33:
827:
2530:
1206:
576:
822:{\displaystyle {\begin{array}{ccc}GL(\mathbf {d} )\times \operatorname {Rep} (Q,\mathbf {d} )&\longrightarrow &\operatorname {Rep} (Q,\mathbf {d} )\\{\Big (}(g_{i}),(V_{i},V(\alpha )){\Big )}&\longmapsto &(V_{i},g_{t(\alpha )}\cdot V(\alpha )\cdot g_{s(\alpha )}^{-1})\end{array}}}
1047:
1829:
2168:
543:
406:
1623:
2889:
1468:
861:
1634:
1968:
1183:
426:
2233:
1901:
1307:
311:
1523:
2330:
299:
198:
2621:
62:
1335:
239:)) by simultaneous base change. Such action induces one on the ring of functions. The ones which are invariants up to a character of the group are called
1042:{\displaystyle {\begin{array}{ccc}GL(\mathbf {d} )\times k&\longrightarrow &k\\(g,f)&\longmapsto &g\cdot f(-):=f(g^{-1}.-)\end{array}}}
1824:{\displaystyle SI(Q,\mathbf {d} )_{\sigma }:=\{f\in k:g\cdot f=\prod _{i\in Q_{0}}\det(g_{i})^{\sigma _{i}}f,\forall g\in GL(\mathbf {d} )\}.}
84:
2163:{\displaystyle (g_{1},g_{2})\cdot {\det }^{u}(B)={\det }^{u}(g_{2}^{-1}Bg_{1})={\det }^{u}(g_{1}){\det }^{-u}(g_{2}){\det }^{u}(B)}
538:{\displaystyle \bigoplus _{\alpha \in Q_{1}}\operatorname {Hom} _{k}(k^{\mathbf {d} (s(\alpha ))},k^{\mathbf {d} (t(\alpha ))})}
3036:
1095:
110:
2252:
2179:
1862:
3031:
45:
3026:
55:
49:
41:
2914:
401:{\displaystyle \operatorname {Rep} (Q,\mathbf {d} ):=\{V\in \operatorname {Rep} (Q):V_{i}=\mathbf {d} (i)\}}
1618:{\displaystyle SI(Q,\mathbf {d} )=\bigoplus _{\sigma \in \mathbb {Z} ^{Q_{0}}}SI(Q,\mathbf {d} )_{\sigma }}
2529:
66:
1234:
272:
244:
98:
2313:
282:
181:
3006:
1205:
2949:"A classification of irreducible prehomogeneous vector spaces and their relative invariants."
2990:
2960:
2926:
2884:{\displaystyle SI(Q,\mathbf {d} )={\frac {k}{D_{1,2}D_{3,4}+D_{1,4}D_{2,3}-D_{1,3}D_{2,4}}}}
3002:
2974:
2940:
2463:
Skowronski–Weyman provided a geometric characterization of the class of tame quivers (i.e.
2998:
2970:
2936:
2355:
Furthermore, we have an interpretation for the generators of this polynomial algebra. Let
221:
17:
866:
581:
243:. They form a ring whose structure reflects representation-theoretical properties of the
2521:
2487:
2483:
2468:
2464:
2264:
2244:
1486:-dimensional semi-simple representation, therefore any invariant function is constant.
3020:
3010:
2401:
s are arranged in increasing order with respect to the codimension so that the first
1463:{\displaystyle \det(A-t\mathbb {I} )=t^{n}-c_{1}(A)t^{n-1}+\cdots +(-1)^{n}c_{n}(A)}
2915:"Semi-invariants of quivers and saturation for Littlewood–Richardson coefficients."
2900:
2981:
Skowronski, A.; Weyman, J. (2000), "The algebras of semi-invariants of quivers.",
2931:
2948:
174:) are, respectively, the starting and the ending vertices of α. Given an element
2965:
2173:
The ring of semi-invariants equals the polynomial ring generated by det, i.e.
