1555:
1311:
1788:
228:— Many important Lie groups are compact, so the results of compact representation theory apply to them. Other techniques specific to Lie groups are used as well. Most of the groups important in physics and chemistry are Lie groups, and their representation theory is crucial to the application of group theory in those fields. See
200:— Many of the results of finite group representation theory are proved by averaging over the group. These proofs can be carried over to infinite groups by replacement of the average with an integral, provided that an acceptable notion of integral can be defined. This can be done for locally compact groups, using the
1365:
259:. Although linear algebraic groups have a classification that is very similar to that of Lie groups, and give rise to the same families of Lie algebras, their representations are rather different (and much less well understood). The analytic techniques used for studying Lie groups must be replaced by techniques from
1121:
1602:
38:
149:
The representation theory of groups divides into subtheories depending on the kind of group being represented. The various theories are quite different in detail, though some basic definitions and concepts are similar. The most important divisions are:
1550:{\displaystyle \sigma \left(1\right)={\begin{bmatrix}1&0\\0&1\\\end{bmatrix}}\qquad \sigma \left(u\right)={\begin{bmatrix}u&0\\0&1\\\end{bmatrix}}\qquad \sigma \left(u^{2}\right)={\begin{bmatrix}u^{2}&0\\0&1\\\end{bmatrix}}.}
1306:{\displaystyle \rho \left(1\right)={\begin{bmatrix}1&0\\0&1\\\end{bmatrix}}\qquad \rho \left(u\right)={\begin{bmatrix}1&0\\0&u\\\end{bmatrix}}\qquad \rho \left(u^{2}\right)={\begin{bmatrix}1&0\\0&u^{2}\\\end{bmatrix}}.}
583:
1893:
1783:{\displaystyle \tau \left(1\right)={\begin{bmatrix}1&0\\0&1\\\end{bmatrix}}\qquad \tau \left(u\right)={\begin{bmatrix}a&-b\\b&a\\\end{bmatrix}}\qquad \tau \left(u^{2}\right)={\begin{bmatrix}a&b\\-b&a\\\end{bmatrix}}}
2779:
2205:
In the example above, the first two representations given (ρ and σ) are both decomposable into two 1-dimensional subrepresentations (given by span{(1,0)} and span{(0,1)}), while the third representation (τ) is irreducible.
303:
on which the group acts. One distinguishes between finite-dimensional representations and infinite-dimensional ones. In the infinite-dimensional case, additional structures are important (e.g. whether or not the space is a
863:
1047:
449:
2447:
273:— The class of non-compact groups is too broad to construct any general representation theory, but specific special cases have been studied, sometimes using ad hoc techniques. The
1975:
1594:
1357:
1113:
121:
is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a
460:
2332:
2117:
2085:
1799:
671:
2004:
2053:
2470:
2024:
1919:
2868:
141:
for the special case of linear representations. The bulk of this article describes linear representation theory; see the last section for generalizations.
159:— Group representations are a very important tool in the study of finite groups. They also arise in the applications of finite group theory to
791:
2167:. The representation of dimension zero is considered to be neither reducible nor irreducible, just as the number 1 is considered to be neither
983:
2895:
406:
393:
2621:, this definition is equivalent to a linear representation. Likewise, a set-theoretic representation is just a representation of
95:
In chemistry, a group representation can relate mathematical group elements to symmetric rotations and reflections of molecules.
2754:
185:
2836:
343:
of the field is also significant; many theorems for finite groups depend on the characteristic of the field not dividing the
17:
2749:
2338:
2744:
233:
2890:
2808:
2655:
756:
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229:
180:
2724:
1924:
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2518:
2142:
2128:
292:
1570:
1333:
1089:
2759:
2685:
2614:
2202:, since the characteristic of the complex numbers is zero, which never divides the size of a group.
578:{\displaystyle \rho (g_{1}g_{2})=\rho (g_{1})\rho (g_{2}),\qquad {\text{for all }}g_{1},g_{2}\in G.}
2179:
697:
340:
168:
1888:{\displaystyle a={\text{Re}}(u)=-{\tfrac {1}{2}},\qquad b={\text{Im}}(u)={\tfrac {\sqrt {3}}{2}}.}
872:
31:
2872:. Translated from the 1985 Russian-language edition (Kharkov, Ukraine). Birkhäuser Verlag. 1988.
