, Cambridge Tracts in Theoretical Computer Science, vol. 57, Cambridge University Press, Definition 1.2, p. 14,
82:
1216:
1184:
1149:
1106:
773:
69:, equivariance under statistical transformations of data is an important property of various estimation methods; see
755:
747:
106:: the centroid of a transformed triangle is the same point as the transformation of the centroid of the triangle.
1309:
913:
751:
886:
from ρ to σ. Using natural transformations as morphisms, one can form the category of all representations of
711:
1133:
Relativity, groups, particles: Special relativity and relativistic symmetry in field and particle physics
154:(combinations of translation, rotation, reflection, and scaling), and the centroid is equivariant under
383:
362:
308:
197:
44:
859:
480:
If one or both of the actions are right actions the equivariance condition may be suitably modified:
245:, a vector space equipped with a group that acts by linear transformations of the space is called a
1101:, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, p. 70,
736:
123:
126:: translating, rotating, or reflecting a triangle does not change its area or perimeter. However,
740:
59:
216:
against certain kinds of changes to a data set, and that (unlike the mean) it is meaningful for
883:
787:
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185:
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52:
8:
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70:
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209:
208:
of a sample is equivariant for a much larger group of transformations, the (strictly)
1314:
1270:
1212:
1180:
1145:
1102:
965:
925:
376:
213:
102:
The centroid of a triangle (where the three red segments meet) is equivariant under
1262:
1137:
1044:
1000:
986:
895:
851:
346:
193:
16:
Maps whose domain and codomain are acted on by the same group, and the map commutes
698:
162:
these changes happen in a predictable way: if a triangle is scaled by a factor of
1280:
1237:
1155:
1112:
1052:
1016:
127:
36:
1048:
73:
for details. In pure mathematics, equivariance is a central object of study in
48:
1266:
1141:
1098:
Symmetries, Lie algebras and representations: A graduate course for physicists
227:
and equivariant estimator have been used to formalize this style of analysis.
1298:
1035:
Neumann, B. H. (1939), "On some affine invariants of closed convex regions",
350:
839:
571:
217:
135:
608:
143:
20:
1012:
989:(1994), "Central Points and Central Lines in the Plane of a Triangle",
316:
289:
250:
1234:
Actes du Congrès
International des Mathématiciens (Nice, 1970), Tome 2
1083:
Disseminations of the
International Statistical Applications Institute
315:) only exists if the two representations are equivalent (that is, are
616:
212:
of the real numbers. This analysis indicates that the median is more
119:
1004:
725:
98:
1024:. "Similar triangles have similarly situated centers", p. 164.
835:
795:
622:
The equivariance condition can also be understood as the following
312:
139:
131:
111:
28:
1211:, Dover Books on Mathematics, Dover Publications, pp. 86–87,
819:
858:, and a linear representation is equivalent to a functor to the
716:
Category of representations § Category-theoretic definition
882:, an equivariant map between those representations is simply a
205:
237:
Representation theory § Equivariant maps and isomorphisms
324:
1085:(4th ed.), vol. 1, 1995, Wichita: ACG Press, pp. 61–66.