1517:), with a richer structure called ring of semi-invariants. It decomposes as
2271:
a dimension vector. Let Σ be the set of weights σ such that there exists
548:
Such affine variety is endowed with an action of the algebraic group GL(
2994:
2448:) consisting of invertible matrices. Then we immediately recover SI(Q,(
2255:. Sato and Kimura described the ring of semi-invariants in such case.
2239:
Characterization of representation type through semi-invariant theory
1874:
2288:
non-zero and irreducible. Then the following properties hold true.
2479:
Let Q be a finite connected quiver. The following are equivalent:
1489:
Elements which are invariants with respect to the subgroup SL(
26:
2243:
For quivers of finite representation-type, that is to say
2615:). Such functions generate the ring of semi-invariants:
1329:), as the coefficients of the characteristic polynomial
1228:) is given by usual conjugation. The invariant ring is
1178:{\displaystyle I(Q,\mathbf {d} ):=k^{GL(\mathbf {d} )}}
1482:
has a unique closed orbit corresponding to the unique
2624:
2316:
2182:
1971:
1865:
1637:
1526:
1338:
1237:
1098:
864:
579:
429:
314:
285:
184:
2512:) is either a polynomial algebra or a hypersurface.
2251:
admits an open dense orbit. In other words, it is a
2310:ii) All weights in Σ are linearly independent over
1922:is congruent to the set of square matrices of size
1478:In case Q has neither loops nor cycles the variety
2883:
2392:is closed and irreducible. We can assume that the
2324:
2227:
2162:
1895:
1823:
1617:
1462:
1301:
1177:
1041:
821:
537:
400:
293:
192:
851:We have an induced action on the coordinate ring
720:
663:
2228:{\displaystyle {\mathsf {SI}}(Q,\mathbf {d} )=k}
2219:
2141:
2110:
2082:
2030:
2006:
1896:{\displaystyle 1{\xrightarrow {\ \ \alpha \ }}2}
1750:
1339:
200:, the set of representations of Q with dim
54:but its sources remain unclear because it lacks
570:)) by simultaneous base change on each vertex:
2409:is the zero-set of the irreducible polynomial
220:It is naturally endowed with an action of the
420:this can be identified with the vector space
8:
1815:
1671:
395:
341:
1065:is called an invariant (with respect to GL(
2339:) is the polynomial ring generated by the
2964:
2930:
2866:
2850:
2831:
2815:
2796:
2780:
2759:
2740:
2721:
2702:
2683:
2664:
2651:
2640:
2623:
2318:
2317:
2315:
2202:
2184:
2183:
2181:
2145:
2140:
2130:
2114:
2109:
2099:
2086:
2081:
2068:
2052:
2047:
2034:
2029:
2010:
2005:
1992:
1979:
1970:
1869:
1864:
1807:
1775:
1770:
1760:
1742:
1731:
1701:
1662:
1653:
1636:
1609:
1600:
1575:
1570:
1566:
1565:
1557:
1542:
1525:
1445:
1435:
1401:
1382:
1369:
1355:
1354:
1337:
1290:
1271:
1250:
1236:
1165:
1155:
1143:
1111:
1097:
1017:
949:
910:
878:
865:
863:
844:) are isomorphic if and only if their GL(
803:
789:
752:
739:
719:
718:
694:
675:
662:
661:
649:
619:
593:
580:
578:
507:
506:
474:
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457:
445:
434:
428:
381:
372:
330:
313:
305:-dimensional representations is given by
287:
286:
284:
186:
185:
183:
85:Learn how and when to remove this message
1844:is called semi-invariant of weight
411:Once fixed bases for each vector space
2432:In the example above the action of GL(
2188:
2185:
7:
1302:{\displaystyle I(Q,\mathbf {d} )=k}
1954:)) is a semi-invariant of weight (
1789:
1220:) the representation space is End(
25:
2291:i) For every weight σ we have dim
1934:). The function defined, for any
2913:Derksen, H.; Weyman, J. (2000),
2641:
2548:can be identified with a 4-ple (
2528:
2203:
1808:
1702:
1654:
1601:
1543:
1251:
1204:
1166:
1144:
1112:
950:
911:
879:
650:
620:
594:
508:
475:
382:
331:
122:to each vertex and a linear map
31:
2504:iii) For each dimension vector
2471:) in terms of semi-invariants.
