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Fq(R)285 2842 y Fp(3)445 2901 y Fr(b)-6 b(y)69 b(computer)224 3317 y(Symb)6 b(ols:)165 b Fo(a;)30 b(b;)f(:)g(:)g(:)166 b Fr(vecto)-6 b(r)68 b(\014elds)1152 3516 y Fo(f)8 b(;)30 b(:)f(:)g(:)320 b Fr(fuctions)270 3968 y(input)69 b(fo)-6 b(rm)p 1240 4026 5 200 v 165 w(p)g(rint)69 b(fo)-6 b(rm)p 2255 4026 V 166 w(explanation)p 187 4031 3595 5 v 187 4056 V 270 4197 a Fn(SP[a,b])p 1240 4255 5 200 v 439 w Fm(h)p Fo(a;)30 b(b)p Fm(i)p 2255 4255 V 633 w Fr(scala)-6 b(r)68 b(p)-6 b(ro)6 b(duct)270 4396 y Fn(VP[a,b])p 1240 4454 V 439 w Fo(a)39 b Fm(\002)g Fo(b)p 2255 4454 V 633 w Fr(vecto)-6 b(r)68 b(p)-6 b(ro)6 b(duct)270 4596 y Fn(grad[f])p 1240 4654 V 439 w Fm(r)p Fo(f)p 2255 4654 V 786 w Fr(gradient)270 4817 y Fn(grad[a,b])p 1240 4899 5 246 v 263 w Fm(h)p Fo(a;)30 b Fm(ri)p Fr([)p Fo(b)p Fr(])p 2255 4899 V 371 w(derivative)67 b Fo(a)3306 4759 y Fl(i)3434 4749 y(@)p 3370 4778 214 6 v 3370 4899 a(@)8 b(x)3537 4860 y Fk(i)3623 4817 y Fo(b)270 5040 y Fn(rot[a])p 1240 5099 5 200 v 527 w Fr(rot[)p 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Fr(Such)69 b(p)-6 b(roblems)69 b(app)6 b(ea)-6 b(red)68 b(in)h(calculations)f(fo)-6 b(r)141 3822 y(Gittel,)62 b(Kijo)-6 b(wski,)63 b(Zeidler,)h Fg(The)e(relativistic)d(dy-)141 4054 y(namics)102 b(of)h(the)g (combined)f(pa)-6 b(rticle-\014eld)102 b(sys-)141 4287 y(tem)66 b(in)h(reno)-6 b(rmalized)66 b(classical)h(electro)6 b(dynam-)141 4519 y(ics)p Fr(,)71 b(Commun.)95 b(Math.)h(Phys.)h(198,) 72 b(711)f(-)f(736)141 4752 y(\(1998\).)141 5233 y(Mathematica)65 b(pack)-6 b(age)p 2060 5079 885 5 v 2060 5275 5 197 v 115 w(VECT)-17 b(AN)p 2940 5275 V 2060 5280 885 5 v 141 5466 a(MathSource,)68 b(W)-6 b(olfram)67 b(Resea)-6 b(rch)69 b(Inc.)3396 5763 y Fi(2)p eop %%Page: 3 3 3 2 bop 141 451 a Fs(De\014nitions)224 679 y Fm(\017)166 b(A)49 b Fr(=)g Fm(f)p Fo(a)1058 713 y Fp(1)1148 679 y Fo(;)29 b(:)g(:)g(:)g(;)g(a)1631 717 y Fl(N)1766 679 y Fm(g)69 b Fr(alphab)6 b(et)68 b(\(\014nite)h(set\))224 879 y Fm(\017)166 b Fq(K)69 b Fr(\014eld,)g Fq(K)48 b Fr(=)h Fq(R)p Fo(;)29 b Fq(C)224 1078 y 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y(additional)69 b(conditions)155 3102 y(\(A\))266 b Fm(h)p Fo(a;)30 b(b;)f(c)p Fm(i)98 b Fr(=)f Fm(\000h)p Fo(a;)30 b(c;)f(b)p Fm(i)98 b Fr(=)f Fm(\000h)p Fo(b;)29 b(a;)g(c)p Fm(i)98 b Fr(=)714 3359 y Fm(h)p Fo(b;)30 b(c;)f(a)p Fm(i)98 b Fr(=)f Fm(h)p Fo(c;)30 b(a;)f(b)p Fm(i)98 b Fr(=)f Fm(\000h)p Fo(c;)29 b(b;)g(a)p Fm(i)153 3664 y Fr(\(B\))266 b Fo(a)40 b Fm(\002)e Fr(\()p Fo(b)i Fm(\002)e Fo(c)p Fr(\))98 b(=)g Fm(h)p Fo(a;)29 b(c)p Fm(i)h Fo(b)39 b Fm(\000)g(h)p Fo(a;)29 b(b)p Fm(i)h Fo(c)156 3969 y Fr(\(C\))266 b Fm(h)p Fo(a)40 b Fm(\002)f Fo(b;)29 b(c)39 b Fm(\002)f Fo(d)p Fm(i)99 b Fr(=)e Fm(h)p Fo(a;)29 b(c)p Fm(i)h(h)p Fo(b;)g(d)p Fm(i)39 b(\000)g(h)p Fo(b;)29 b(c)p Fm(i)h(h)p Fo(a;)g(d)p Fm(i)141 4441 y Fr(\(D\))266 b Fm(h)p Fo(a;)30 b(b;)f(c)p Fm(i)h(h)p Fo(u;)f(v)6 b(;)28 b(w)t Fm(i)99 b Fr(=)2253 4099 y Fh(\014)2253 4185 y(\014)2253 4271 y(\014)2253 4357 y(\014)2253 4443 y(\014)2253 4529 y(\014)2253 4615 y(\014)2367 4242 y Fm(h)p Fo(a;)30 b(u)p Fm(i)132 b(h)p Fo(a;)30 b(v)6 b Fm(i)133 b(h)p Fo(a;)29 b(w)t Fm(i)2376 4442 y(h)p Fo(b;)h(u)p Fm(i)151 b(h)p Fo(b;)29 b(v)6 b Fm(i)151 b(h)p Fo(b;)29 b(w)t Fm(i)2376 4641 y(h)p Fo(c;)g(u)p Fm(i)150 b(h)p Fo(c;)29 b(v)6 b Fm(i)150 b(h)p Fo(c;)30 b(w)t Fm(i)3946 4099 y Fh(\014)3946 4185 y(\014)3946 4271 y(\014)3946 4357 y(\014)3946 4443 y(\014)3946 4529 y(\014)3946 4615 y(\014)141 5199 y Fs(Example:)91 b Fq(R)1229 5141 y Fp(3)1388 5199 y Fr(with)69 b(usual)g Fm(h)30 b Fo(:)f(;)58 b(:)29 b Fm(i)p Fr(,)69 b(")p Fm(\002)p Fr(".)3396 5763 y Fi(3)p eop %%Page: 4 4 4 3 bop 141 344 a Fs(Simple)76 b(conclusions)141 826 y Fr(\(A\))70 b(=)-29 b Fm(\))141 1058 y Fs(Co)-6 b(r.1:)84 b Fm(8)28 b Fo(a;)i(b)48 b Fm(2)h(L)55 b Fr(with)f Fo(a)15 b Fm(\002)g Fo(b)48 b Fm(6)p Fr(=)h(0)54 b Fm(\))h Fo(a)15 b Fm(\002)g Fo(b)48 b Fm(?)g Fo(a;)30 b(b)p Fr(.)141 1540 y(Co)-6 b(r.1,)70 b(\(B\),)h(\(C\))e(=)-29 b Fm(\))141 1772 y Fs(Prop.1:)177 b Fr(If)112 b Fm(L)f Fr(contains)h(elements)d Fo(a;)30 b(b)109 b Fm(2)g(L)141 2005 y Fr(which)69 b(ful\014l)g Fo(a)39 b Fm(\002)g Fo(b)49 b Fm(6)p Fr(=)f(0)70 b(then)1450 2362 y(dim)28 b Fm(L)133 b Fr(=)f(3)30 b Fo(:)141 2926 y Fr(Co)-6 b(r.1,)70 b(Prop.1,)h(\(C\))e(=)-29 b Fm(\))141 3159 y Fs(Prop.2:)93 b Fr(If)69 b Fm(f)p Fo(n)1298 3193 y Fp(1)1388 3159 y Fo(;)29 b(n)1573 3193 y Fp(2)1663 3159 y Fo(;)g(n)1848 3193 y Fp(3)1937 3159 y Fm(g)70 b Fr(o)-6 b(rthono)g(rmal)68 b(basis)i(of)141 3391 y Fm(L)f Fr(and)g Fm(f)p Fo(i;)28 b(j;)h(k)t Fm(g)70 b Fr(p)6 b(ermutation)67 b(of)i Fm(f)p Fr(1)p Fo(;)29 b Fr(2)p Fo(;)g Fr(3)p Fm(g)69 b Fr(then)720 3748 y Fo(n)827 3782 y Fl(i)920 3748 y Fm(\002)38 b Fo(n)1202 3782 y Fl(j)1371 3748 y Fr(=)97 b Fo(\027)40 b(n)1862 3787 y Fl(k)2079 3748 y Fr(and)133 b Fm(j)p Fo(\027)11 b Fm(j)97 b Fr(=)g(1)30 b Fo(:)141 4313 y Fr(\(A\))70 b(=)-29 b Fm(\))141 4545 y Fs(Co)-6 b(r.2:)113 b Fr(A)80 b(de\014nition)g(of)h Fo(n)2249 4579 y Fp(1)2384 4545 y Fm(\002)44 b Fo(n)2672 4579 y Fp(2)2842 4545 y Fr(de\014nes)80 b(also)141 4778 y(unique)70 b(values)e(of)i Fo(n)1682 4812 y Fp(1)1810 4778 y Fm(\002)39 b Fo(n)2093 4812 y Fp(3)2183 4778 y Fr(,)69 b Fo(n)2416 4812 y Fp(2)2545 4778 y Fm(\002)38 b Fo(n)2827 4812 y Fp(3)2917 4778 y Fr(.)p 141 5068 3105 5 v 141 5342 5 274 v 188 5259 a Fm(\))68 b Fr(Up)h(to)g(o)-6 b(rientation:)92 b Fq(K)2193 5201 y Fp(3)2331 5259 y Fo(;)78 b Fm(h)30 b Fo(:)f(;)58 b(:)29 b Fm(i)49 b Fo(;)78 b Fm(\002)p 3241 5342 V 141 5347 3105 5 v 3396 5763 a Fi(4)p eop %%Page: 5 5 5 4 bop 141 135 a Fs(Standa)-6 b(rd)77 b(exp)-6 b(ressions)141 382 y Fo(E)58 b Fm(2)49 b(L)69 b Fr(exp)-6 b(ression)70 b(fo)-6 b(rmed)69 b(from)f(the)h Fo(a)3158 416 y Fl(i)3261 382 y Fm(2)49 b(A)70 b Fr(b)-6 b(y)141 582 y(a)69 b(\014nite)f(use)h (of)h Fm(h)29 b Fo(:)g(;)58 b(:)29 b Fm(i)70 b Fr(and)f(")p Fm(\002)p Fr(".)141 829 y(Example:)90 