1232:
Segal, G. B. (1971), "Equivariant stable homotopy theory",
189:
115:
1136:, Springer Physics, Vienna: Springer-Verlag, p. 165,
1073:
Measurement theory: Frequently asked questions (Version 3)
948:
between representations which commutes with the action of
192:
of a sample (a set of real numbers) is commonly used as a
361:
Equivariance can be formalized using the concept of a
675:
655:
632:
1176:
Nominal Sets: Names and
Symmetry in Computer Science
1255:Adhikari, Mahima Ranjan; Adhikari, Avishek (2014),
1204:
794:can be viewed as a category with a single object (
687:
661:
641:
786:Equivariant maps can be generalized to arbitrary
35:from one space with symmetry to another (such as
1296:
1254:
375:. This is a mathematical object consisting of a
1094:
936:. An equivariant map is then a continuous map
1205:Auslander, Maurice; Buchsbaum, David (2014),
1130:Sexl, Roman U.; Urbantke, Helmuth K. (2001),
1095:Fuchs, Jürgen; Schweigert, Christoph (1997),
1129:
712:Representation theory § Generalizations
184:Another class of simple examples comes from
1236:, Gauthier-Villars, Paris, pp. 59–63,
754:. Unsourced material may be challenged and
335:). These properties hold when the image of
253:that commutes with the action is called an
58:Equivariant maps generalize the concept of
1037:Journal of the London Mathematical Society
985:
798:in this category are just the elements of
790:in a straightforward manner. Every group
774:Learn how and when to remove this message
230:
97:
1070:Sarle, Warren S. (September 14, 1997),
1034:
874:Given two representations, ρ and σ, of
196:of the sample. It is equivariant under
1297:
1258:Basic modern algebra with applications
830:. Such a functor selects an object of
649:denotes the map that takes an element
311:, then an intertwiner (other than the
243:representation theory of finite groups
93:
1231:
1200:
1198:
1196:
1172:
1069:
846:-set is equivalent to a functor from
1261:, New Delhi: Springer, p. 142,
752:adding citations to reliable sources
719:
327:a multiplicative factor (a non-zero
323:). That intertwiner is then unique
122:of a triangle are invariants under
13:
1193:
697:
83:equivariant stable homotopy theory
14:
1326:
705:
341:is a simple algebra, with centre
43:when its domain and codomain are
724:
586:). Hence they are also known as
356:
842:of that object. For example, a
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166:, the perimeter also scales by
39:). A function is said to be an
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1166:
1123:
1088:
1063:
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979:
914:category of topological spaces
1:
972:
962:Curtis–Hedlund–Lyndon theorem
424:is said to be equivariant if
179:
968:in terms of equivariant maps
7:
1305:Group actions (mathematics)
1081:. Revision of a chapter in
955:
309:irreducible representations
88:
10:
1331:
904:For another example, take
709:
299:Under some conditions, if
234:
152:similarity transformations
1267:10.1007/978-81-322-1599-8
1173:Pitts, Andrew M. (2013),
1142:10.1007/978-3-7091-6234-7
860:category of vector spaces
406:-sets for the same group
124:Euclidean transformations
1049:10.1112/jlms/s1-14.4.262
964:, a characterization of
688:{\displaystyle g\cdot z}
51:, and when the function
642:{\displaystyle g\cdot }
268:is the same thing as a
170:and the area scales by
1208:Groups, Rings, Modules
916:. A representation of
884:natural transformation
702:
689:
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198:linear transformations
186:statistical estimation
156:affine transformations
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104:affine transformations
79:equivariant cohomology
1310:Representation theory
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570:Equivariant maps are
247:linear representation
231:Representation theory
101:
67:statistical inference
1079:, SAS Institute Inc.
992:Mathematics Magazine
748:improve this section
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75:equivariant topology
894:. This is just the
624:commutative diagram
582:-sets (for a fixed
345:(by what is called
270:module homomorphism
225:invariant estimator
223:The concepts of an
210:monotonic functions
110:In the geometry of
94:Elementary geometry
71:invariant estimator
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685:
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639:
619:equivariant maps.
410:, then a function
108:
77:and its subtopics
1276:978-81-322-1598-1
1039:, Second Series,
987:Kimberling, Clark
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926:topological space
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662:{\displaystyle z}
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128:triangle centers
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626:. Note that
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351:simple module
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27:is a form of
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934:continuously
929:
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746:Please help
734:
669:and returns
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609:Isomorphisms
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222:
218:ordinal data
203:
183:
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136:circumcenter
130:such as the
109:
64:
57:
47:by the same
40:
25:equivariance
24:
18:
255:intertwiner
144:orthocenter
21:mathematics
1299:Categories
973:References
788:categories
764:April 2016
710:See also:
591:-morphisms
317:isomorphic
290:group ring
251:linear map
235:See also:
180:Statistics
148:congruence
60:invariants
928:on which
796:morphisms
735:does not
680:⋅
637:⋅
617:bijective
402:are both
307:are both
120:perimeter
112:triangles
33:functions
1315:Symmetry
956:See also
940: :
836:subgroup
576:category
467:and all
457:for all
415: :
313:zero map
282:, where
140:incenter
132:centroid
89:Examples
53:commutes
45:acted on
29:symmetry
1285:3155599
1242:0423340
1160:1798479
1117:1473220
1057:0000978
1021:1573021
1013:2690608
850:to the
820:functor
756:removed
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574:in the
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288:is the
280:modules
261:over a
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329:scalar
214:robust
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