2771:
2657:
2645:
2631:
2493:ii) For each dimension vector
2222:
2216:
2207:
2193:
2157:
2151:
2136:
2123:
2105:
2092:
2074:
2040:
2022:
2016:
1998:
1972:
1812:
1804:
1767:
1753:
1709:
1706:
1692:
1683:
1659:
1644:
1606:
1591:
1547:
1533:
1457:
1451:
1432:
1422:
1394:
1388:
1359:
1342:
1296:
1264:
1255:
1241:
1200:Consider the 1-loop quiver Q:
1188:is in general a subalgebra of
1170:
1162:
1152:
1148:
1134:
1125:
1116:
1102:
1032:
1010:
1001:
995:
981:
976:
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957:
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931:
923:
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915:
901:
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883:
875:
812:
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793:
779:
773:
762:
756:
732:
727:
715:
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687:
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654:
640:
629:
624:
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532:
527:
524:
518:
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494:
491:
485:
479:
466:
392:
386:
362:
356:
335:
321:
275:. Consider a dimension vector
217:has a vector space structure.
1:
2947:Sato, M.; Kimura, T. (1977),
2932:10.1090/S0894-0347-00-00331-3
2593:the function defined on each
1834:A function belonging to SI(Q,
2501:) is complete intersection.
2325:{\displaystyle \mathbb {Q} }
294:{\displaystyle \mathbb {N} }
193:{\displaystyle \mathbb {N} }
113:of Q assigns a vector space
3053:
2540:= (1,1,1,1,2). An element
2536:Pick the dimension vector
2405:have codimension one and Z
2253:prehomogenous vector space
832:By definition two modules
18:Semi-invariants of quivers
2966:10.1017/S0027763000017633
2475:Skowronski–Weyman theorem
1089:). The set of invariants
2359:be the open orbit, then
279:, that is an element in
101:Q with set of vertices Q
97:In mathematics, given a
40:This article includes a
2440:) has an open orbit on
1856:Consider the quiver Q:
1321:s are defined, for any
1224:) and the action of GL(
69:more precise citations.
2885:
2326:
2229:
2164:
1897:
1825:
1619:
1464:
1303:
1179:
1043:
823:
539:
402:
295:
194:
3037:Representation theory
2886:
2327:
2230:
2165:
1898:
1826:
1620:
1513:)) form a ring, SI(Q,
1465:
1304:
1180:
1053:Polynomial invariants
1044:
824:
540:
403:
296:
195:
2622:
2314:
2180:
1969:
1863:
1635:
1524:
1336:
1235:
1096:
862:
577:
427:
312:
283:
182:
2919:J. Amer. Math. Soc.
2508:, the algebra SI(Q,
2497:, the algebra SI(Q,
2259:Sato–Kimura theorem
2247:, the vector space
2060:
1887:
848:)-orbits coincide.
811:
105:and set of arrows Q
2995:10.1007/bf01234798
2881:
2322:
2225:
2160:
2043:
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1749:
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819:
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785:
535:
452:
398:
291:
190:
42:list of references
2983:Transform. Groups
2879:
2576:) of matrices in
2482:i) Q is either a
2469:Euclidean quivers
1888:
1886:
1880:
1877:
1727:
1553:
430:
154:)) to each arrow
95:
94:
87:
16:(Redirected from
3044:
3032:Invariant theory
3013:
2977:
2968:
2943:
2934:
2890:
2888:
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2732:
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2713:
2712:
2694:
2693:
2675:
2674:
2652:
2644:
2532:
2522:Euclidean quiver
2488:Euclidean quiver
2331:
2329:
2328:
2323:
2321:
2234:
2232:
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2206:
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2191:
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2015:
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2009:
1997:
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1984:
1983:
1918:). In this case
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65:this article by
56:inline citations
35:
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3027:Directed graphs
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2953:Nagoya Math. J.