2293:
2534:
2242:
2090:
2058:
650:
624:
240:
217:
2163:; if it has a proper subrepresentation of nonzero dimension, the representation is said to be
281:
Lie groups cannot be classified in the same way. The general theory for Lie groups deals with
2885:
2699:
196:
89:
73:
58:
2846:
2195:
2191:
1982:
344:
336:
2032:
8:
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374:
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66:
42:
251:) — These are the analogues of Lie groups, but over more general fields than just
2739:
2590:
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2455:
2009:
1921:
be the space of homogeneous degree-3 polynomials over the complex numbers in variables
1904:
381:
282:
260:
209:
126:
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2147:
689:
673:
213:
205:
85:
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2168:
264:
2842:
2729:
2711:
2707:
2501:
2199:
160:
319:
over which the vector space is defined. The most important case is the field of
114:
of a physical system affects the solutions of equations describing that system.
320:
111:
103:
46:
2828:
2198:). This holds in particular for any representation of a finite group over the
612:
itself as the representation when the homomorphism is clear from the context.
2879:
2854:
2671:
2155:
has exactly two subrepresentations, namely the zero-dimensional subspace and
332:
305:
287:
192:
2681:
Two types of representations closely related to linear representations are:
858:{\displaystyle \ker \rho =\left\{g\in G\mid \rho (g)=\mathrm {id} \right\}.}
2703:
2187:
2172:
1083:
367:
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324:
309:
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247:
201:
155:
122:
99:
81:
77:
2485:
950:
54:
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2496:. Thus we may equivalently define a permutation representation to be a
224:
37:
2823:, Readings in Mathematics. Vol. 129. New York: Springer-Verlag.
2481:
1317:
887:
70:
339:
fields are easier to handle than non-algebraically closed ones. The
277:
have a deep theory, building on the compact case. The complementary
27:
Group homomorphism into the general linear group over a vector space
2645:
2538:
84:); in particular, they can be used to represent group elements as
2562:
107:
49:, consisting of reflections and rotations, transform the polygon.
1042:{\displaystyle \alpha \circ \rho (g)\circ \alpha ^{-1}=\pi (g).}
2869:
212:
describes the theory for commutative groups, as a generalised
2693:
890:; in other words, one whose kernel is the trivial subgroup {
444:{\displaystyle \rho \colon G\to \mathrm {GL} \left(V\right)}
1086:
under multiplication. This group has a representation ρ on
2186:
does not divide the size of the group, representations of
299:
Representation theory also depends heavily on the type of
144:
2859:. Introduction to representation theory with emphasis on
2476:. This condition and the axioms for a group imply that ρ(
608:
of the representation. It is common practice to refer to
129:
of an object. If the object is a vector space we have a
184:; this special case has very different properties. See
2524:
2517:
For more information on this topic see the article on
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2012:
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1124:
1092:
986:
794:
653:
463:
409:
285:
of the two types, by means of general results called
2692:. These can be described as "linear representations
178:
divides the order of the group, then this is called
2464:
2441:
2326:
2214:
2111:
2079:
2047:
2018:
1998:
1969:
1913:
1887:
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1351:
1305:
1107:
1041:
894:} consisting only of the group's identity element.
857:
665:
577:
443:
88:so that the group operation can be represented by
1359:, isomorphic to the previous one, is σ given by:
45:"acts" on an object. A simple example is how the
2877:
2442:{\displaystyle \rho (g_{1}g_{2})=\rho (g_{1})],}
1316:This representation is faithful because ρ is a
2159:itself, then the representation is said to be
323:. The other important cases are the field of
2807:
204:. The resulting theory is a central part of
137:for the general notion and reserve the term
2541:in this category are just the elements of
1576:
1339:
1095:
36:
2194:of irreducible subrepresentations (see
2026:by permutation of the three variables.
400:. That is, a representation is a map
145:Branches of group representation theory
14:
2878:
2755:Representation theory of finite groups
1567:may also be faithfully represented on
186:Representation theory of finite groups
102:problems to be reduced to problems in
2817:Representation theory. A first course
774:is defined as the normal subgroup of
98:Representations of groups allow many
2750:List of representation theory topics
267:causes many technical complications.