b Fo(E)58 b Fr(=)48 b Fm(h)p Fo(a)30 b(;)58 b(b)p Fm(ih)p Fo(c)30 b(;)59 b(d)39 b Fm(\002)g Fr(\()p Fo(e)g Fm(\002)g Fo(f)18 b Fr(\))p Fm(i)31 b Fr(\()p Fo(a)39 b Fm(\002)g Fo(g)6 b Fr(\))141 1076 y(Consider)80 b(exp)-6 b(ression)79 b Fm(h)30 b Fo(E)38 b(;)59 b(x)29 b Fm(i)p Fr(,)82 b(where)c Fo(x)63 b Fm(2)g(L)79 b Fr(is)141 1275 y(an)69 b(a)-6 b(rbitra)g(ry)68 b(test)g(vecto)-6 b(r.)446 1575 y Fo(E)107 b Fr(=)97 b(0)231 b Fm(,)f(8)28 b Fo(x)48 b Fm(2)h(L)g Fr(:)146 b Fm(h)30 b Fo(E)38 b(;)58 b(x)30 b Fm(i)97 b Fr(=)h(0)141 2008 y Fs(Theo)-6 b(rem)74 b(1:)141 2207 y Fr(Assume,)j Fo(E)1052 2246 y Fl(k)1193 2207 y Fm(2)58 b(L)76 b Fr(is)f(fo)-6 b(rmed)75 b(from)f(a)h(sequence)141 2407 y Fm(f)p Fo(c)305 2441 y Fp(1)395 2407 y Fo(;)29 b(:)g(:)g(:)g(;)g(c)861 2446 y Fl(k)946 2407 y Fm(g)p Fr(,)69 b Fo(c)1236 2441 y Fl(i)1338 2407 y Fm(2)49 b(A)p Fr(,)70 b(of)f(length)g Fo(k)t Fr(.)141 2654 y Fs(\(I\):)58 b Fr(If)f Fo(k)d Fm(\025)48 b Fr(1)58 b(o)6 b(dd,)61 b(then)c Fm(h)30 b Fo(E)2252 2692 y Fl(k)2365 2654 y Fo(;)58 b(x)29 b Fm(i)58 b Fr(can)f(b)6 b(e)57 b(trans-)141 2853 y(fo)-6 b(rmed)68 b(into)p 141 2946 3656 5 v 141 3466 5 521 v 256 3111 a Fm(h)30 b Fo(E)483 3150 y Fl(k)596 3111 y Fo(;)58 b(x)29 b Fm(i)98 b Fr(=)256 3231 y Fh(P)408 3383 y Fl(i)491 3344 y Fo(\013)604 3378 y Fl(i)687 3344 y Fm(h)30 b Fo(b)860 3258 y Fp(\()p Fl(i)p Fp(\))860 3401 y(1)1077 3344 y Fo(;)58 b(b)1259 3258 y Fp(\()p Fl(i)p Fp(\))1259 3401 y(2)1477 3344 y Fm(ih)29 b Fo(b)1717 3258 y Fp(\()p Fl(i)p Fp(\))1717 3401 y(3)1935 3344 y Fo(;)58 b(b)2117 3258 y Fp(\()p Fl(i)p Fp(\))2117 3401 y(4)2334 3344 y Fm(i)30 b Fo(:)f(:)g(:)g Fm(h)h Fo(b)2839 3258 y Fp(\()p Fl(i)p Fp(\))2839 3405 y Fl(k)3056 3344 y Fo(;)58 b(b)3238 3258 y Fp(\()p Fl(i)p Fp(\))3238 3405 y Fl(k)t Fp(+1)3572 3344 y Fm(i)p 3792 3466 V 141 3471 3656 5 v 141 3749 a Fr(where)83 b Fo(\013)831 3783 y Fl(i)953 3749 y Fm(2)69 b Fq(K)82 b Fr(and)i Fm(f)p Fo(b)1924 3664 y Fp(\()p Fl(i)p Fp(\))1924 3806 y(1)2111 3749 y Fo(;)29 b(:)g(:)g(:)h(;)f(b)2577 3664 y Fp(\()p Fl(i)p Fp(\))2577 3811 y Fl(k)t Fp(+1)2881 3749 y Fm(g)82 b Fr(p)6 b(ermuta-)141 3949 y(tion)69 b(of)g Fm(f)p Fo(c)936 3983 y Fp(1)1026 3949 y Fo(;)30 b(:)f(:)g(:)g(;)g(c)1493 3988 y Fl(k)1577 3949 y Fo(;)g(x)p Fm(g)p Fr(.)141 4219 y Fs(\(I)6 b(I\):)104 b Fr(If)e Fo(k)g Fm(\025)96 b Fr(2)104 b(even,)111 b(then)103 b Fm(h)29 b Fo(E)2710 4258 y Fl(k)2823 4219 y Fo(;)59 b(x)29 b Fm(i)103 b Fr(can)f(b)6 b(e)141 4419 y(transfo)-6 b(rmed)68 b(into)p 141 4511 V 141 5032 5 521 v 256 4677 a Fm(h)30 b Fo(E)483 4715 y Fl(k)596 4677 y Fo(;)58 b(x)29 b Fm(i)98 b Fr(=)256 4796 y Fh(P)408 4948 y Fl(i)491 4909 y Fo(\013)604 4943 y Fl(i)687 4909 y Fm(h)30 b Fo(b)860 4824 y Fp(\()p Fl(i)p Fp(\))860 4966 y(1)1077 4909 y Fo(;)58 b(b)1259 4824 y Fp(\()p Fl(i)p Fp(\))1259 4966 y(2)1477 4909 y Fm(i)29 b Fo(:)g(:)g(:)g Fm(h)h(\003)f Fo(;)58 b Fm(\003)29 b(ih)h Fo(b)2488 4824 y Fp(\()p Fl(i)p Fp(\))2488 4971 y Fl(k)t Fj(\000)p Fp(1)2801 4909 y Fo(;)59 b(b)2984 4824 y Fp(\()p Fl(i)p Fp(\))2984 4971 y Fl(k)3201 4909 y Fo(;)f(b)3383 4824 y Fp(\()p Fl(i)p Fp(\))3383 4971 y Fl(k)t Fp(+1)3717 4909 y Fm(i)p 3792 5032 V 141 5037 3656 5 v 141 5315 a Fr(where)83 b Fo(\013)831 5349 y Fl(i)953 5315 y Fm(2)69 b Fq(K)82 b Fr(and)i Fm(f)p Fo(b)1924 5229 y Fp(\()p Fl(i)p Fp(\))1924 5372 y(1)2111 5315 y Fo(;)29 b(:)g(:)g(:)h(;)f(b)2577 5229 y Fp(\()p Fl(i)p Fp(\))2577 5376 y Fl(k)t Fp(+1)2881 5315 y Fm(g)82 b Fr(p)6 b(ermuta-)141 5514 y(tion)69 b(of)g Fm(f)p Fo(c)936 5548 y Fp(1)1026 5514 y Fo(;)30 b(:)f(:)g(:)g(;)g(c)1493 5553 y Fl(k)1577 5514 y Fo(;)g(x)p Fm(g)p Fr(.)3396 5763 y Fi(5)p eop %%Page: 6 6 6 5 bop 141 135 a Fs(T)-19 b(enso)-6 b(rial)76 b(notation)141 412 y(Prop.:)141 617 y Fr(F)-6 b(o)g(r)89 b(every)f Fo(E)1167 656 y Fl(k)1328 617 y Fm(2)77 b(L)89 b Fr(there)f(exists)g(a)h Fm(P)h(\022)77 b(S)3522 656 y Fl(k)t Fp(+1)141 823 y Fr(such)93 b(that)f(the)g(fo)-6 b(rmulas)91 b(of)i(Theo)-6 b(rem)92 b(1)h(can)141 1028 y(b)6 b(e)69 b(written)f(as)208 1290 y(\()p Fq(I)360 1232 y Fj(0)404 1290 y Fr(\))98 b Fo(k)53 b Fm(\025)c Fr(1)p Fo(;)29 b Fr(o)6 b(dd)50 b(:)97 b Fm(\))146 b Fo(l)52 b Fr(=)c(\()p Fo(k)c Fr(+)39 b(1\))p Fo(=)p Fr(2)p 141 1401 3824 5 v 139 1600 5 200 v 208 1542 a Fm(h)29 b Fo(E)434 1581 y Fl(k)547 1542 y Fo(;)58 b(x)30 b Fm(i)97 b Fr(=)p 3962 1600 V 139 1970 5 370 v 274 1667 a Fh(P)208 1929 y Fl(p)p Fj(2P)522 1780 y Fo(\013)635 1801 y Fl(p)761 1780 y Fr(\()p Fo(p)30 b Fr([)p Fo(c)1093 1814 y Fp(1)1223 1780 y Fm(\012)38 b Fo(:)29 b(:)h(:)38 b Fm(\012)h Fo(c)1894 1819 y Fl(k)2017 1780 y Fm(\012)g Fo(x)p Fr(]\))2552 1691 y Fl(i)2602 1719 y Fi(1)2800 1691 y Fl(i)2850 1719 y Fi(2)3047 1691 y Fl(:::)121 b(i)3341 1723 y Fk(l)2430 1837 y Fl(i)2480 1866 y Fi(1)2678 1837 y Fl(i)2728 1866 y Fi(2)2925 1837 y Fl(:::)g(i)3219 1869 y Fk(l)p 3962 1970 V 141 1975 3824 5 v 208 2315 a Fr(\()p Fq(I)5 b(I)437 2257 y Fj(0)481 2315 y Fr(\))98 b Fo(k)54 b Fm(\025)48 b Fr(2)p Fo(;)30 b Fr(even)48 b(:)98 b Fm(\))146 b Fo(l)51 b Fr(=)e(\()p Fo(k)44 b Fm(\000)38 b Fr(2\))p Fo(=)p Fr(2)p 141 2425 V 139 2625 5 200 v 208 2567 a Fm(h)29 b Fo(E)434 2605 y Fl(k)547 2567 y Fo(;)58 b(x)30 b Fm(i)97 b Fr(=)p 3962 2625 V 139 2994 5 370 v 274 2692 a Fh(P)208 2953 y Fl(p)p Fj(2P)522 2805 y Fo(\013)635 2826 y Fl(p)761 2805 y Fr(\()p Fo(p)30 b Fr([)p Fo(c)1093 2839 y Fp(1)1223 2805 y Fm(\012)38 b Fo(:)29 b(:)h(:)38 b Fm(\012)h Fo(c)1894 2844 y Fl(k)2017 2805 y Fm(\012)g Fo(x)p Fr(]\))2552 2716 y Fl(i)2602 2744 y Fi(1)2718 2716 y Fl(:::)h(i)2931 2748 y Fk(l)2973 2716 y Fl(j)3032 2744 y Fi(1)3108 2716 y Fl(j)3167 2744 y Fi(2)3243 2716 y Fl(j)3302 2744 y Fi(3)2430 2862 y Fl(i)2480 2890 y Fi(1)2596 2862 y Fl(:::)g(i)2809 2894 y Fk(l)3382 2805 y Fo(S)3489 2839 y Fl(j)3548 2867 y Fi(1)3624 2839 y Fl(j)3683 2867 y Fi(2)3759 2839 y Fl(j)3818 2867 y Fi(3)p 3962 2994 V 141 2999 3824 5 v 557 3343 a Fm(h)p Fo(a;)30 b(b)p Fm(i)162 b Fr(=)f Fo(a)1517 3377 y Fl(i)1571 3343 y Fo(b)1646 3274 y Fl(i)1798 3343 y Fr(=)97 