2946:
2925:(13): 467–479,
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1474:Semi-invariants
1441:
1431:
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1233:
1232:
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241:semi-invariants
230:
229:
222:algebraic group
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46:related reading
36:
32:
23:
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2989:(4): 361–402,
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2245:Dynkin quivers
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2224:
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92:
50:external links
39:
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2520:Consider the
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2500:
2496:
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2484:Dynkin quiver
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2265:Dynkin quiver
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2019:
2011:
2001:
1993:
1989:
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886:
872:
869:
858:
857:
856:
855:by defining:
854:
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843:
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790:
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308:
307:
306:
304:
301:. The set of
278:
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246:
242:
238:
234:
223:
218:
216:
213:(i) for each
212:
207:
203:
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149:
145:
141:
137:
133:
129:
125:
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116:
112:
100:
89:
86:
78:
68:
64:
58:
57:
51:
47:
43:
38:
29:
28:
19:
2986:
2982:
2956:
2952:
2922:
2918:
2901:Wild problem
2611:
2607:
2602:
2598:
2594:
2589:
2585:
2581:
2580:(1,2). Call
2577:
2570:
2563:
2556:
2549:
2545:
2541:
2537:
2535:
2519:
2509:
2505:
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2462:
2457:
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2449:
2445:
2441:
2437:
2433:
2431:
2421:
2417:
2416:, then SI(Q,
2410:
2402:
2397:
2393:
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2356:
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2268:
2262:
2248:
2242:
2172:
1959:
1955:
1951:
1947:
1943:
1939:
1935:
1931:
1927:
1923:
1919:
1915:
1911:
1907:
1905:
1855:
1845:
1840:
1835:
1833:
1627:
1514:
1510:
1506:
1496:
1490:
1488:
1483:
1479:
1477:
1326:
1322:
1317:
1313:
1311:
1225:
1221:
1217:
1213:
1211:
1199:
1189:
1187:
1086:
1082:
1078:
1074:
1070:
1066:
1062:
1058:
1056:
852:
850:
845:
841:
837:
833:
831:
567:
563:
554:
549:
547:
416:
412:
410:
302:
276:
268:
264:
254:
240:
236:
232:
219:
214:
210:
205:
201:
175:
171:
167:
163:
159:
155:
151:
147:
143:
139:
135:
131:
127:
123:
118:
114:
96:
81:
72:
61:Please help
53:
2383:where each
2263:Let Q be a
1493:) := Π
1057:An element
552:) := Π
251:Definitions
67:introducing
3021:Categories
2907:References
2335:iii) SI(Q,
1962:) in fact
1946:), as det(
1312:where the
255:Let Q = (Q
3011:120708005
2959:: 1–155,
2844:−
2116:−
2054:−
2002:⋅
1882:α
1796:∈
1790:∀
1773:σ
1736:∈
1729:∏
1719:⋅
1690:
1678:∈
1664:σ
1611:σ
1562:∈
1559:σ
1555:⨁
1426:−
1417:⋯
1406:−
1376:−
1349:−
1281:…
1132:
1030:−
1019:−
999:−
990:⋅
982:⟼
938:
924:⟶
899:
887:×
805:−
797:α
783:⋅
777:α
768:⋅
760:α
728:⟼
710:α
638:
630:⟶
608:
602:×
522:α
489:α
464:
439:∈
436:α
432:⨁
354:
348:∈
319:
75:June 2020
2895:See also
2374:∪ ... ∪
1872:→
1081:for any
840:∈ Rep(Q,
158:, where
3003:1800533
2975:0430336
2941:1758750
2597:as det(
2516:Example
2428:Example
2280:∈ SI(Q,
1852:Example
1196:Example
271:) be a
63:improve
3009:
3001:
2973:
2939:
2465:Dynkin
1885:
1879:
1876:
1628:where
1325:∈ End(
1069:)) if
273:quiver
245:quiver
99:quiver
3007:S2CID
2486:or a
2456:)) =
2352:∈ Σ.
