COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

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118 J. S. Frame pair of characters of degrees fi, fi or f3, f4 such that 120(56)(63)/fifi is a square, and fi+f2 = 119, (4.1) 135(64)(7O)/f3f, is a square, and f3+f4 = 134. The unique positive integral solutions of these equations are fi, fi = 35 and 84 in 120,, (4.2) fs, f4 = 50 and 84 in 135,. (4.3) It is easily verified that 120, contains the character 35, already computed, so 84, = 120,- lx-35,, (4.4) 50, = 135,- 120,+ 35,. (4.5) It can be checked that 84, is also a constituent of the character 960, induced by the symmetric subgroup S = &. Hence the character 960,- lx- 84, of degree 875 splits into two irreducible constituents whose degrees divide 213 35 52 7. The splitting into 175, and 700, is versed when it is found that this character has a constituent in common with the Murnaghan character [6] of degree 1386 for the orthogonal group Os, which splits as follows into three known constituents and one new constituent 700,: [6] = 700,+567,+84,+35,. (4.6) The other constituent 175, of 960, is 175, = 960,-lx-84,-700,. (4.7) To check this character 175,, we note that it splits in H into the sum of the two characters 105b+70i. Thus its values for the 46 classes of A in H can be computed directly from these subgroup characters without evaluating [6]. Both degrees 175 and 700 are divisible by 52 and 7, so the characters 175, and 700, both vanish for the 11 classes of elements whose orders are divisible by 5 or 7. Thus it may be easier to determine 175, first for most classes, and then find 700, from (4.7) rather than from (4.6). Having the additional characters 84,, 50,, 175, and 700,, we can now compute several more characters of types x and y quite simply as follows: 1400, = 28,(50,) (4.8) 1050, = 35,(50,) - 700, 1575, = 28,(84,)-567,-210, = 8,(560,)-35,(84,-lx) 1344, = 35,(84,)-8,(112,)-700, 2100, = 28,(210,-84,)-84,- 1344, 2268, = 35,(210,-84,)- 567,- 1575, 525, = 84:“1-567,- 1050,- 1344, 700,, = 5Og’l- 525 4200, = 28,( 175,) y700x,. The characters of the Weyl group Es 119 The symbol 700,, denotes a second character of type x and degree 700, distinct from 700,. Other new characters are defined by 972, = 50,(84,) - 50, - 84, - 1050, - 1344, - 700,, (4.9) 4096, = (35x- 1,)(300,-28,)-840,-700,- 1344,-2268x. The character 972, of degree 2235 is of highest type modulo 3, so it must vanish in all 3-singular classes. By decomposing Kronecker products involving characters of relatively small degree such as 70,, 50,, 84,, 28, and 35, we can now solve for ten more of the 13 self-associated characters of type y as follows : 1134, = 70,(28x) - 70, - (28, + 28;) - (350, + 350;) (4.10) 1680, = 70,(35,)-70,-(350,+350;) = 70,(84x)-(2100,+2100~) 168, = SOP1 - 50, - 84,- 972 420, = 70,(70,+lx)-28x(28~+28~)-(840,+840~)-1134Y-168, 3150, = 28,(168,)- 1134,-420, 4200, = 35,( 168,) - (840,f 840;) 2688, = 28,(420, - 168,) - 168, - 4200, 2100, = 50x(168,- lx-I:)-(700,+700;,)-(972,+972;)- 168,-2688, 1400, = 50,(70,) - 2100, 4536, = 28,(525,)-300,-700,- 1400,-2268,-4096,- 1400,. 5. Kronecker products with the character 8,. Products of the character 8, with any constituent of an even (odd) Kronecker power will split into constituents of powers of the opposite parity. By splitting products of 8, with the even-power constituents of types x and y already found we can complete the list of irreducible characters of types z and w as follows. We start with a second character 1400, of type z and degree 1400, not to be confused with 1400, already obtained. CPA 9 1400,, = 8,(175,) (5.1) 4200, = 8,(700,x- 175,) 400, = 8,(50x) 3240, = 8,(700, - 50,) - 1400, - 560, 4536, = 8,(972,j - 3240, 2400, = 56,(84,) - 1296, - 1008, 3360, = 8,(1400,-525,-50,)-3240, 2800, = 56,(50x) 4096, = 8,( 1575,) - 56,(84x) - 560, - 3240, 5600, = 8,(2268,-525,)- 1008,-3240,-2800, 448, = 8,(1344,- 1575,)f 1296,- 1400,+2400,