b(\()p Fo(a)40 b Fm(\012)f Fo(b)p Fr(\))2681 3274 y Fl(i)2600 3388 y(i)201 3573 y Fr(\()p Fo(pT)24 b Fr(\))575 3607 y Fl(i)625 3636 y Fi(1)702 3607 y Fl(:::)o(i)874 3625 y Fk(r)1073 3573 y Fr(:=)133 b Fo(T)1526 3607 y Fl(i)1576 3648 y Fk(p)p Fi(\(1\))1825 3607 y Fl(:::)o(i)1997 3648 y Fk(p)p Fi(\()p Fk(r)s Fi(\))263 3851 y Fo(S)9 b Fr(\()p Fo(a;)30 b(b;)f(c)p Fr(\))133 b(:=)g Fm(h)30 b Fo(a)f(;)59 b(b)29 b(;)58 b(c)30 b Fm(i)146 b Fr(sk)-6 b(ew)42 b Fm(\000)c Fr(symmetric)228 4081 y Fm(h)29 b Fo(a)h(;)58 b(b)30 b(;)58 b(c)29 b Fm(i)162 b Fr(=)f Fo(S)1531 4115 y Fl(j)1590 4144 y Fi(1)1666 4115 y Fl(j)1725 4144 y Fi(2)1801 4115 y Fl(j)1860 4144 y Fi(3)1970 4081 y Fo(a)2063 4013 y Fl(j)2122 4041 y Fi(1)2231 4081 y Fo(b)2306 4013 y Fl(j)2365 4041 y Fi(2)2475 4081 y Fo(c)2551 4013 y Fl(j)2610 4041 y Fi(3)141 4538 y Fm(\))62 b Fr(W)-6 b(e)62 b(can)g(use)g(no)-6 b(rmal)61 b(fo)-6 b(rm)62 b(metho)6 b(ds)61 b(fo)-6 b(r)63 b(ex-)141 4743 y(p)-6 b(ressions)254 5053 y Fo(\034)332 5087 y Fl(i)382 5120 y Fi(2)p Fk(l)q Fi(+1)678 5087 y Fl(:::)o(i)850 5105 y Fk(r)1049 5053 y Fr(:=)1439 4911 y Fh(X)1400 5245 y Fl(p)p Fj(2P)1715 5053 y Fo(\013)1828 5074 y Fl(p)1935 5053 y Fr(\()p Fo(pT)24 b Fr(\))2432 4964 y Fl(j)2491 4992 y Fi(1)2649 4964 y Fl(j)2708 4992 y Fi(2)2865 4964 y Fl(:::)80 b(j)3127 4996 y Fk(l)2309 5110 y Fl(j)2368 5139 y Fi(1)2527 5110 y Fl(j)2586 5139 y Fi(2)2743 5110 y Fl(:::)g(j)3005 5143 y Fk(l)3129 5110 y Fl(i)3179 5143 y Fi(2)p Fk(l)q Fi(+1)3475 5110 y Fl(:::)o(i)3647 5128 y Fk(r)1400 5445 y Fo(\013)1513 5466 y Fl(p)1640 5445 y Fm(2)49 b Fq(K)29 b Fo(;)58 b Fm(P)k(\022)49 b(S)2571 5466 y Fl(r)2674 5445 y Fo(;)59 b(T)73 b Fm(2)49 b(T)3220 5466 y Fl(r)3294 5445 y Fo(V)3396 5763 y Fi(6)p eop %%Page: 7 7 7 6 bop 141 135 a Fs(Determination)70 b(of)h(linea)-6 b(r)72 b(tenso)-6 b(r)71 b(identities)522 437 y Fo(\034)600 471 y Fl(i)650 499 y Fi(1)725 471 y Fl(:::)o(i)897 488 y Fk(r)1097 437 y Fr(:=)1486 295 y Fh(X)1447 628 y Fl(p)p Fj(2P)1762 437 y Fo(c)1838 457 y Fl(p)1945 437 y Fo(T)2047 471 y Fl(i)2097 511 y Fk(p)p Fi(\(1\))2346 471 y Fl(:::)o(i)2518 511 y Fk(p)p Fi(\()p Fk(r)s Fi(\))1447 829 y Fo(T)j Fm(2)49 b(T)1886 849 y Fl(r)1960 829 y Fo(V)68 b(;)59 b(c)2314 849 y Fl(p)2440 829 y Fm(2)49 b Fq(K)29 b Fo(;)58 b Fm(P)63 b(\022)48 b(S)3371 849 y Fl(r)141 1088 y Fs(Def.:)126 b Fr(Every)85 b(pair)h Fo(b)73 b Fr(=)g(\()p Fo(v)2237 1122 y Fp(1)2327 1088 y Fo(;)29 b(:)g(:)g(:)g(;)g(v)2803 1109 y Fl(r)2877 1088 y Fr(\))73 b Fm(2)g Fo(V)3362 1030 y Fl(r)3522 1088 y Fr(and)141 1289 y Fo(T)h Fm(2)48 b(T)579 1309 y Fl(r)654 1289 y Fo(V)108 b Fr(induces)69 b(a)g(group)h(ring)f (element)338 1590 y Fo(T)440 1628 y Fl(b)640 1590 y Fr(:=)1048 1448 y Fh(X)991 1781 y Fl(p)p Fj(2S)1252 1798 y Fk(r)1342 1590 y Fo(T)24 b Fr(\()p Fo(v)1634 1638 y Fl(p)p Fp(\(1\))1932 1590 y Fo(;)30 b(:)f(:)g(:)g(;)g(v)2409 1638 y Fl(p)p Fp(\()p Fl(r)t Fp(\))2690 1590 y Fr(\))h Fo(p)49 b Fm(2)g Fq(K)p Fr([)p Fm(S)3419 1610 y Fl(r)3493 1590 y Fr(])30 b Fo(:)p 141 1903 3918 5 v 141 2604 5 702 v 256 2059 a Fr(If)94 b Fo(T)108 b Fm(2)83 b(T)973 2080 y Fl(r)1047 2059 y Fo(V)133 b Fr(b)6 b(elongs)94 b(to)f(a)g(symmetry)d(class)j Fm(T)256 2292 y Fr(then)81 b(sea)-6 b(rch)81 b(fo)-6 b(r)81 b(the)g(smallest)e(linea)-6 b(r)80 b(subspace)256 2524 y Fo(W)73 b Fm(\022)49 b Fq(K)p Fr([)p Fm(S)997 2545 y Fl(r)1071 2524 y Fr(])69 b(s.t.:)93 b Fm(8)28 b Fo(T)73 b Fm(2)49 b(T)43 b Fo(;)29 b(b)49 b Fm(2)g Fo(V)2733 2466 y Fl(r)2856 2524 y Fr(:)g Fo(T)3064 2563 y Fl(b)3180 2524 y Fm(2)g Fo(W)24 b Fr(.)p 4054 2604 V 141 2609 3918 5 v 141 2835 a(Consider)141 3036 y Fo(W)331 2977 y Fj(?)498 3036 y Fr(:=)49 b Fm(f)p Fo(x)f Fm(2)h Fq(K)p Fr([)p Fm(S)1482 3056 y Fl(r)1557 3036 y Fr(])1615 2977 y Fl(?)1741 3036 y Fm(j)f(8)28 b Fo(w)54 b Fm(2)48 b Fo(W)73 b Fr(:)98 b Fm(h)p Fo(x;)29 b(w)t Fm(i)50 b Fr(=)e(0)p Fm(g)141 3413 y Fr(Then)69 b(every)f Fo(x)49 b Fm(2)g Fo(W)1665 3355 y Fj(?)1852 3413 y Fr(yields)141 3714 y(0)133 b(=)g Fm(h)p Fo(x;)29 b(T)1018 3753 y Fl(b)1085 3714 y Fm(i)49 b Fr(=)1469 3573 y Fh(X)1411 3906 y Fl(p)p Fj(2S)1672 3923 y Fk(r)1763 3714 y Fo(x)1862 3735 y Fl(p)1969 3714 y Fo(T)2071 3753 y Fl(b)2138 3714 y Fr(\()p Fo(p)p Fr(\))79 b(=)2762 3573 y Fh(X)2704 3906 y Fl(p)p Fj(2S)2965 3923 y Fk(r)3056 3714 y Fo(x)3155 3735 y Fl(p)3262 3714 y Fo(T)3364 3748 y Fl(i)3414 3789 y Fk(p)p Fi(\(1\))3662 3748 y Fl(:::)o(i)3834 3789 y Fk(p)p Fi(\()p Fk(r)s Fi(\))671 4112 y Fo(x)770 4133 y Fl(p)896 4112 y Fr(:=)49 b Fm(h)p Fo(x;)29 b(p)p Fm(i)h Fo(;)59 b(p)49 b Fm(2)g(S)2111 4133 y Fl(r)141 4549 y Fm(f)p Fo(h)330 4583 y Fp(1)420 4549 y Fo(;)29 b(:)g(:)h(:)f(;)g(h)912 4570 y Fl(m)1045 4549 y Fm(g)69 b Fr(basis)g(of)g Fo(W)24 b Fr(.)93 b Fm(\))68 b Fr(system)f(fo)-6 b(r)70 b Fo(x)3539 4570 y Fl(p)3616 4549 y Fr(:)p 143 4795 2432 5 v 143 5551 5 756 v 201 5194 a Fm(h)p Fo(x;)29 b(h)547 5228 y Fl(i)601 5194 y Fm(i)98 b Fr(=)1083 5052 y Fh(X)1025 5385 y Fl(p)p Fj(2S)1286 5402 y Fk(r)1377 5194 y Fo(h)1478 5228 y Fl(i)1532 5194 y Fr(\()p Fo(p)p Fr(\))30 b Fo(x)1909 5214 y Fl(p)2120 5194 y Fr(=)132 b(0)p 2569 5551 V 143 5556 2432 5 v 2738 5214 a(\(1)49 b Fm(\024)g Fo(i)f Fm(\024)h Fo(m)p Fr(\))2738 5447 y Fo(m)22 b Fm(\002)g Fo(r)5 b Fr(!