2573:. Such a functor selects an object
2525:Representations in other categories
315:One must also consider the type of
167:of scalars of the vector space has
24:
2644:, the objects obtained are called
2209:
1970:{\displaystyle x_{1},x_{2},x_{3}.}
843:
840:
426:
423:
25:
2907:
2654:For another example consider the
2223:(also known as a group action or
875:is one in which the homomorphism
2745:List of harmonic analysis topics
1589:{\displaystyle \mathbb {R} ^{2}}
1352:{\displaystyle \mathbb {C} ^{2}}
1108:{\displaystyle \mathbb {C} ^{2}}
2896:Representation theory of groups
2215:Set-theoretical representations
2122:
1844:
1716:
1659:
1476:
1422:
1232:
1178:
949:if there exists a vector space
782:is the identity transformation:
537:
291:, which is a generalization of
234:Representations of Lie algebras
47:symmetries of a regular polygon
2772:
2656:category of topological spaces
2581:and a group homomorphism from
2545:. Given an arbitrary category
2433:
2430:
2424:
2421:
2408:
2402:
2399:
2386:
2377:
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2368:
2345:
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2300:
2178:Under the assumption that the
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1002:
996:
833:
827:
531:
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467:
419:
350:
271:Non-compact topological groups
13:
1:
2821:Graduate Texts in Mathematics
2801:
2674:group of a topological space
230:Representations of Lie groups
181:modular representation theory
80:to itself (i.e. vector space
2253:, the set of functions from
2221:set-theoretic representation
2141:that is invariant under the
1059:Consider the complex number
263:, where the relatively weak
7:
2725:Irreducible representations
2718:
2472:is the identity element of
1323:Another representation for
1063:= e which has the property
1054:
10:
2912:
2686:projective representations
2642:category of abelian groups
2327:{\displaystyle \rho (1)=x}
2225:permutation representation
2129:Irreducible representation
2126:
716:such that the application
29:
2829:10.1007/978-1-4612-0979-9
2760:Semisimple representation
2615:category of vector spaces
2190:can be decomposed into a
2112:{\displaystyle x_{2}^{3}}
2080:{\displaystyle x_{1}^{3}}
702:continuous representation
666:{\displaystyle n\times n}
623:it is common to choose a
119:representation of a group
2765:
2696:scalar transformations".
698:topological vector space
163:and to geometry. If the
110:, they describe how the
30:Not to be confused with
2666:are homomorphisms from
873:faithful representation
619:is of finite dimension
293:Wigner's classification
241:Linear algebraic groups
32:Presentation of a group
2780:"1.4: Representations"
2700:affine representations
2537:with a single object;
2466:
2443:
2328:
2113:
2081:
2049:
2020:
2000:
1971:
1915:
1889:
1784:
1590:
1551:
1353:
1307:
1109:
1043:
911:, two representations
859:
667:
579:
445:
197:locally compact groups
125:from the group to the
74:linear transformations
50:
41:A representation of a
2891:Representation theory
2702:: in the category of
2688:: in the category of
2662:. Representations in
2467:
2444:
2329:
2114:
2082:
2050:
2021:
2001:
1999:{\displaystyle S_{3}}
1972:
1916:
1890:
1785:
1591:
1552:
1354:
1308:
1110:
1044:
860:
668:
596:and the dimension of
580:
446:
275:semisimple Lie groups
131:linear representation
90:matrix multiplication
63:group representations
59:representation theory
40:
18:Group representations
2784:Chemistry LibreTexts
2456:
2339:
2294:
2261:, such that for all
2091:
2059:
2048:{\displaystyle (12)}
2033:
2010:
1983:
1925:
1905:
1800:
1603:
1571:
1366:
1334:
1122:
1090:
984:
792:
766:of a representation
712:is a representation
651:
594:representation space
461:
407:
394:general linear group
337:algebraically closed
244:(or more generally
2710:acts affinely upon
2706:. For example, the
2533:can be viewed as a
2108:
2076:
674:invertible matrices
283:semidirect products
86:invertible matrices
2866:Yurii I. Lyubich.