2307:≤ 1.
2297:SI(Q,
1085:∈ GL(
142:)) →
48:, or
2467:and
2420:) =
2348:'s,
1906:Fix
1212:For
109:, a
2991:doi
2961:doi
2927:doi
2524:Q:
2220:det
2142:det
2111:det
2083:det
2031:det
2007:det
1910:= (
1751:det
1687:Rep
1505:SL(
1499:∈ Q
1340:det
1216:= (
1129:Rep
935:Rep
896:Rep
635:Rep
605:Rep
562:GL(
557:∈ Q
455:Hom
351:Rep
316:Rep
231:GL(
226:i∈Q
166:),
130:):
3023::
3005:,
2999:MR
2997:,
2985:,
2971:MR
2969:,
2957:65
2955:,
2951:,
2937:MR
2935:,
2921:,
2917:,
2569:,
2562:,
2555:,
2544:∈
2490:.
2460:.
2424:.
2367:=
2363:\
2332:.
2267:,
1958:,−
1938:∈
1926::
1848:.
1669::=
1192:.
1120::=
1077:=
1061:∈
1005::=
339::=
259:,Q
247:.
209:=
178:∈
52:,
44:,
2993::
2987:5
2963::
2929::
2923:3
2874:4
2871:,
2868:2
2864:D
2858:3
2855:,
2852:1
2848:D
2839:3
2836:,
2833:2
2829:D
2823:4
2820:,
2817:1
2813:D
2809:+
2804:4
2801:,
2798:3
2794:D
2788:2
2785:,
2782:1
2778:D
2772:]
2767:4
2764:,
2761:2
2757:D
2753:,
2748:3
2745:,
2742:1
2738:D
2734:,
2729:3
2726:,
2723:2
2719:D
2715:,
2710:4
2707:,
2704:1
2700:D
2696:,
2691:4
2688:,
2685:3
2681:D
2677:,
2672:2
2669:,
2666:1
2662:D
2658:[
2655:k
2649:=
2646:)
2642:d
2638:,
2635:Q
2632:(
2629:I
2626:S
2612:j
2608:A
2606:,
2603:i
2599:A
2595:V
2590:j
2588:,
2586:i
2582:D
2578:M
2574:4
2571:A
2567:3
2564:A
2560:2
2557:A
2553:1
2550:A
2546:k
2542:V
2538:d
2510:d
2506:d
2499:d
2495:d
2458:k
2454:n
2452:,
2450:n
2446:n
2444:(
2442:M
2438:n
2436:,
2434:n
2422:k
2418:d
2414:1
2411:f
2407:i
2403:l
2398:i
2394:Z
2389:i
2385:Z
2380:t
2376:Z
2372:1
2369:Z
2365:O
2361:k
2357:O
2350:σ
2345:σ
2341:f
2337:d
2319:Q
2304:σ
2301:)
2299:d
2294:k
2286:σ
2284:)
2282:d
2277:σ
2273:f
2269:d
2249:k
2223:]
2217:[
2214:k
2211:=
2208:)
2204:d
2200:,
2197:Q
2194:(
2189:I
2186:S
2158:)
2155:B
2152:(
2147:u
2137:)
2132:2
2128:g
2124:(
2119:u
2106:)
2101:1
2097:g
2093:(
2088:u
2078:=
2075:)
2070:1
2066:g
2062:B
2057:1
2049:2
2045:g
2041:(
2036:u
2026:=
2023:)
2020:B
2017:(
2012:u
1999:)
1994:2
1990:g
1986:,
1981:1
1977:g
1973:(
1960:u
1956:u
1952:α
1950:(
1948:B
1944:n
1942:(
1940:M
1936:B
1932:n
1930:(
1928:M
1924:n
1920:k
1916:n
1914:,
1912:n
1908:d
1891:2
1867:1
1846:σ
1841:σ
1838:)
1836:d
1819:.