-matrix)3396 5763 y Fi(7)p eop %%Page: 8 8 8 7 bop 141 135 a Fs(T)-19 b(enso)-6 b(rs)76 b(with)h(index)g (contractions)141 565 y Fr(Let)68 b Fo(g)55 b Fm(2)49 b(T)895 599 y Fp(2)985 565 y Fo(V)108 b Fr(b)6 b(e)69 b(a)f(p)6 b(ositive)68 b(de\014nit)h(metric.)141 994 y Fm(B)53 b Fr(=)c Fm(f)p Fo(n)714 1028 y Fp(1)803 994 y Fo(;)29 b(:)g(:)g(:)g(;)g(n)1300 1033 y Fl(d)1379 994 y Fm(g)69 b Fr(o)-6 b(rthono)g(rmal)68 b(basis)141 1210 y Fo(r)5 b Fr(,)70 b Fo(l)h Fr(integers)d(with)h(2)49 b Fm(\024)f Fr(2)p Fo(l)k(<)c(r)141 1427 y(b)216 1461 y Fp(0)355 1427 y Fr(=)h(\()p Fo(v)731 1465 y Fp(2)p Fl(l)r Fp(+1)1086 1427 y Fo(;)29 b(:)g(:)g(:)g(;)g(v)1562 1447 y Fl(r)1636 1427 y Fr(\))49 b Fm(2)g(B)2050 1368 y Fl(r)t Fj(\000)p Fp(2)p Fl(l)141 1856 y Fq(B)279 1895 y Fl(b)342 1923 y Fi(0)490 1856 y Fr(the)69 b(set)f(of)i(all)e Fo(r)5 b Fr(-tuples)206 2185 y(\()p Fo(w)412 2219 y Fp(1)502 2185 y Fo(;)29 b(w)706 2219 y Fp(1)796 2185 y Fo(;)g(w)1000 2219 y Fp(2)1091 2185 y Fo(;)g(w)1295 2219 y Fp(2)1385 2185 y Fo(;)g(:)g(:)g(:)g(;)g(w)1901 2224 y Fl(l)1951 2185 y Fo(;)g(w)2155 2224 y Fl(l)2204 2185 y Fo(;)g(v)2368 2224 y Fp(2)p Fl(l)r Fp(+1)2723 2185 y Fo(;)g(:)g(:)g(:)g(;)h(v)3200 2206 y Fl(r)3273 2185 y Fr(\))49 b Fm(2)g(B)3687 2116 y Fl(r)141 2472 y Fr(with)69 b(\()p Fo(w)778 2506 y Fp(1)868 2472 y Fo(;)29 b(:)g(:)g(:)g(;)g(w)1384 2511 y Fl(l)1434 2472 y Fr(\))49 b Fm(2)g(B)1848 2414 y Fl(l)1898 2472 y Fr(.)141 2901 y Fs(Prop.:)118 b Fr(All)81 b(tenso)-6 b(r)82 b(co)6 b(o)-6 b(rdinates)81 b(with)g(resp)6 b(ect)141 3118 y(to)55 b(o)-6 b(rthono)g(rmal)54 b(basis)g Fm(B)f Fr(=)c Fm(f)p Fo(n)2482 3152 y Fp(1)2571 3118 y Fo(;)29 b(:)g(:)g(:)g(;)g(n)3068 3157 y Fl(d)3147 3118 y Fm(g)p Fr(.)88 b(Then)342 3305 y Fh(X)284 3638 y Fl(p)p Fj(2S)545 3655 y Fk(r)636 3447 y Fr(\()p Fo(pT)24 b Fr(\))1134 3358 y Fl(j)1193 3386 y Fi(1)1350 3358 y Fl(j)1409 3386 y Fi(2)1566 3358 y Fl(:::)80 b(j)1828 3390 y Fk(l)1010 3504 y Fl(j)1069 3532 y Fi(1)1228 3504 y Fl(j)1287 3532 y Fi(2)1444 3504 y Fl(:::)h(j)1707 3536 y Fk(l)1871 3504 y Fl(i)1921 3536 y Fi(2)p Fk(l)q Fi(+1)2217 3504 y Fl(:::)o(i)2389 3521 y Fk(r)2485 3447 y Fo(p)133 b Fr(=)3100 3305 y Fh(X)3000 3642 y Fl(b)p Fj(2)p Ff(B)3276 3675 y Fk(b)3328 3711 y Fi(0)3436 3447 y Fo(T)3538 3485 y Fl(b)3634 3447 y Fo(:)530 4081 y Fr(0)g(=)g Fm(h)p Fo(x;)1405 3940 y Fh(X)1305 4277 y Fl(b)p Fj(2)p Ff(B)1581 4309 y Fk(b)1633 4346 y Fi(0)1741 4081 y Fo(T)1843 4120 y Fl(b)1910 4081 y Fm(i)766 4582 y Fr(=)1117 4440 y Fh(X)1060 4773 y Fl(p)p Fj(2S)1321 4790 y Fk(r)1411 4582 y Fr(\()p Fo(pT)24 b Fr(\))1909 4492 y Fl(j)1968 4521 y Fi(1)2125 4492 y Fl(j)2184 4521 y Fi(2)2342 4492 y Fl(:::)80 b(j)2604 4525 y Fk(l)1785 4639 y Fl(j)1844 4667 y Fi(1)2003 4639 y Fl(j)2062 4667 y Fi(2)2220 4639 y Fl(:::)g(j)2482 4671 y Fk(l)2646 4639 y Fl(i)2696 4671 y Fi(2)p Fk(l)q Fi(+1)2993 4639 y Fl(:::)o(i)3165 4656 y Fk(r)3261 4582 y Fo(x)3360 4602 y Fl(p)141 5229 y Fm(\))c Fr(Sea)-6 b(rch)76 b(fo)-6 b(r)76 b(the)g(smallest)f Fo(W)100 b Fr(that)75 b(contains)141 5446 y(all)399 5333 y Fh(P)551 5487 y Fl(b)p Fj(2)p Ff(B)827 5520 y Fk(b)879 5556 y Fi(0)991 5446 y Fo(T)1093 5485 y Fl(b)1229 5446 y Fr(fo)-6 b(r)70 b(all)e Fo(T)73 b Fm(2)49 b(T)43 b Fr(.)3396 5763 y Fi(8)p eop %%Page: 9 9 9 8 bop 141 135 a Fs(The)76 b(linea)-6 b(r)76 b(subspaces)h Fo(W)141 558 y Fs(Def.:)92 b Fo(T)73 b Fm(2)49 b(T)1132 579 y Fl(r)1206 558 y Fo(V)40 b Fr(,)69 b Fo(a)49 b Fr(=)1826 445 y Fh(P)1977 597 y Fl(p)p Fj(2S)2238 615 y Fk(r)2333 558 y Fo(a)p Fr(\()p Fo(p)p Fr(\))30 b Fo(p)50 b Fm(2)e Fq(K)p Fr([)p Fm(S)3323 579 y Fl(r)3398 558 y Fr(])98 b Fm(\))602 884 y Fr(\()p Fo(aT)24 b Fr(\))981 918 y Fl(i)1031 946 y Fi(1)1108 918 y Fl(:::)o(i)1280 935 y Fk(r)1479 884 y Fr(:=)1888 742 y Fh(X)1830 1075 y Fl(p)p Fj(2S)2091 1092 y Fk(r)2182 884 y Fo(a)p Fr(\()p Fo(p)p Fr(\))30 b Fo(T)2655 918 y Fl(i)2705 958 y Fk(p)p Fi(\(1\))2954 918 y Fl(:::)o(i)3126 958 y Fk(p)p Fi(\()p Fk(r)s Fi(\))197 1473 y Fm(\003)48 b Fr(:)98 b Fo(a)49 b Fr(=)898 1331 y Fh(X)840 1664 y Fl(p)p Fj(2S)1101 1682 y Fk(r)1192 1473 y Fo(a)p Fr(\()p Fo(p)p Fr(\))30 b Fo(p)133 b Fm(7!)g Fo(a)2186 1404 y Fj(\003)2361 1473 y Fr(:=)2734 1331 y Fh(X)2677 1664 y Fl(p)p Fj(2S)2938 1682 y Fk(r)3028 1473 y Fo(a)p Fr(\()p Fo(p)p Fr(\))31 b Fo(p)3488 1404 y Fj(\000)p Fp(1)3722 1473 y Fo(:)141 2126 y Fs(Def.:)87 b Fe(R)48 b Fm(\022)h Fq(K)p Fr([)p Fm(S)1328 2147 y Fl(r)1402 2126 y Fr(])59 b(right)g(ideal)f(fo)-6 b(r)59 b(which)f Fm(9)29 b Fo(a)49 b Fm(2)g Fe(R)p Fr(,)141 2341 y Fo(T)74 b Fm(2)48 b(T)579 2361 y Fl(r)654 2341 y Fo(V)108 b Fr(with)68 b Fo(aT)74 b Fm(6)p Fr(=)48 b(0.)93 b Fg(Symmetry)65 b(class:)271 2666 y Fm(T)368 2697 y Fd(R)578 2666 y Fr(:=)133 b Fm(f)p Fo(aT)74 b Fm(j)48 b Fo(a)h Fm(2)g Fe(R)29 b Fo(;)58 b(T)74 b Fm(2)49 b(T)2355 2687 y Fl(r)2429 2666 y Fo(V)39 b Fm(g)p Fo(:)236 b Fr(\()p Fo(eT)74 b Fr(=)49 b Fo(T)24 b Fr(\))141 3158 y Fs(Prop.:)92 b Fo(e)50 b Fm(2)e Fq(K)p Fr([)p Fm(S)1415 3179 y Fl(r)1490 3158 y Fr(])69 b(idemp)6 b(otent,)68 b Fo(T)73 b Fm(2)49 b(T)3140 3179 y Fl(r)3214 3158 y Fo(V)39 b Fr(.)185 3483 y Fo(eT)74 b Fr(=)48 b Fo(T)158 b Fm(,)132 b(8)28 b Fo(b)49 b Fm(2)g Fo(V)1777 3415 y Fl(r)1900 3483 y Fr(:)98 b Fo(T)2157 3522 y Fl(b)2322 3483 y Fm(2)f Fe(L)49 b Fr(:=)g Fq(K)p Fr([)p Fm(S)3257 3504 y Fl(r)3331 3483 y Fr(])39 b Fm(\001)g Fo(e)3597 3415 y Fj(\003)3704 3483 y Fo(;)197 3723 y(T)73 b Fm(2)49 b(T)635 3758 y Fd(L)709 3719 y Fc(\003)911 3723 y Fm(,)132 b(8)28 b Fo(b)49 b Fm(2)g Fo(V)1777 3654 y Fl(r)1900 3723 y Fr(:)79 b Fo(T)2138 3761 y Fl(b)2253 3723 y Fm(2)49 b Fe(L)29 b Fo(:)141 4215 y Fs(Prop.:)139 b Fr(If)93 b(dim)28 b Fo(V)121 b Fm(\025)82 b Fo(r)5 b Fr(,)99 b(then)92 b(every)g(left)g(ideal)141 4429 y Fe(L)49 b Fm(\022)f Fq(K)p Fr([)p Fm(S)780 4450 y Fl(r)854 4429 y Fr(])69 b(ful\014ls)744 4755 y Fe(L)132 b Fr(=)g Fm(L)1379 4796 y Ff(K)1510 4755 y Fm(f)p Fo(T)1700 4793 y Fl(b)1815 4755 y Fm(j)49 b Fo(T)73 b Fm(2)49 b(T)2351 4790 y Fd(L)2425 4751 y Fc(\003)2523 4755 y Fo(;)59 b(b)49 b Fm(2)f Fo(V)3062 4686 y Fl(r)3136 4755 y Fm(g)1258 4994 y(L)1379 5036 y Ff(K)1539 4994 y Fo(:)29 b(:)g(:)g Fr(linea)-6 b(r)69 b(closure.)p 141 5330 2040 5 v 141 5551 5 221 v 188 5486 a(In)g(simple)e(cases:)92 b Fo(W)73 b Fr(=)48 b Fe(L)p 2175 5551 V 141 5556 2040 5 v 3396 5763 a Fi(9)p eop %%Page: 10 10 10 9 bop 141 780 a Fs(The)76 b(subspaces)h Fo(W)100 b Fs(fo)-6 b(r)2175 667 y Fh(P)2326 822 y Fl(b)p Fj(2)p Ff(B)2602 854 y Fk(b)2654 890 y Fi(0)2767 780 y Fo(T)2869 819 y Fl(b)141 1282 y Fs(Def.:)92 b Fo(\025)49 b Fr(=)g(\()p Fo(\025)1237 1316 y Fp(1)1328 1282 y Fo(;)29 b(:)g(:)g(:)g(;)g(\025) 1820 1321 y Fl(k)1904 1282 y Fr(\))50 b Fm(`)f Fo(r)5 b Fr(,)69 b(length)g Fm(j)p Fo(\025)p Fm(j)49 b Fr(=)f Fo(k)694 1515 y Fr(\()p Fo(v)860 1549 y Fp(1)950 