2740:Molecular symmetry
2600:In the case where
2591:automorphism group
2498:group homomorphism
2462:
2439:
2324:
2109:
2094:
2077:
2062:
2045:
2016:
1996:
1967:
1911:
1885:
1880:
1839:
1780:
1774:
1710:
1653:
1586:
1547:
1538:
1470:
1416:
1349:
1303:
1294:
1226:
1172:
1105:
1039:
855:
778:whose image under
663:
615:In the case where
575:
441:
382:group homomorphism
345:order of the group
261:algebraic geometry
218:Peter–Weyl theorem
210:Pontryagin duality
133:. Some people use
127:automorphism group
65:describe abstract
51:
2838:978-0-387-97495-8
2690:projective spaces
2465:{\displaystyle 1}
2196:Maschke's theorem
2148:subrepresentation
2019:{\displaystyle V}
1914:{\displaystyle V}
1898:Another example:
1879:
1875:
1854:
1838:
1812:
690:topological group
541:
214:Fourier transform
206:harmonic analysis
16:(Redirected from
2903:
2858:
2795:
2794:
2792:
2791:
2776:
2735:Character theory
2627:category of sets
2471:
2469:
2468:
2463:
2448:
2446:
2445:
2440:
2420:
2419:
2398:
2397:
2367:
2366:
2357:
2356:
2333:
2331:
2330:
2325:
2118:
2116:
2115:
2110:
2107:
2102:
2086:
2084:
2083:
2078:
2075:
2070:
2054:
2052:
2051:
2046:
2025:
2023:
2022:
2017:
2005:
2003:
2002:
1997:
1995:
1994:
1976:
1974:
1973:
1968:
1963:
1962:
1950:
1949:
1937:
1936:
1920:
1918:
1917:
1912:
1894:
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1831:
1813:
1810:
1789:
1787:
1786:
1781:
1779:
1778:
1737:
1733:
1732:
1715:
1714:
1673:
1658:
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1619:
1595:
1593:
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1556:
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1518:
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1497:
1493:
1492:
1475:
1474:
1436:
1421:
1420:
1382:
1358:
1356:
1355:
1350:
1348:
1347:
1342:
1312:
1310:
1309:
1304:
1299:
1298:
1291:
1290:
1253:
1249:
1248:
1231:
1230:
1192:
1177:
1176:
1138:
1114:
1112:
1111:
1106:
1104:
1103:
1098:
1048:
1046:
1045:
1040:
1020:
1019:
966:so that for all
965:
940:
925:
885:
864:
862:
861:
856:
851:
847:
846:
754:
730:
672:
670:
669:
664:
646:
631:and identify GL(
584:
582:
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576:
565:
564:
552:
551:
542:
539:
530:
529:
511:
510:
489:
488:
479:
478:
450:
448:
447:
442:
440:
429:
331:, and fields of
265:Zariski topology
21:
2911:
2910:
2906:
2905:
2904:
2902:
2901:
2900:
2876:
2875:
2839:
2809:Fulton, William
2804:
2799:
2798:
2789:
2787:
2778:
2777:
2773:
2768:
2730:Character table
2721:
2712:Euclidean space
2708:Euclidean group
2612:
2527:
2509:
2502:symmetric group
2457:
2454:
2453:
2415:
2411:
2393:
2389:
2362:
2358:
2352:
2348:
2340:
2337:
2336:
2295:
2292:
2291:
2274:
2267:
2217:
2212:
2210:Generalizations
2200:complex numbers
2131:
2125:
2103:
2098:
2092:
2089:
2088:
2071:
2066:
2060:
2057:
2056:
2034:
2031:
2030:
2011:
2008:
2007:
1990:
1986:
1984:
1981:
1980:
1958:
1954:
1945:
1941:
1932:
1928:
1926:
1923:
1922:
1906:
1903:
1902:
1868:
1851:
1829:
1809:
1801:
1798:
1797:
1773:
1772:
1767:
1758:
1757:
1752:
1742:
1741:
1728:
1724:
1720:
1709:
1708:
1703:
1697:
1696:
1688:
1678:
1677:
1663:
1652:
1651:
1646:
1640:
1639:
1634:
1624:
1623:
1609:
1604:
1601:
1600:
1596:by τ given by:
1580:
1575:
1574:
1572:
1569:
1568:
1566:
1537:
1536:
1531:
1525:
1524:
1519:
1513:
1509:
1502:
1501:
1488:
1484:
1480:
1469:
1468:
1463:
1457:
1456:
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1441:
1440:
1426:
1415:
1414:
1409:
1403:
1402:
1397:
1387:
1386:
1372:
1367:
1364:
1363:
1343:
1338:
1337:
1335:
1332:
1331:
1329:
1293:
1292:
1286:
1282:
1280:
1274:
1273:
1268:
1258:
1257:
1244:
1240:
1236:
1225:
1224:
1219:
1213:
1212:
1207:
1197:
1196:
1182:
1171:
1170:
1165:
1159:
1158:
1153:
1143:
1142:
1128:
1123:
1120:
1119:
1099:
1094:
1093:
1091:
1088:
1087:
1073:
1057:
1012:
1008:
985:
982:
981:
953:
941:are said to be
927:
912:
876:
839:
811:
807:
793:
790:
789:
732:
717:
652:
649:
648:
647:, the group of
636:
560:
556:
547:
543:
538:
525:
521:
506:
502:
484:
480:
474:
470:
462:
459:
458:
430:
422:
408:
405:
404:
353:
321:complex numbers
161:crystallography
147:
100:group-theoretic
35:
28:
23:
22:
15:
12:
11:
5:
2909:
2899:
2898:
2893:
2888:
2874:
2873:
2864:
2837:
2803:
2800:
2797:
2796:
2770:
2769:
2767:
2764:
2763:
2762:
2757:
2752:
2747:
2742:
2737:
2732:
2727:
2720:
2717:
2716:
2715:
2697:
2608:
2551:representation
2526:
2523:
2505:
2500:from G to the
2461:
2450:
2449:
2438:
2435:
2432:
2429:
2426:
2423:
2418:
2414:
2410:
2407:
2404:
2401:
2396:
2392:
2388:
2385:
2382:
2379:
2376:
2373:
2370:
2365:
2361:
2355:
2351:
2347:
2344:
2334:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2299:
2272:
2265:
2241:is given by a
2216:
2213:
2211:
2208:
2180:characteristic
2127:Main article:
2124:
2121:
2106:
2101:
2097:
2074:
2069:
2065:
2044:
2041:
2038:
2029:For instance,
2015:
1993:
1989:
1966:
1961:
1957:
1953:
1948:
1944:
1940:
1935:
1931:
1910:
1896:
1895:
1884:
1878:
1874:
1867:
1864:
1861:
1858:
1850:
1847:
1843:
1837:
1834:
1828:
1825:
1822:
1819:
1816:
1808:
1805:
1791:
1790:
1777:
1771:
1768:
1766:
1763:
1760:
1759:
1756:
1753:
1751:
1748:
1747:
1745:
1740:
1736:
1731:
1727:
1723:
1719:
1713:
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903:vector spaces
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357:representation
352:
349:
341:characteristic
335:. In general,
333:p-adic numbers
297:
296:
268:
237:
221:
193:Compact groups
189:
169:characteristic
146:
143:
139:representation
112:symmetry group
104:linear algebra
26:
9:
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2:
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2188:finite groups
2185:
2182:of the field
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2158:
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1067:= 1. The set
1066:
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682:
681:
679:
676:on the field
675:
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611:
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603:
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540:for all
534:
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330:
329:finite fields
326:
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313:
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306:Hilbert space
302:
294:
290:
289:
288:Mackey theory
284:
280:
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266:
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248:group schemes
243:
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156:Finite groups
153:
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151:
142:
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132:
128:
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109:
105:
101:
96:
93:
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83:
82:automorphisms
79:
75:
72:
68:
64:
60:
56:
48:
44:
39:
33:
19:
2886:Group theory
2867:
2816:
2788:. Retrieved
2786:. 2019-09-04
2783:
2774:
2680:
2675:
2667:
2663:
2659:
2653:
2646:
2637:
2633:
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2530:
2529:Every group
2528:
2519:group action
2516:
2511:
2506:
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2224:
2220:
2218:
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2183:
2177:
2164:
2160:
2156:
2152:
2146:
2145:is called a
2143:group action
2138:
2134:
2132:
2123:Reducibility
2028:
1978:
1900:
1897:
1792:
1561:
1559:
1324:
1322:
1315:
1084:cyclic group
1079:
1075:
1068:
1064:
1060:
1058:
971:
967:
962:
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954:
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601:
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593:
589:
587:
453:
397:
389:
385:
377:
370:
368:vector space
363:
356:
354:
325:real numbers
314:
310:Banach space
301:vector space
298:
286:
278:
274:
270:
256:
252:
245:
239:
223:
216:. See also:
202:Haar measure
191:
179:
175:
171:
154:
148:
138:
134:
130:
123:homomorphism
118:
116:
97:
94:
78:vector space
69:in terms of
62:
55:mathematical
52:
2813:Harris, Joe
2486:permutation
2161:irreducible
2133:A subspace
951:isomorphism
770:of a group
731:defined by
351:Definitions
135:realization
2880:Categories
2861:Lie groups
2802:References
2790:2021-06-23
2488:) for all
2192:direct sum
1560:The group
1115:given by:
1082:} forms a
947:isomorphic
943:equivalent
899:Given two
757:continuous
454:such that
225:Lie groups
2855:246650103
2539:morphisms
2482:bijection
2406:ρ
2384:ρ
2343:ρ
2298:ρ
2245:ρ :
2169:composite
2165:reducible
1827:−
1762:−
1718:τ
1691:−
1661:τ
1607:τ
1478:σ
1424:σ
1370:σ
1234:ρ
1180:ρ
1126:ρ
1025:π
1014:−
1010:α
1006:∘
994:ρ
991:∘
988:α
888:injective
825:ρ
822:∣
816:∈
802:ρ
799:
718:Φ :
658:×
602:dimension
567:∈
516:ρ
497:ρ
465:ρ
420:→
414::
411:ρ
312:, etc.).
174:, and if
117:The term
71:bijective
57:field of
2815:(1991).
2719:See also
2649:-modules
2535:category
2279:and all
2243:function
2006:acts on
1055:Examples
957: :
931: :
916: :
295:methods.
279:solvable
2847:1153249
2670:to the
2625:in the
2589:), the
2585:to Aut(
2563:functor
2480:) is a
2227:) of a
635:) with
392:), the
373:over a
246:affine
108:physics
53:In the
2853:
2845:
2835:
2640:, the
2613:, the
2452:where
2055:sends
1793:where
1074:= {1,
764:kernel
606:degree
388:to GL(
208:. The
67:groups
2766:Notes
2694:up to
2632:When
2565:from
2561:is a
2234:on a
2229:group
2173:prime
2151:. If
1979:Then
935:→ GL(
920:→ GL(
880:→ GL(
696:is a
688:is a
625:basis
588:Here
384:from
380:is a
375:field
366:on a
361:group
359:of a
317:field
165:field
106:. In
76:of a
43:group
2851:OCLC
2833:ISBN
2606:Vect
2549:, a
2484:(or
2171:nor
1901:Let
926:and
907:and
762:The
741:) =
700:, a
692:and
627:for
232:and
2825:doi
2664:Top
2660:Top
2636:is
2604:is
2593:of
2577:in
2569:to
2557:in
2553:of
2510:of
2492:in
2283:in
2275:in
2257:to
2236:set
2137:of
2087:to
1330:on
970:in
945:or
886:is
796:ker
755:is
708:on
704:of
684:If
637:GL(
604:or
396:on
255:or
195:or
2882::
2849:.
2843:MR
2841:.
2831:.
2819:.
2811:;
2782:.
2678:.
2658:,
2651:.
2638:Ab
2629:.
2597:.
2521:.
2514:.
2287::
2268:,
2249:→
2219:A
2175:.
2119:.
2040:12
1853:Im
1811:Re
1320:.
1078:,
961:→
871:A
749:)(
737:,
733:Φ(
726:→
722:×
680:.
641:,
355:A
347:.
327:,
308:,
92:.
61:,
2863:.