1816:}
1813:)
1809:d
1805:(
1802:L
1799:G
1793:g
1787:,
1784:f
1777:i
1768:)
1762:i
1758:g
1754:(
1744:0
1740:Q
1733:i
1725:=
1722:f
1716:g
1713::
1710:]
1707:)
1703:d
1699:,
1696:Q
1693:(
1684:[
1681:k
1675:f
1672:{
1660:)
1655:d
1651:,
1648:Q
1645:(
1642:I
1639:S
1607:)
1602:d
1598:,
1595:Q
1592:(
1589:I
1586:S
1577:0
1573:Q
1567:Z
1551:=
1548:)
1544:d
1540:,
1537:Q
1534:(
1531:I
1528:S
1515:d
1511:i
1509:(
1507:d
1503:}
1501:0
1497:i
1495:{
1491:d
1484:d
1480:k
1458:)
1455:A
1452:(
1447:n
1443:c
1437:n
1433:)
1429:1
1423:(
1420:+
1414:+
1409:1
1403:n
1399:t
1395:)
1392:A
1389:(
1384:1
1380:c
1371:n
1367:t
1363:=
1360:)
1356:I
1352:t
1346:A
1343:(
1327:k
1323:A
1318:i
1314:c
1297:]
1292:n
1288:c
1284:,
1278:,
1273:1
1269:c
1265:[
1262:k
1259:=
1256:)
1252:d
1248:,
1245:Q
1242:(
1239:I
1226:n
1222:k
1218:n
1214:d
1190:k
1171:)
1167:d
1163:(
1160:L
1157:G
1153:]
1149:)
1145:d
1141:,
1138:Q
1135:(
1126:[
1123:k
1117:)
1113:d
1109:,
1106:Q
1103:(
1100:I
1087:d
1083:g
1079:f
1075:f
1073:⋅
1071:g
1067:d
1063:k
1059:f
1033:)
1027:.
1022:1
1015:g
1011:(
1008:f
1002:)
996:(
993:f
987:g
977:)
974:f
971:,
968:g
965:(
958:]
955:)
951:d
947:,
944:Q
941:(
932:[
929:k
919:]
916:)
912:d
908:,
905:Q
902:(
893:[
890:k
884:)
880:d
876:(
873:L
870:G
853:k
846:d
842:d
838:N
836:,
834:M
813:)
808:1
800:)
794:(
791:s
787:g
780:)
774:(
771:V
763:)
757:(
754:t
750:g
746:,
741:i
737:V
733:(
721:)
716:)
713:)
707:(
704:V
701:,
696:i
692:V
688:(
685:,
682:)
677:i
673:g
669:(
664:(
655:)
651:d
647:,
644:Q
641:(
625:)
621:d
617:,
614:Q
611:(
599:)
595:d
591:(
588:L
585:G
568:i
566:(
564:d
559:0
555:i
550:d
533:)
528:)
525:)
519:(
516:t
513:(
509:d
504:k
500:,
495:)
492:)
486:(
483:s
480:(
476:d
471:k
467:(
459:k
447:1
443:Q
417:i
413:V
396:}
393:)
390:i
387:(
383:d
379:=
374:i
370:V
366::
363:)
360:Q
357:(
345:V
342:{
336:)
332:d
328:,
325:Q
322:(
303:d
288:N
277:d
269:t
267:,
265:s
263:,
261:1
257:0
237:i
235:(
233:d
228:0
224:Π
215:i
211:d
206:i
202:V
187:N
176:d
172:α
170:(
168:t
164:α
162:(
160:s
156:α
152:α
150:(
148:t
146:(
144:V
140:α
138:(
136:s
134:(
132:V
128:α
126:(
124:V
119:i
115:V
107:1
103:0
88:)
82:(
77:)
73:(
59:.
20:)
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