1515 y Fo(;)29 b(:)g(:)g(:)g(;)g(v)1426 1553 y Fl(k)1510 1515 y Fr(\))49 b Fm(2)g Fo(V)1946 1456 y Fl(k)456 1866 y Fm(h)p Fo(\025)p Fr(;)31 b Fo(v)800 1900 y Fp(1)889 1866 y Fo(;)e(:)g(:)g(:)g(;)h(v)1366 1904 y Fl(k)1449 1866 y Fm(i)78 b Fr(:=)456 2123 y(\()p Fo(v)622 2157 y Fp(1)712 2123 y Fo(;)29 b(:)g(:)g(:)g(;)h(v)1189 2157 y Fp(1)536 2199 y Fh(|)p 601 2199 242 18 v 242 w({z)p 973 2199 V 242 w(})827 2344 y Fl(\025)912 2372 y Fi(1)1278 2123 y Fo(;)f(v)1442 2157 y Fp(2)1532 2123 y Fo(;)g(:)g(:)g(:)g(;)g(v)2008 2157 y Fp(2)1356 2199 y Fh(|)p 1421 2199 V 242 w({z)p 1793 2199 V 242 w(})1647 2344 y Fl(\025)1732 2372 y Fi(2)2098 2123 y Fo(;)g(:)g(:)g(:)g(;)g(v)2574 2162 y Fl(k)2658 2123 y Fo(;)g(:)g(:)g(:)g(;)g(v)3134 2162 y Fl(k)2488 2204 y Fh(|)p 2553 2204 236 18 v 236 w({z)p 2919 2204 V 236 w(})2775 2348 y Fl(\025)2860 2381 y Fk(k)3218 2123 y Fr(\))49 b Fm(2)f Fo(V)3654 2054 y Fl(r)3728 2123 y Fo(:)141 2785 y Fm(8)28 b Fo(b)49 b Fr(=)f(\()p Fo(v)767 2819 y Fp(1)857 2785 y Fo(;)29 b(:)g(:)h(:)f(;)g(v)1334 2806 y Fl(r)1407 2785 y Fr(\))50 b Fm(2)e Fo(V)1844 2727 y Fl(r)1967 2785 y Fm(9)p Fr(!)28 b Fo(\025)49 b Fm(`)g Fo(r)93 b Fm(^)88 b(9)29 b Fo(q)55 b Fm(2)49 b(S)3368 2806 y Fl(r)3491 2785 y Fr(:)141 3018 y Fo(b)g Fr(=)f Fo(q)6 b Fm(h)p Fo(\025)p Fr(;)31 b Fo(w)942 3052 y Fp(1)1032 3018 y Fo(;)e(:)h(:)f(:)g(;)g(w)1549 3064 y Fj(j)p Fl(\025)p Fj(j)1720 3018 y Fm(i)187 b Fo(w)2101 3052 y Fl(i)2224 3018 y Fr(pairwise)68 b(di\013erent)141 3463 y Fm(h)p Fo(\025)p Fr(;)31 b Fo(w)525 3497 y Fp(1)615 3463 y Fo(;)e(:)g(:)g(:)g (;)g(w)1131 3509 y Fj(j)p Fl(\025)p Fj(j)1302 3463 y Fm(i)69 b Fr(.)35 b(.)g(.)f Fg(grouping)71 b Fr(of)e Fo(b)141 3697 y(\025)49 b Fr(=)g Fo(b)601 3639 y Fj(`)1439 3697 y Fr(.)35 b(.)g(.)f Fg(grouping)71 b(pa)-6 b(rtition)68 b Fr(of)i Fo(b)141 4179 y Fs(Example:)141 4536 y Fr(\()p Fo(a;)30 b(b;)f(a;)g(a;)h(c;)f(c;)g(b)p Fr(\))133 b Fm(\))g Fr(\()p Fo(a;)29 b(a;)h(a;)f(b;)g(b;)g(c;)g(c)p Fr(\))49 b(=)g Fm(h)p Fo(\025)p Fr(;)30 b Fo(a;)g(b;)f(c)p Fm(i)1794 4793 y Fo(\025)49 b Fr(=)f(\(3)30 b(2)2470 4725 y Fp(2)2561 4793 y Fr(\))3396 5763 y Fi(10)p eop %%Page: 11 11 11 10 bop 141 481 a Fr(\()p Fo(a;)30 b(b;)f(a;)g(a;)h(c;)f(c;)g(b)p Fr(\))133 b Fm(\))g Fr(\()p Fo(a;)29 b(a;)h(a;)f(b;)g(b;)g(c;)g(c)p Fr(\))49 b(=)g Fm(h)p Fo(\025)p Fr(;)30 b Fo(a;)g(b;)f(c)p Fm(i)2266 732 y(^)505 983 y Fo(\025)49 b Fr(=)f(\(3)30 b(2)1181 915 y Fp(2)1272 983 y Fr(\))442 b(\()p Fo(a;)29 b(a;)h(a;)f(c;)g(c;)g(b;)g(b)p Fr(\))49 b(=)g Fm(h)p Fo(\025)p Fr(;)30 b Fo(a;)g(c;)f(b)p Fm(i)141 1523 y(8)f Fo(b)102 b Fm(2)g(B)783 1464 y Fl(r)959 1523 y Fm(9)29 b Fo(p)1174 1561 y Fl(b)1343 1523 y Fm(2)101 b(S)1667 1543 y Fl(r)1844 1523 y Fr(:)h Fo(b)g Fr(=)f Fo(p)2530 1561 y Fl(b)2597 1523 y Fm(h)p Fo(\025)p Fr(;)31 b Fo(w)2981 1557 y Fp(1)3071 1523 y Fo(;)e(:)g(:)g(:)g(;)g(w)3587 1569 y Fj(j)p Fl(\025)p Fj(j)3758 1523 y Fm(i)141 1749 y Fr(where)69 b Fm(h)p Fo(\025)p Fr(;)30 b Fo(w)1087 1783 y Fp(1)1177 1749 y Fo(;)f(:)g(:)g(:)g(;)g(w)1693 1795 y Fj(j)p Fl(\025)p Fj(j)1864 1749 y Fm(i)69 b Fr(smallest)e (grouping.)141 2210 y Fq(p)49 b Fr(:)g Fm(B)523 2151 y Fl(r)646 2210 y Fm(!)f(S)976 2230 y Fl(r)1051 2210 y Fo(;)29 b(b)49 b Fm(7!)f Fq(p)p Fr(\()p Fo(b)p Fr(\))h(:=)g Fo(p)2224 2248 y Fl(b)141 2671 y Fo(b)216 2705 y Fp(0)355 2671 y Fm(2)g(B)640 2612 y Fl(r)t Fj(\000)p Fp(2)p Fl(l)1029 2671 y Fr(\014xed.)141 2897 y(\003)263 2935 y Fl(b)326 2964 y Fi(0)454 2897 y Fr(:=)g Fm(f)p Fo(\025)g Fm(`)g Fo(r)54 b Fm(j)49 b(9)28 b Fo(b)49 b Fm(2)g Fq(B)1901 2935 y Fl(b)1964 2964 y Fi(0)2092 2897 y Fr(:)98 b Fo(\025)49 b Fr(=)g Fo(b)2707 2838 y Fj(`)2801 2897 y Fm(g)141 3123 y Fo(\025)g Fm(2)g Fr(\003)580 3162 y Fl(b)643 3190 y Fi(0)141 3349 y Fm(M)350 3388 y Fl(b)413 3416 y Fi(0)489 3388 y Fl(;\025)668 3349 y Fr(:=)f Fm(f)p Fq(p)p Fr(\()p Fo(b)p Fr(\))1365 3291 y Fj(\000)p Fp(1)1570 3349 y Fo(b)h Fm(2)f(B)1978 3291 y Fl(r)2102 3349 y Fm(j)g Fo(b)h Fm(2)g Fq(B)2627 3388 y Fl(b)2690 3416 y Fi(0)2857 3349 y Fm(^)88 b Fo(b)3161 3291 y Fj(`)3304 3349 y Fr(=)48 b Fo(\025)p Fm(g)p 328 3557 3311 5 v 326 4006 5 449 v 395 3728 a(L)516 3769 y Ff(K)646 3563 y Fh(n)871 3586 y(X)771 3924 y Fl(b)p Fj(2)p Ff(B)1047 3956 y Fk(b)1099 3992 y Fi(0)1207 3728 y Fo(T)1309 3766 y Fl(b)1425 3557 y Fh(\014)1425 3643 y(\014)1425 3729 y(\014)1522 3728 y Fo(T)73 b Fm(2)49 b(T)1960 3763 y Fd(L)2034 3724 y Fc(\003)2103 3563 y Fh(o)2296 3728 y Fr(=)p 3637 4006 V 326 4446 5 440 v 499 4021 a Fh(X)395 4355 y Fl(\025)p Fj(2)p Fp(\003)679 4388 y Fk(b)731 4424 y Fi(0)1222 4021 y Fh(X)870 4363 y Fj(h)p Fl(\025)p Fp(;)p Fl(w)1165 4391 y Fk(i)1210 4363 y Fj(i2M)1539 4395 y Fk(b)1591 4431 y Fi(0)1667 4395 y(;)p Fk(\025)1811 4162 y Fo(a)1904 4209 y Fj(h)p Fl(\025)p Fp(;)p Fl(w)2199 4237 y Fk(i)2245 4209 y Fj(i)2346 4162 y Fm(\001)38 b Fr(1)2536 4200 y Fj(H)2661 4223 y Fk(t)2705 4263 y(\025)2828 4162 y Fm(\001)h Fq(K)p Fr([)p Fm(S)3232 4183 y Fl(r)3306 4162 y Fr(])g Fm(\001)g Fo(e)p 3637 4446 V 328 4451 3311 5 v 326 5042 5 592 v 755 4610 a(a)848 4657 y Fj(h)p Fl(\025)p Fp(;)p Fl(w)1143 4685 y Fk(i)1189 4657 y Fj(i)1348 4610 y Fr(:=)2046 4469 y Fh(X)1969 4790 y Fk(b)p Fc(2)p Fb(B)2198 4830 y Fk(b)2250 4866 y Fi(0)1673 4992 y Fb(p)p Fi(\()p Fk(b)p Fi(\))1912 4959 y Fc(\000)p Fi(1)2082 4992 y Fk(b)21 b Fi(=)f Fc(h)p Fk(\025)p Fi(;)p Fk(w)2533 5028 y(i)2578 4992 y Fc(i)2665 4610 y Fq(p)p Fr(\()p Fo(b)p Fr(\))3008 4542 y Fj(\000)p Fp(1)p 3637 5042 V 328 5047 3311 5 v 326 5472 5 426 v 1338 5203 a Fr(1)1441 5241 y Fj(H)1566 5264 y Fk(t)1610 5304 y(\025)1790 5203 y Fr(:=)2209 5062 y Fh(X)2106 5395 y Fl(s)p Fj(2H)2397 5418 y Fk(t)2441 5458 y(\025)2549 5203 y Fo(s)p 3637 5472 V 328 5477 3311 5 v 3396 5763 a Fi(11)p eop %%Page: 12 12 12 11 bop 141 135 a Fs(Linea)-6 b(r)75 b(identities)j(if)e Fo(r)54 b Fr(=)49 b Fo(k)43 b Fr(+)c(1)76 b Fs(even)714 306 y Fh(X)614 644 y Fl(b)p Fj(2)p Ff(B)890 676 y Fk(b)942 712 y Fi(0)1050 448 y Fo(T)1152 487 y Fl(b)1352 448 y Fr(=)1703 306 y Fh(X)1645 639 y Fl(p)p Fj(2S)1906 656 