2857:.
2827::
2793:.
2714:.
2676:X
2668:G
2647:G
2634:C
2623:G
2619:K
2610:K
2602:C
2595:X
2587:X
2583:G
2579:C
2575:X
2571:C
2567:G
2559:C
2555:G
2547:C
2543:G
2531:G
2512:X
2507:X
2504:S
2494:G
2490:g
2478:g
2474:G
2460:1
2437:,
2434:]
2431:]
2428:x
2425:[
2422:)
2417:2
2413:g
2409:(
2403:[
2400:)
2395:1
2391:g
2387:(
2381:=
2378:]
2375:x
2372:[
2369:)
2364:2
2360:g
2354:1
2350:g
2346:(
2322:x
2319:=
2316:]
2313:x
2310:[
2307:)
2304:1
2301:(
2285:X
2281:x
2277:G
2273:2
2270:g
2266:1
2263:g
2259:X
2255:X
2251:X
2247:G
2239:X
2232:G
2184:K
2157:V
2153:V
2139:V
2135:W
2105:3
2100:2
2096:x
2073:3
2068:1
2064:x
2043:)
2037:(
2014:V
1992:3
1988:S
1965:.
1960:3
1956:x
1952:,
1947:2
1943:x
1939:,
1934:1
1930:x
1909:V
1883:.
1877:2
1873:3
1866:=
1863:)
1860:u
1857:(
1849:=
1846:b
1842:,
1836:2
1833:1
1824:=
1821:)
1818:u
1815:(
1807:=
1804:a
1776:]
1770:a
1765:b
1755:b
1750:a
1744:[
1739:=
1735:)
1730:2
1726:u
1722:(
1712:]
1706:a
1701:b
1694:b
1686:a
1680:[
1675:=
1671:)
1668:u
1665:(
1655:]
1649:1
1644:0
1637:0
1632:1
1626:[
1621:=
1617:)
1614:1
1611:(
1582:2
1577:R
1565:3
1562:C
1545:.
1540:]
1534:1
1529:0
1522:0
1515:2
1511:u
1504:[
1499:=
1495:)
1490:2
1486:u
1482:(
1472:]
1466:1
1461:0
1454:0
1449:u
1443:[
1438:=
1434:)
1431:u
1428:(
1418:]
1412:1
1407:0
1400:0
1395:1
1389:[
1384:=
1380:)
1377:1
1374:(
1345:2
1340:C
1328:3
1325:C
1301:.
1296:]
1288:2
1284:u
1278:0
1271:0
1266:1
1260:[
1255:=
1251:)
1246:2
1242:u
1238:(
1228:]
1222:u
1217:0
1210:0
1205:1
1199:[
1194:=
1190:)
1187:u
1184:(
1174:]
1168:1
1163:0
1156:0
1151:1
1145:[
1140:=
1136:)
1133:1
1130:(
1101:2
1096:C
1080:u
1076:u
1072:3
1069:C
1065:u
1061:u
1037:.
1034:)
1031:g
1028:(
1022:=
1017:1
1003:)
1000:g
997:(
974:,
972:G
968:g
963:W
959:V
955:α
939:)
937:W
933:G
929:π
924:)
922:V
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914:ρ
909:W
905:V
901:K
892:e
884:)
882:V
878:G
853:.
849:}
844:d
841:i
837:=
834:)
831:g
828:(
819:G
813:g
809:{
805:=
780:ρ
776:G
772:G
768:ρ
759:.
753:)
751:v
747:g
745:(
743:ρ
739:v
735:g
728:V
724:V
720:G
714:ρ
710:V
706:G
694:V
686:G
678:K
661:n
655:n
645:)
643:K
639:n
633:V
629:V
621:n
617:V
610:V
598:V
590:V
573:.
570:G
562:2
558:g
554:,
549:1
545:g
535:,
532:)
527:2
523:g
519:(
513:)
508:1
504:g
500:(
494:=
491:)
486:2
482:g
476:1
472:g
468:(
438:)
435:V
432:(
427:L
424:G
417:G
398:V
390:V
386:G
378:K
371:V
364:G
257:C
253:R
236:.
220:.
188:.
176:p
172:p
34:.
20:)
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