120 J. S. Frame The characters of the Weyl group Es I III XI xv XXXVII IV XIV IX XXVII xxx11 X XXVI XL LVIII XXXVIII XL1 XL11 LVI LIX XII XXIX XXX LX LXV XXXIV XXIII XVIII LIII XLVI LIV LV XLVIII TABLE 1. Three permutation characters in Type a classes the 67 classes of A = F/C Type b classes Class symbols 1 A l/l ck 1 120, 135, 960, Class symbols 1 A l/l ck / 120, 18 1422 1524 142” 1224 153 141 3V’ 1 223 1223 I 223v 153z 1232 12913 s/i 33/1 1 3 3211’ 1 3 612 34114 3%/P2 135 121 5 l,:,* 1.15/3*5 17 213355=1 120 211325 32 2’3 8 21032 12 26 2 2’/355 36 2733 12 2632 8 243 2 2532 - 2434 6 2432 2 2 33 3 - 33 2435 3 2433 3 2a32 3 2”355 - - 2432 23352 10 23 5 2 235 1 2”35% - 35 - 27 1 Type d classes 224v 293a 2% 28 224 283 2 3 3v 2332 116 233 8w 26 8u 24 448, = 8,(70,) - 56,- 56: 1344, = 8,(840,) - 840,-4536, 12 4 - - - 2 - 5600, = 8,(1134,)-840,-840;-448~ 1344, 2016, = 8,(420,) - 1344, 7168, = 8,(2688,.)-3360,-3360;-5600,~2016, 135 31 7 3 1 21 3 7 1 3 9 1 - - - - - - - 5 1 2 - - 2 19 3 7 1 1 5 1 - 960 96 8 - - 72 - 12 2 - 12 - - 3 24 - - - - 5 1 2 - - 1 - 8 - - - - 2 (5.2) XIII XXXVI VIII XXVIII xxxv .LXI LXII xXx1x xxx111 LX111 LVII LXVI LXVII LXIV II V X X I VII xxv VI XVI XXIV XXIX XLVII XVII XXII LI XLIX L LII xx XLIV XL111 XLV 24v 4% 24u 4% 1124 24 42 1133 2 6v 2 6u 26 62/22v 6”/22 :;;I; 162 1223 122 22 144 131 4 132 3 111 2 3 132 3 134 134 2 32 126 2 913 2 33113 6 311 2 3 6/T 2 125 - - 125 2 3 5/P 2 711 . 21333 210 2113 27 26 210325 263 2333 2433 243 2338 2633 2532 233 2” 5 Type c classes 21”3457 21032 2&3 293 5 263 25335 2532 2532 253 233 2333 2332 2 32 2434 243’ 233 2z3 5 2=5 23 5 27 24 4 8 - 2 - - - 6 2 - - - - - 64 16 4 20 6 16 4 6 2 - 4 4 1 1 1 1 4 - 1 1 121 135, 960, 39 11 7 3 1 15 3 3 9 1 3 - - - - - - 16 4 - - - - - 4 - - - - - 63 288 1 5 32 3 - 1 1 16 1 - 15 30 3 2 1 - 5 4 1 - 3 6 3 2 - - - - - - 8 - 3 3 1 1 - - - 1 6. Blocks of defect one. The Brauer theory of modular characters, used to determine several of the characters of the subgroup Ho of A [4], can be applied in like manner to find some new characters and check some of those already computed. In a block of defect one (mod p), where p” but not pafl divides the group order, the characters have degrees divisible by pa-l but not by pa. Since all characters of A and F are rationa& there 9*
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COMPUTATIONAL PROBLEMS IN ABSTRACT
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vi Contents E. KRAUSE and K. WESTON
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X Preface I am indebted to the auth
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4 J. Neubiiser 2.3.5. Added in proo
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8 J. Neubiiser Investigations of gr
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12 J. Neubiiser For 1 es 4 r, ws is
Page 13 and 14: 16 J. Neubiiser Investigations of g
Page 23 and 24: Some examples using coset enumerati
Page 25 and 26: 40 C. M. Campbell Some examples usi
Page 27 and 28: 44 N. S. Mendelsohn for example, in
Page 29 and 30: 48 H. Jiirgensen Calculation with e
Page 35 and 36: 60 V. Fe&h and J. Neubiiser 2. The
Page 37 and 38: 64 L. Gerhards and E. Altmann Autom
Page 39 and 40: 68 L. Gerhards and E. Altmann Autom
Page 41 and 42: 72 L. Gerhards and E. Altmann ni E
Page 43 and 44: 76 W. Lindenberg and L. Gerhards Se
Page 45 and 46: 80 W. Lindenberg and L. Gerhards Se
Page 47 and 48: 84 K. Ferber and H. Jiirgensen The
Page 49 and 50: 7 The construction of the character
Page 51 and 52: 92 John McKay defining multiplicati
Page 53 and 54: 96 John McKay Construction of chara
Page 55 and 56: 100 John McKay In the table, c indi
Page 57 and 58: 104 C. Brott and J. Neubiiser irred
Page 59 and 60: 108 C. Brott and J. Neubiiser eleme
Page 61 and 62: 112 J. S. Frame The characters of t
Page 63: 116 J. S. Frame 3. The decompositio
Page 67 and 68: 124 J. S. Frame The characters of t
Page 69 and 70: TABLE 5. The Z, character block for
Page 71 and 72: 132 R. Biilow and J. Neubiiser Deri
Page 73 and 74: A search for simple groups of order
Page 75 and 76: 140 Marshall Hall Jr. Simple groups
Page 79 and 80: 148 Marshall Hall Jr. otherwise no
Page 81 and 82: 152 Marshall Hall Jr. easily found
Page 85 and 86: 160 Marshall Hail Jr. This leads to
Page 87 and 88: 164 Marshall Hall Jr. b= (OO)(Ol, 0
Page 89 and 90: 168 Marshall Hall Jr. 11. R. BRAUER
Page 91 and 92: 172 Charles C. Sims THEOREM 2.3. Th
Page 93 and 94: 176 Charles C. Sims has a set of im
Page 95 and 96: 180 Degrc - No. Order t N (-63 Gene
Page 97 and 98: An algorithm related to the restric
Page 99 and 100: A module-theoretic computation rela
Page 101 and 102: 192 A. L. Tritter expressed in the
Page 103 and 104: 196 A. L. Tritter a question is mos
Page 105 and 106: 200 John J. Cannon stack. The Cayle
Page 107 and 108: , I The computation of irreducible
Page 109 and 110: 208 P. G. Ruud and R. Keown product
Page 111 and 112: 212 P. G. Ruud and R. Keown TX wher
Page 113 and 114: 216 P. G. Ruud and R. Keown 4. L. G
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220 N. S. Mendelsohn where it is kn
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224 Robert J. Plemmons such class [
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228 Robert J. Plemmons ! TOTALS Sem
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232 Takayuki Tamura 8” = 0B if an
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236 Takayuki Tamura Case ,Q = {e).
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240 Takayuki Tamura The calculation
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244 Takayuki Tamura We calculate46
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248 Takayuki Tamura Immediately we
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252 Takayuki Tamura Let U be a subs
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256 Takayuki Tamura TABLE 8. AI1 Se
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I The author and R. Dickinson have
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264 Donald E. Knuth and Peter B. Be
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300 Lowell J. Paige Non-associative
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304 Lowell J. Paige We may lineariz
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308 T(n) U(n) 0) [VI R-4 C. M. Glen
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312 C. M. Glennie where in each cas
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316 A. D. Keedwell of the elements
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A projective coqfiguration J. W. P.
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The uses of computers in Galois the
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328 W. D. Maurer mod p; for a facto
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332 J. H. Conway Enumeration of kno
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J. H. Conway Enumeration of knots a
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340 J. H. Conway -thus our symmetry
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344 J. H. Conway 2 string links to
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348 J. H. Conway 3 and 4 string lin
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352 Non-alternating J. H. Conway L
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356 J. H. Conway Alternating 11 cro
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H. F. Trotter Computations in knot
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364 H. F. Trotter to a. Schubert [1
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368 Shen Lin sequence of squares, t
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372 Harvey Cohn We also consider GZ
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376 Harvey Cohn In the case of Fig.
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380 Harvey Cohn always dictated by
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384 Hans Zassenhaus A real root cal
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388 Hans Zassenhaus the elements A,
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392 Hans Zassenhaus It is clear fro
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396 R. E. Kalman Invariant factors
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400 List of participants DR. J. J.
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