y Fk(r)1997 448 y Fr(\()p Fo(pT)24 b Fr(\))2495 359 y Fl(j)2554 387 y Fi(1)2711 359 y Fl(j)2770 387 y Fi(2)2927 359 y Fl(:::)81 b(j)3190 391 y Fk(l)2371 505 y Fl(j)2430 533 y Fi(1)2589 505 y Fl(j)2648 533 y Fi(2)2805 505 y Fl(:::)g(j)3068 537 y Fk(l)3265 448 y Fo(p)1092 881 y(T)158 b Fr(=)132 b Fo(c)1721 915 y Fp(1)1851 881 y Fm(\012)38 b Fo(:)29 b(:)g(:)39 b Fm(\012)g Fo(c)2522 920 y Fl(k)2645 881 y Fm(\012)g Fo(x)141 1257 y(W)157 b Fr(=)862 1115 y Fh(X)758 1450 y Fl(\025)p Fj(2)p Fp(\003)1042 1482 y Fk(b)1094 1518 y Fi(0)1585 1115 y Fh(X)1233 1458 y Fj(h)p Fl(\025)p Fp(;)p Fl(w)1528 1486 y Fk(i)1573 1458 y Fj(i2M)1902 1490 y Fk(b)1954 1526 y Fi(0)2030 1490 y(;)p Fk(\025)2174 1257 y Fo(a)2267 1303 y Fj(h)p Fl(\025)p Fp(;)p Fl(w)2562 1332 y Fk(i)2608 1303 y Fj(i)2708 1257 y Fm(\001)39 b Fr(1)2899 1294 y Fj(H)3024 1317 y Fk(t)3068 1357 y(\025)3191 1257 y Fm(\001)g Fq(K)p Fr([)p Fm(S)3595 1278 y Fl(r)3669 1257 y Fr(])g Fm(\001)g Fo(e)212 1706 y Fm(B)137 b Fr(=)c Fm(f)p Fo(n)953 1740 y Fp(1)1042 1706 y Fo(;)29 b(n)1227 1740 y Fp(2)1317 1706 y Fo(;)g(n)1502 1740 y Fp(3)1592 1706 y Fm(g)146 b Fo(;)175 b(e)98 b Fr(=)g(id)141 2104 y(1.)93 b Fq(B)532 2143 y Fl(b)595 2171 y Fi(0)723 2104 y Fr(=)48 b Fm(f)p Fr(\()p Fo(w)1226 2138 y Fp(1)1317 2104 y Fo(;)29 b(w)1521 2138 y Fp(1)1611 2104 y Fo(;)g(w)1815 2138 y Fp(2)1905 2104 y Fo(;)g(w)2109 2138 y Fp(2)2199 2104 y Fo(;)h(:)f(:)g(:)g(;)g(w)2716 2143 y Fl(l)2765 2104 y Fo(;)g(w)2969 2143 y Fl(l)3019 2104 y Fr(\))g Fm(j)g Fo(w)3332 2138 y Fl(i)3435 2104 y Fm(2)49 b(B)t(g)394 2311 y Fo(b)469 2345 y Fp(0)608 2311 y Fr(=)f Fm(;)141 2518 y Fr(2.)93 b(\003)516 2557 y Fl(b)579 2585 y Fi(0)707 2518 y Fr(=)48 b Fm(f)p Fo(\025)h Fm(`)g Fo(r)54 b Fm(j)49 b(9)29 b Fo(b)48 b Fm(2)h Fq(B)2096 2557 y Fl(b)2159 2585 y Fi(0)2287 2518 y Fr(:)98 b Fo(\025)50 b Fr(=)e Fo(b)2902 2460 y Fj(`)2996 2518 y Fm(g)141 2725 y Fr(3.)93 b(F)-6 b(o)g(r)69 b(every)f Fo(\025)49 b Fm(2)g Fr(\003)1690 2764 y Fl(b)1753 2792 y Fi(0)394 2932 y Fm(M)603 2971 y Fl(b)666 2999 y Fi(0)741 2971 y Fl(;\025)920 2932 y Fr(:=)g Fm(f)p Fq(p)p Fr(\()p Fo(b)p Fr(\))1618 2874 y Fj(\000)p Fp(1)1822 2932 y Fo(b)g Fm(2)g(B)2231 2874 y Fl(r)2354 2932 y Fm(j)g Fo(b)g Fm(2)g Fq(B)2880 2971 y Fl(b)2943 2999 y Fi(0)3110 2932 y Fm(^)88 b Fo(b)3414 2874 y Fj(`)3556 2932 y Fr(=)49 b Fo(\025)p Fm(g)394 3139 y Fr(1)497 3176 y Fj(H)622 3199 y Fk(t)666 3239 y(\025)798 3139 y Fr(=)1007 3026 y Fh(P)1159 3178 y Fl(s)p Fj(2H)1450 3201 y Fk(t)1494 3241 y(\025)1605 3139 y Fo(s)141 3382 y Fr(4.)93 b(F)-6 b(o)g(r)69 b(every)f Fm(h)p Fo(\025)p Fr(;)30 b Fo(w)1634 3417 y Fl(i)1688 3382 y Fm(i)49 b(2)g(M)2180 3421 y Fl(b)2243 3449 y Fi(0)2319 3421 y Fp(;)p Fl(\025)394 3589 y Fo(a)487 3636 y Fj(h)p Fl(\025)p Fp(;)p Fl(w)782 3664 y Fk(i)828 3636 y Fj(i)986 3589 y Fr(:=)1302 3476 y Fh(P)1453 3647 y Ff(p)p Fp(\()p Fl(b)p Fp(\))1740 3608 y Fc(\000)p Fi(1)1910 3647 y Fl(b)24 b Fp(=)h Fj(h)p Fl(\025)p Fp(;)p Fl(w)2451 3676 y Fk(i)2496 3647 y Fj(i)2587 3589 y Fq(p)p Fr(\()p Fo(b)p Fr(\))2930 3531 y Fj(\000)p Fp(1)141 3823 y Fr(5.)93 b(Generating)68 b(idemp)6 b(otent)67 b Fo(f)87 b Fr(fo)-6 b(r)70 b(right)f(ideal)394 4030 y Fo(W)j Fr(=)842 3917 y Fh(P)30 b(P)1203 4030 y Fo(a)1296 4076 y Fj(h)p Fl(\025)p Fp(;)p Fl(w)1591 4105 y Fk(i)1637 4076 y Fj(i)1737 4030 y Fm(\001)39 b Fr(1)1928 4067 y Fj(H)2053 4090 y Fk(t)2097 4130 y(\025)2220 4030 y Fm(\001)g Fq(K)p Fr([)p Fm(S)2624 4050 y Fl(r)2698 4030 y Fr(])141 4271 y(6.)93 b(Basis)69 b Fm(f)p Fo(h)1089 4305 y Fp(1)1179 4271 y Fo(;)29 b(:)g(:)g(:)g(;)g(h)1670 4292 y Fl(m)1804 4271 y Fm(g)69 b Fr(of)g Fo(W)141 4478 y Fr(7.)93 b(Solve)910 4365 y Fh(P)1062 4517 y Fl(p)p Fj(2S)1323 4535 y Fk(r)1417 4478 y Fo(h)1518 4512 y Fl(i)1573 4478 y Fr(\()p Fo(p)p Fr(\))30 b Fo(x)1950 4499 y Fl(p)2076 4478 y Fr(=)49 b(0)98 b Fo(;)126 b(i)49 b Fr(=)f(1)p Fo(;)30 b(:)f(:)g(:)g(;)g(m)141 4877 y Fm(\017)69 b Fr(5.,)h(6.)92 b(b)-6 b(y)69 b(algo)-6 b(rithms)68 b(from)g(Habilitationsschrift)141 5084 y Fm(\017)h Fr(Use)f(FFT)h Fo(D)54 b Fr(:)49 b Fq(K)p Fr([)p Fm(S)1754 5104 y Fl(r)1828 5084 y Fr(])g Fm(3)g Fo(a)g Fm(7!)g Fo(D)5 b Fr(\()p Fo(a)p Fr(\))49 b(=)3129 4919 y Fh(\020)3224 5008 y Fa(2)3349 5149 y(2)3464 4919 y Fh(\021)141 5291 y Fm(\017)69 b Fr(LR)g(rule)f(&)h(plethysms)3396 5763 y Fi(12)p eop %%Page: 13 13 13 12 bop 141 135 a Fs(Linea)-6 b(r)75 b(identities)j(if)e Fo(k)44 b Fr(+)39 b(1)76 b Fs(o)6 b(dd)714 311 y Fh(X)614 648 y Fl(b)p Fj(2)p Ff(B)890 680 y Fk(b)942 716 y Fi(0)1050 452 y Fo(T)1152 491 y Fl(b)1352 452 y Fr(=)1703 311 y Fh(X)1645 644 y Fl(p)p Fj(2S)1906 661 y Fk(r)1997 452 y Fr(\()p Fo(pT)24 b Fr(\))2495 363 y Fl(j)2554 391 y Fi(1)2711 363 y Fl(j)2770 391 y Fi(2)2927 363 y Fl(:::)81 b(j)3190 395 y Fk(l)2371 509 y Fl(j)2430 538 y Fi(1)2589 509 y Fl(j)2648 538 y Fi(2)2805 509 y Fl(:::)g(j)3068 542 y Fk(l)3265 452 y Fo(p)1092 885 y(T)158 b Fr(=)132 b Fo(c)1721 919 y Fp(1)1851 885 y Fm(\012)38 b Fo(:)29 b(:)g(:)39 b Fm(\012)g Fo(c)2522 924 y Fl(k)2645 885 y Fm(\012)g Fo(x)f Fm(\012)h Fo(S)141 1268 y(W)157 b Fr(=)862 1126 y Fh(X)758 1461 y Fl(\025)p Fj(2)p Fp(\003)1042 1493 y Fk(b)1094 1529 y Fi(0)1585 1126 y Fh(X)1233 1469 y Fj(h)p Fl(\025)p Fp(;)p Fl(w)1528 1497 y Fk(i)1573 1469 y Fj(i2M)1902 1501 y Fk(b)1954 1537 y Fi(0)2030 1501 y(;)p Fk(\025)2174 1268 y Fo(a)2267 1314 y Fj(h)p Fl(\025)p Fp(;)p Fl(w)2562 1343 y Fk(i)2608 1314 y Fj(i)2708 1268 y Fm(\001)39 b Fr(1)2899 1305 y Fj(H)3024 1328 y Fk(t)3068 1368 y(\025)3191 1268 y Fm(\001)g Fq(K)p Fr([)p Fm(S)3595 1289 y Fl(r)3669 1268 y Fr(])g Fm(\001)g Fo(e)141 1795 y(S)58 b Fm(2)49 b(T)569 1829 y Fp(3)659 1795 y Fo(V)186 b Fm(\))145 b Fq(K)p Fr([)p Fm(S)1584 1816 y Fl(r)1659 1795 y Fr(])69 b(with)g Fo(r)53 b Fr(=)c Fo(k)44 b Fr(+)38 b(4)p 3066 1639 386 5 v 3066 1836 5 198 v 116 w(!!!!!)p 3446 1836 V 3066 1841 386 5 v 141 2238 a Fo(T)74 b Fr(=)48 b Fo(c)602 2272 y Fp(1)731 2238 y Fm(\012)39 b Fo(:)29 b(:)g(:)39 b Fm(\012)f Fo(c)1402 2277 y Fl(k)1526 2238 y Fm(\012)g Fo(x)266 b(S)2182 