Improvements to the ATLAS – II

This list contains improvements not in Appendix 2 of “An Atlas of Brauer characters” (by Jansen, Lux, Parker & Wilson, OUP 1995), hitherto referred to as ABC. While some of them are new (i.e. not discovered at the time of publication of ABC), others were deliberately left out of ABC because they were not thought significant enough, or because they did not fit the format of ABC; but we believe that they are worth keeping on record.

As in ABC, we denote mathematical errors by *** and new information by NEW. We use C to denote improvements concerning grammar or notational consistency; and M to refer to misprints or cases of illegibility. Changes are shown in order of appearance. New references are not shown. The inclusion or otherwise of any modification is not intended to imply anything about its inclusion in any future ATLAS publication.

A character table in ATLAS format has been obtained for ^{2}E_{6}(2).3.

Introduction

C Page iv  : Top, insert “page” before “number”.
M Page vii : Top left, change head of Section 1 to Preliminaries.
C  : Section 2, change “to” to “from” after “differently”.

NEW Page xi : Middle right, for sake of completeness append description of orthogonal groups in

dimensions 1 and 2.

C Page xiv : Middle left, omit “to” in “also to specify”.
M Page xxv : Section 5, insert i in “contans”.
C Page xxix : Middle right, singularize “act”.

NEW : In Section 18, specify notational convention where both coset and cohort are

“squashed”.

C Page xx : Middle right, insert “it” between “which” and “becomes”.
M Page xxxi : Section 1, replace “product” by “produce”.

The Groups

NEW Page 4 A_{6}  : In Linear section line 2, replace G.2 by G.2_{2}.
C Page 12 L_{2}(16) : Singularize “Presentations”.
C Page 14 U_{3}(3) : In 3rd Max, replace 4.S_{4} :2 by 2_{+}^{1+4}.S_{3}.
C Page 21 L_{2}(31) : Singularize “Presentations”.
C Page 29 L_{2}(32) : Singularize “Presentations”.
NEW Page 30 U_{3}(4) : In 3rd Max, N(5^{2}) is also N(5A–D_{3}EF_{3}).
C Page 31 M_{12}  : In MINIMOG section line 6, close bracket.
NEW Page 33  : In 9th Max, replace H.2 by M_{8}.(S_{4} × 2).
M Page 36 J_{1}  : Raise most of bottom line in Graph section.
C  : Replace = by ≅ in presentation.
C  : In 6th Max, replace D_{6} by S_{3}.
NEW Page 42 J_{2}  : In 4th Max, replace H.2 by 2^{2+4}.(S_{3} × S_{3}).
NEW Page 53 U_{4}(3) : 1st Max is N(3^{4}) = N(3A_{10}B_{15}C_{15}).
NEW Page 70 L_{5}(2) : In 3rd and 4th Maxes, N(2^{6}) is also N(2A_{21}B_{42}).
C Page 73 U_{5}(2) : In 4th Max, remove brackets round AB and CD.
NEW Page 74 L_{3}(8) : In 3rd Max, N(7^{2}) is also N(7A–F_{3}GH_{2}IJK_{3}).
NEW Page 79 U_{3}(9) : In 4th Max, N(5^{2}) is also N(5A–D_{3}EF_{3}).

C Page 80 HS : 2nd, 3rd, 6th, 7th, 9th and 11th Maxes respectively fix 233−2323, 233−2332, 233 − 2433, 233 − 2433, 233 − 2343 and 233 − 2433 points. For the 6th, 7th and 11th this is a further change, in all cases designed to conform with a convention that in abc − defg the sums of (the vector corresponding to) d with a, b, c are e, f, g respectively.

C Page 83 J_{3}  : Replace r6 ∗ by r6 −r6 in χ_{9}.
C Page 89 O_{8}^{−}(2) : Pluralize “Presentation”.

*** ^{3}D_{4}(2) : Replace top row (but not first column) occurrences of \frac{1}{4}s by \frac{1}{4} \overline{s} in 768 matrix

orbit.

M  : Delete comma before slash in line after display of roots.
NEW  : In 7th Max, N(7^{2}) is also N(7A–C_{4}D_{4}).

C Page 100 McL : 3rd, 7th and 10th Maxes respectively fix 223−2324, 223−2443 and 223−2434

points. For the 7th this is a further change. See amendment to Page 80 above.

C Page 109 O_{7}(3) : Pluralize “Presentation”.
***  : In 14th Max, replace 2E by 2F.
C Page 111 S_{6}(3) : Move “Specifications”.
NEW Page 113  : 2nd Max is N(3^{6}) = N(3AB_{13}C_{39}D_{78}E_{234}).
NEW  : 3rd Max is N(3^{3}) = N(3AB_{4}C_{3}D_{6}).
***  : In 7th Max, extensions are split.
NEW Page 115 U_{6}(2) : N(2^{9}) is also N(2A_{21}B_{210}C_{280}).

C : 5th, 6th, 11th and 12th Maxes respectively fix 222 − 2323, 222 − 2332, 222 −

2433 and 222 − 2343 points. See amendment to Page 80 above.

C Page 122 R(27) : Insert separator between R(27) and S_{8}(2).
C  S_{8}(2) : Pluralize “a presentation”.
NEW  : 7th Max is N(2^{6}) = N(2A_{7}B_{7}C_{21}E_{28}).
C Page 129 Suz  : Replace r10 ∗ by r10 −r10 in χ_{40}.
C Page 131  : Pluralize “Presentation”.
C Page 140 O_{8}^{+}(3) : Pluralize “a presentation”.
NEW  : Maxes 7-9 are also N(3^{6}).
***  : In 8th and 9th Maxes, replace 3^{6} :(L_{4}(3) × 2) by 3^{6} :(L_{4}(3):2_{1}).
NEW Page 146 O_{10}^{+}(2) : In 2nd Max, N(2^{8}) is also N(2A_{135}B_{120}).
NEW  : In 3rd and 4th Maxes, N(2^{10}) is also N(2A_{155}C_{868}).
NEW Page 147 O_{10}^{−}(2) : In 1st Max, N(2^{8}) is also N(2A_{119}B_{136}).
NEW  : In 5th Max, N(2^{6}) is also N(2A_{35}C_{28}).
NEW  : In 6th Max, replace N(2^{3}) by N(2A^{3}).
C  : In last Max, N(3A_{2}D_{2}) is also N(3^{2}).
NEW Page 154 Co_{2}  : In 2nd Max, N(2^{10}) is also N(2A_{77}B_{330}C_{616}).
NEW  : In 8th Max, replace N(2^{4}) by N(2B^{4}).
NEW Page 177 F i_{23}  : In 6th Max, N(2^{11}) is also N(2A_{23}B_{253}C_{1771}).
NEW  : In 11th Max, N(2^{6}) is also N(2A_{7}B_{21}C_{35}).
C Page 182 Co_{1}  : Near bottom, pluralize “portion”, “group” and “parenthesis”.
C Page 191 ^{2}E_{6}(2) : Replace 3^{2}Q_{8} by 3^{2} :Q_{8}.

NEW Page 217 B : New Maxes M_{11}, L_{3}(3), L_{2}(17):2 and L_{2}(11):2. With the classification of all 2-locals, the list is now complete except possibly for groups of type U_{3}(3) or U_{3}(3) : 2. Furthermore, in the group (S_{6} × L_{3}(4) : 2) : 2 quoted in the original list of amendments, the first “:2” is in fact “:2_{2}”.

C Page 230 M  : Pluralize “Presentation”.
M  : Replace , by . in last line before list of groups.

NEW Page 234 : The list of Maxes that are 2-local or normalizers of products of two or more simple groups is complete.

Additional Information

C Page 239 : Pluralize “Order”.

NEW : Insert sentence: groups whose full character tables are in the Atlas are starred, while a bracketed star is appended to other Atlas groups and to those for which all or part of the character table exists in Atlas format. Also, insert relevant stars, with or without brackets, in pages 239-242.

Bibliography

M Page 247 : line 3: Replace 243 by 323.

The number of changes of each type is as follows: 4*** 26NEW 30C 7M. Last update 28 August 1996; this file has 136 lines.