2169 y Fl(j)2241 2198 y Fi(1)2317 2169 y Fl(j)2376 2198 y Fi(2)2452 2169 y Fl(j)2511 2198 y Fi(3)2640 2238 y Fr(=)2849 1987 y Fh(\()3031 2136 y Fo(\026)29 b Fr(sgn)3481 1971 y Fh(\020)3601 2060 y Fp(1)3577 2201 y Fl(j)3636 2229 y Fi(1)3756 2060 y Fp(2)3732 2201 y Fl(j)3791 2229 y Fi(2)3911 2060 y Fp(3)3887 2201 y Fl(j)3946 2229 y Fi(3)4032 1971 y Fh(\021)3031 2370 y Fr(0)361 2632 y Fh(X)208 2965 y Fl(p)p Fj(2S)469 2997 y Fk(k)s Fi(+1)751 2774 y Fo(p)g Fr(\()p Fo(pT)24 b Fr(\))1366 2685 y Fl(i)1416 2713 y Fi(1)1573 2685 y Fl(i)1623 2713 y Fi(2)1780 2685 y Fl(:::)80 b(i)2033 2717 y Fk(l)1242 2831 y Fl(i)1292 2859 y Fi(1)1451 2831 y Fl(i)1501 2859 y Fi(2)1658 2831 y Fl(:::)g(i)1911 2863 y Fk(l)2034 2831 y Fl(j)2093 2859 y Fi(1)2169 2831 y Fl(j)2228 2859 y Fi(2)2304 2831 y Fl(j)2363 2859 y Fi(3)2473 2774 y Fo(S)2589 2705 y Fl(j)2648 2733 y Fi(1)2724 2705 y Fl(j)2783 2733 y Fi(2)2859 2705 y Fl(j)2918 2733 y Fi(3)208 3208 y Fr(=)48 b Fo(\026)734 3066 y Fh(X)581 3399 y Fl(p)p Fj(2S)842 3432 y Fk(k)s Fi(+1)1187 3066 y Fh(X)1124 3399 y Fl(q)5 b Fj(2S)1381 3428 y Fi(3)1486 3208 y Fo(p)29 b Fr(\()p Fo(pT)24 b Fr(\))2101 3119 y Fl(i)2151 3147 y Fi(1)2308 3119 y Fl(i)2358 3147 y Fi(2)2515 3119 y Fl(:::)80 b(i)2768 3151 y Fk(l)1977 3279 y Fl(i)2027 3307 y Fi(1)2186 3279 y Fl(i)2236 3307 y Fi(2)2393 3279 y Fl(:::)g(i)2646 3311 y Fk(l)2769 3279 y Fl(q)5 b Fp(\(1\))p Fl(q)g Fp(\(2\))p Fl(q)g Fp(\(3\))3644 3208 y Fo(\037)3753 3229 y Fl(q)208 3642 y Fr(=)48 b Fo(\026)644 3500 y Fh(X)581 3834 y Fl(q)5 b Fj(2S)838 3862 y Fi(3)943 3642 y Fo(\037)1052 3663 y Fl(q)1309 3500 y Fh(X)1156 3834 y Fl(p)p Fj(2S)1417 3866 y Fk(k)s Fi(+1)1699 3642 y Fo(p)29 b Fr(\()p Fo(pT)24 b Fr(\))2314 3553 y Fl(i)2364 3581 y Fi(1)2521 3553 y Fl(i)2571 3581 y Fi(2)2728 3553 y Fl(:::)80 b(i)2981 3585 y Fk(l)2190 3713 y Fl(i)2240 3742 y Fi(1)2399 3713 y Fl(i)2449 3742 y Fi(2)2606 3713 y Fl(:::)g(i)2859 3746 y Fk(l)2982 3713 y Fl(q)5 b Fp(\(1\))p Fl(q)g Fp(\(2\))p Fl(q)g Fp(\(3\))208 4052 y Fr(=)48 b Fo(\026)644 3910 y Fh(X)585 4268 y Fp(~)-90 b Fl(q)5 b Fj(2)776 4245 y Fp(~)749 4268 y Fj(S)838 4296 y Fi(3)943 4052 y Fo(\037)1056 4091 y Fp(~)-90 b Fl(q)1255 3910 y Fh(X)1156 4248 y Fl(b)p Fj(2)p Ff(B)1432 4280 y Fk(b)1484 4316 y Fi(0)1592 4052 y Fo(T)1698 4091 y Fp(~)g Fl(q)5 b(b)2465 4052 y Fo(b)2540 4086 y Fp(0)2679 4052 y Fr(=)49 b(\()p Fo(n)3076 4086 y Fp(1)3166 4052 y Fo(;)29 b(n)3351 4086 y Fp(2)3441 4052 y Fo(;)g(n)3626 4086 y Fp(3)3716 4052 y Fr(\))208 4487 y(=)48 b Fo(\026)644 4345 y Fh(X)585 4702 y Fp(~)-90 b Fl(q)5 b Fj(2)776 4680 y Fp(~)749 4702 y Fj(S)838 4731 y Fi(3)943 4487 y Fo(\037)1056 4526 y Fp(~)-90 b Fl(q)1255 4345 y Fh(X)1156 4683 y Fl(b)p Fj(2)p Ff(B)1432 4715 y Fk(b)1484 4751 y Fi(0)1597 4487 y Fr(~)-108 b Fo(q)45 b Fm(\001)39 b Fo(T)1905 4526 y Fl(b)208 4936 y Fr(=)48 b Fo(\026)552 4771 y Fh(\020)638 4936 y Fo(\017)39 b Fm(\001)935 4794 y Fh(X)835 5132 y Fl(b)p Fj(2)p Ff(B)1111 5164 y Fk(b)1163 5200 y Fi(0)1272 4936 y Fo(T)1374 4975 y Fl(b)1441 4771 y Fh(\021)2307 4936 y Fo(\017)49 b Fr(:=)2756 4794 y Fh(X)2698 5151 y Fp(~)-91 b Fl(q)6 b Fj(2)2888 5129 y Fp(~)2862 5151 y Fj(S)2951 5179 y Fi(3)3055 4936 y Fr(sgn)q(\()f(~)-108 b Fo(q)7 b Fr(\))34 b(~)-108 b Fo(q)p 141 5319 3400 5 v 141 5551 5 232 v 188 5475 a Fm(\))68 b Fr(Mo)6 b(di\014cation:)92 b(other)69 b Fo(b)2220 5509 y Fp(0)2310 5475 y Fr(,)g(multipl.)91 b(b)-6 b(y)69 b Fo(\017)p 3536 5551 V 141 5556 3400 5 v 3396 5763 a Fi(13)p eop %%Page: 14 14 14 13 bop 141 135 a Fs(Identities)77 b(fo)-6 b(r)75 b(small)g Fo(k)p 2988 -19 770 5 v 2988 177 5 197 v 1013 w Fr(PERMS)p 3752 177 V 2988 182 770 5 v 141 366 3782 5 v 139 565 5 200 v 224 507 a Fo(k)p 408 565 V 178 w(\025)p Fr(/blo)6 b(ck)p 1227 565 V 175 w Fn(R)p 1497 565 V 173 w Fr(dim)p 1973 565 V 165 w(terms)p 2614 565 V 371 w(id.)p 3193 565 V 167 w(summ.)p 3920 565 V 141 570 3782 5 v 141 595 V 139 794 5 200 v 224 736 a(3)p 408 794 V 166 w(\(4\))p 1227 794 V 557 w(1)p 1497 794 V 373 w(1)p 1973 794 V 538 w(3)p 2614 794 V 476 w(0)p 3193 794 V 167 w({)p 3920 794 V 139 1002 5 208 v 408 1002 V 493 944 a(\(2)676 885 y Fp(2)767 944 y Fr(\))p 1227 1002 V 466 w(1)p 1497 1002 V 373 w(2)p 1973 1002 V 2614 1002 V 3193 1002 V 3920 1002 V 141 1007 3782 5 v 139 1214 5 208 v 224 1156 a(4)p 408 1214 V 166 w(\(3)30 b(1)809 1098 y Fp(2)900 1156 y Fr(\))p 1227 1214 V 333 w(1)p 1497 1214 V 373 w(6)p 1973 1214 V 435 w(10)p 2614 1214 V 476 w(4)p 3193 1214 V 167 w(4)p 3920 1214 V 141 1219 3782 5 v 139 1418 5 200 v 224 1360 a(5)p 408 1418 V 166 w(\(6\))p 1227 1418 V 557 w(1)p 1497 1418 V 373 w(1)p 1973 1418 V 435 w(15)p 2614 1418 V 476 w(0)p 3193 1418 V 167 w({)p 3920 1418 V 139 1618 V 408 1618 V 493 1559 a(\(4)g(2\))p 1227 1618 V 424 w(1)p 1497 1618 V 373 w(9)p 1973 1618 V 2614 1618 V 3193 1618 V 3920 1618 V 139 1825 5 208 v 408 1825 V 493 1767 a(\(2)676 1708 y Fp(3)767 1767 y Fr(\))p 1227 1825 V 466 w(1)p 1497 1825 V 373 w(5)p 1973 1825 V 2614 1825 V 3193 1825 V 3920 1825 V 141 1830 3782 5 v 139 2037 5 208 v 224 1979 a(6)p 408 2037 V 166 w(\(5)g(1)809 1921 y Fp(2)900 1979 y Fr(\))p 1227 2037 V 333 w(1)p 1497 2037 V 270 w(15)p 1973 2037 V 331 w(105)p 2614 2037 V 374 w(69)p 3193 2037 V 167 w(4,10,)p 3920 2037 V 139 2244 V 408 2244 V 493 2186 a(\(3)676 2128 y Fp(2)797 2186 y Fr(1\))p 1227 2244 V 333 w(1)p 1497 2244 V 270 w(21)p 1973 2244 V 2614 2244 V 3193 2244 V 1387 w(12,14)p 3920 2244 V 139 2444 5 200 v 408 2444 V 1227 2444 V 1497 2444 V 1973 2444 V 2614 2444 V 3193 2444 V 3279 2386 a(16,18)p 3920 2444 V 141 2449 3782 5 v 139 2648 5 200 v 224 2590 a(7)p 408 2648 V 166 w(\(8\))p 1227 2648 V 557 w(1)p 1497 2648 V 373 w(1)p 1973 2648 V 331 w(105)p 2614 2648 V 374 w(14)p 3193 2648 V 167 w(24,36)p 3920 2648 V 139 2847 V 408 2847 V 493 2789 a(\(6)g(2\))p 1227 2847 V 424 w(1)p 1497 2847 V 270 w(20)p 1973 2847 V 2614 2847 V 3193 2847 V 1387 w(40,44)p 3920 2847 V 139 3054 5 208 v 408 3054 V 493 2996 a(\(4)676 2938 y Fp(2)767 2996 y Fr(\))p 1227 3054 V 466 w(1)p 1497 3054 V 270 w(14)p 1973 3054 V 2614 3054 V 3193 3054 V 1387 w(50,52)p 3920 3054 V 139 3262 V 408 3262 V 493 3204 a(\(4)g(2)809 3145 y Fp(2)900 3204 y Fr(\))p 1227 3262 V 333 w(1)p 1497 3262 V 270 w(56)p 1973 3262 V 2614 3262 V 3193 3262 V 3920 3262 V 141 3267 3782 5 v 139 3474 5 208 v 224 3416 a(8)p 408 3474 V 166 w(\(7)g(1)809 3358 y Fp(2)900 3416 y Fr(\))p 1227 3474 V 333 w(1)p 1497 3474 V 270 w(28)p 1973 3474 V 228 w(1260)p 2614 3474 V 167 w(1028)p 3193 3474 V 168 w(???)p 3920 3474 V 139 3673 5 200 v 408 3673 V 493 3615 a(\(5)g(3)g(1\))p 1227 3673 V 291 w(1)p 1497 3673 V 166 w(162)p 1973 3673 V 2614 3673 V 3193 3673 V 3920 3673 V 139 3881 5 208 v 408 3881 V 493 3823 a(\(3)676 3764 y Fp(3)767 3823 y Fr(\))p 1227 3881 V 466 w(1)p 1497 3881 V 270 w(42)p 1973 3881 V 2614 3881 V 3193 3881 V 3920 3881 V 141 3886 3782 5 v 1512 4235 a Fm(h)p Fo(a;)g(b)p Fm(ih)p Fo(c;)f(d;)h(x)p Fm(i)166 b(h)p Fo(b;)29 b(d)p Fm(ih)p Fo(a;)i(c;)e(x)p Fm(i)407 4434 y(h)p Fo(a;)h(b)p Fm(ih)p Fo(c;)g(x)p Fm(i)332 b(h)p Fo(a;)30 b(c)p Fm(ih)p Fo(b;)f(d;)h(x)p Fm(i)166 b(h)p Fo(b;)29 b(x)p Fm(ih)p Fo(a;)h(c;)f(d)p Fm(i)407 4633 y(h)p Fo(a;)h(c)p Fm(ih)p Fo(b;)g(x)p Fm(i)332 b(h)p Fo(a;)30 b(d)p Fm(ih)p Fo(b;)g(c;)f(x)p Fm(i)166 b(h)p Fo(c;)29 b(d)p Fm(ih)p Fo(a;)i(b;)e(x)p Fm(i)407 4832 y(h)p Fo(a;)h(x)p Fm(ih)p Fo(b;)g(c)p Fm(i)332 b(h)p Fo(a;)30 b(x)p Fm(ih)p Fo(b;)f(c;)g(d)p Fm(i)167 b(h)p Fo(c;)29 b(x)p Fm(ih)p Fo(a;)h(b;)f(d)p Fm(i)1512 5032 y(h)p Fo(b;)g(c)p Fm(ih)p Fo(a;)h(d;)g(x)p Fm(i)166 b(h)p Fo(d;)30 b(x)p Fm(ih)p Fo(a;)f(b;)h(c)p Fm(i)776 5468 y Fo(D)54 b Fr(:)49 b Fq(K)p Fr([)p Fm(S)1396 5489 y Fl(r)1470 5468 y Fr(])g Fm(3)g Fo(a)g Fm(7!)g Fo(D)5 b Fr(\()p Fo(a)p Fr(\))49 b(=)2771 5303 y Fh(\020)2866 5392 y Fa(2)2991 5533 y(2)3106 5303 y Fh(\021)3396 5763 y Fi(14)p eop %%Page: 15 15 15 14 bop 141 135 a Fs(Examples)75 b(of)h(identities)199 295 y Fh(X)141 628 y Fl(p)p Fj(2S)402 646 y Fk(r)493 437 y Fo(h)594 471 y Fl(i)648 437 y Fr(\()p Fo(p)p Fr(\))30 b Fo(x)1025 458 y Fl(p)1152 437 y Fr(=)48 b(0)133 b Fm(\))1963 295 y Fh(X)1906 628 y Fl(p)p Fj(2S)2167 646 y Fk(r)2257 437 y Fr(\()p Fo(pT)24 b Fr(\))2755 348 y Fl(j)2814 376 y Fi(1)2890 348 y Fl(:::)o(j)3071 380 y Fk(l)2631 494 y Fl(j)2690 522 y Fi(1)2768 494 y Fl(:::)o(j)2949 526 y Fk(l)3113 494 y Fl(i)3163 522 y Fi(1)3239 494 y Fl(i)3289 522 y Fi(2)3365 494 y Fl(i)3415 522 y Fi(3)3524 437 y Fo(x)3623 458 y Fl(p)3749 437 y Fr(=)49 b(0)141 1038 y Fo(k)44 b Fr(+)39 b(1)49 b(=)f(5)141 1339 y Fm(h)p Fo(a;)30 b(d)p Fm(i)g Fo(b)39 b Fm(\002)f Fo(c)h Fr(+)g Fm(h)p Fo(b;)30 b(d)p Fm(i)g Fo(c)39 b Fm(\002)f Fo(a)i Fr(+)e Fm(h)p Fo(c;)30 b(d)p Fm(i)g Fo(a)39 b Fm(\002)f Fo(b)133 b Fr(=)g Fm(h)p Fo(a;)30 b(b;)f(c)p Fm(i)g Fo(d)141 1565 y Fm(h)p Fo(a;)h(c)p Fm(i)f Fo(b)40 b Fm(\002)e Fo(d)h Fr(+)g Fm(h)p Fo(b;)30 b(c)p Fm(i)f Fo(d)40 b Fm(\002)e Fo(a)i Fr(+)e Fm(h)p Fo(c;)30 b(d)p Fm(i)g Fo(a)39 b Fm(\002)f Fo(b)133 b Fr(=)g Fm(h)p Fo(a;)30 b(b;)f(d)p Fm(i)h Fo(c)141 1790 y Fm(h)p Fo(a;)g(b)p Fm(i)f Fo(c)40 b Fm(\002)e Fo(d)h Fr(+)g Fm(h)p Fo(b;)30 b(c)p Fm(i)f Fo(d)40 b Fm(\002)e Fo(a)i Fr(+)e Fm(h)p Fo(b;)30 b(d)p Fm(i)g Fo(a)39 b Fm(\002)g Fo(c)132 b Fr(=)h Fm(h)p Fo(a;)30 b(c;)f(d)p Fm(i)h Fo(b)141 2016 y Fm(h)p Fo(a;)g(b)p Fm(i)f Fo(c)40 b Fm(\002)e Fo(d)h Fr(+)g Fm(h)p Fo(a;)30 b(c)p Fm(i)f Fo(d)40 b Fm(\002)e Fo(b)i Fr(+)e Fm(h)p Fo(a;)30 b(d)p Fm(i)g Fo(b)39 b Fm(\002)g Fo(c)132 b Fr(=)h Fm(h)p Fo(b;)29 b(c;)g(d)p Fm(i)h Fo(a)141 2501 y(k)44 b Fr(+)39 b(1)49 b(=)f(7)208 2642 y Fh(\014)208 2728 y(\014)208 2814 y(\014)208 2900 y(\014)208 2986 y(\014)326 2799 y Fm(h)p Fo(a;)30 b(e)p Fm(i)142 b(h)p Fo(b;)30 b(e)p Fm(i)322 2998 y(h)p Fo(a;)f(d)p Fm(i)134 b(h)p Fo(b;)29 b(d)p Fm(i)1299 2642 y Fh(\014)1299 2728 y(\014)1299 2814 y(\014)1299 2900 y(\014)1299 2986 y(\014)1376 2898 y Fm(h)p Fo(c;)h(f)8 b(;)30 b(x)p Fm(i)39 b Fr(+)2176 2642 y Fh(\014)2176 2728 y(\014)2176 2814 y(\014)2176 2900 y(\014)2176 2986 y(\014)2290 2799 y Fm(h)p Fo(a;)29 b(d)p Fm(i)134 b(h)p Fo(a;)30 b(e)p Fm(i)2298 2998 y(h)p Fo(c;)g(d)p Fm(i)151 b(h)p Fo(c;)29 b(e)p Fm(i)3277 2642 y Fh(\014)3277 2728 y(\014)3277 2814 y(\014)3277 2900 y(\014)3277 2986 y(\014)3354 2898 y Fm(h)p Fo(b;)g(f)8 b(;)30 b(x)p Fm(i)208 3213 y Fr(+)208 3278 y Fh(\014)208 3364 y(\014)208 3450 y(\014)208 3536 y(\014)208 3622 y(\014)322 3435 y Fm(h)p Fo(b;)g(e)p Fm(i)134 b(h)p Fo(b;)30 b(d)p Fm(i)322 3634 y(h)p Fo(c;)f(e)p Fm(i)134 b(h)p Fo(c;)29 b(d)p Fm(i)1274 3278 y Fh(\014)1274 3364 y(\014)1274 3450 y(\014)1274 3536 y(\014)1274 3622 y(\014)1351 3534 y Fm(h)p Fo(a;)h(f)8 b(;)30 b(x)p Fm(i)39 b Fr(+)2168 3278 y Fh(\014)2168 3364 y(\014)2168 3450 y(\014)2168 3536 y(\014)2168 3622 y(\014)2282 3435 y Fm(h)p Fo(d;)30 b(f)18 b Fm(i)133 b(h)p Fo(d;)30 b(x)p Fm(i)2286 3634 y(h)p Fo(e;)g(f)18 b Fm(i)143 b(h)p Fo(e;)30 b(x)p Fm(i)3293 3278 y Fh(\014)3293 3364 y(\014)3293 3450 y(\014)3293 3536 y(\014)3293 3622 y(\014)3370 3534 y Fm(h)p Fo(a;)f(b;)g(c)p Fm(i)141 3750 y Fh(|)p 206 3750 1791 18 v 1791 w({z)p 2127 3750 V 1791 w(})2011 3890 y Fp(=)41 b(0)141 4559 y Fo(k)j Fr(+)39 b(1)49 b(=)f(8)603 4711 y Fh(\014)603 4797 y(\014)603 4883 y(\014)603 4969 y(\014)603 5055 y(\014)603 5141 y(\014)603 5227 y(\014)603 5313 y(\014)603 5399 y(\014)725 4841 y Fm(h)p Fo(b;)30 b(f)18 b Fm(i)149 b(h)p Fo(b;)29 b(g)6 b Fm(i)148 b(h)p Fo(b;)29 b(x)p Fm(i)149 b(h)p Fo(b;)29 b(a)p Fm(i)725 5040 y(h)p Fo(c;)g(f)18 b Fm(i)148 b(h)p Fo(c;)29 b(g)6 b Fm(i)147 b(h)p Fo(c;)30 b(x)p Fm(i)147 b(h)p Fo(c;)29 b(a)p Fm(i)718 5239 y(h)p Fo(d;)g(f)18 b Fm(i)134 b(h)p Fo(d;)30 b(g)6 b Fm(i)132 b(h)p Fo(d;)30 b(x)p Fm(i)133 b(h)p Fo(d;)30 b(a)p Fm(i)722 5438 y(h)p Fo(e;)g(f)18 b Fm(i)142 b(h)p Fo(e;)30 b(g)6 b Fm(i)142 b(h)p Fo(e;)29 b(x)p Fm(i)142 b(h)p Fo(e;)30 b(a)p Fm(i)2787 4711 y Fh(\014)2787 4797 y(\014)2787 4883 y(\014)2787 4969 y(\014)2787 5055 y(\014)2787 5141 y(\014)2787 5227 y(\014)2787 5313 y(\014)2787 5399 y(\014)2967 5139 y Fr(=)133 b(0)3396 5763 y Fi(15)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF