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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd"> <html> <meta name="GENERATOR" content="TtH 3.59"> <style type="text/css"> div.p { margin-top: 7pt;}</style> <style type="text/css"><!-- td div.comp { margin-top: -0.6ex; margin-bottom: -1ex;} td div.comb { margin-top: -0.6ex; margin-bottom: -.6ex;} td div.hrcomp { line-height: 0.9; margin-top: -0.8ex; margin-bottom: -1ex;} td div.norm {line-height:normal;} span.roman {font-family: serif; font-style: normal; font-weight: normal;} span.overacc2 {position: relative; left: .8em; top: -1.2ex;} span.overacc1 {position: relative; left: .6em; top: -1.2ex;} --></style> <title>Atlas of Brauer Characters -- Corrections</title> <font color="#AF0000"><h1 align="center">Corrections to the A<small>TLAS</small> of Brauer Chara <div class="p"><!----></div> <br /><br /> <body bgcolor="FFFFFF"> <div class="p"><!----></div> This is the list of known errors in the A<font size="-2">TLAS</font> of Brauer Characters. <div class="p"><!----></div> We denote mathematical errors by ***. We use C to denote improvements concerning grammar or notational consistency. Changes are shown in order of appearance. <div class="p"><!----></div> <br /><br /><b>Introduction</b> <div class="p"><!----></div> <br /><br /> <table> <tr><td valign="top" align="right">***</td><td valign="top"> p.</td><td valign="top" align="right <td valign="top"><table border="0"><tr><td valign="top"></td><td width="558"> Replace the last paragraph of Section 10 by the following. <div class="p"><!----></div> If the indicator is 0, then D(G) may or may not embed in a suitable unitary group. Let *q be the map on the Brauer characters induced by the field automorphism x → x<sup>q</sup>. Then D is unitary if and only if D(G) is defined over a field with q<sup>2</sup> elements and φ<sup>*q</sup> is the complex conju of φ. </td></tr></table><!--vbox--> </td><td valign="top"></td></tr></td></tr></table><!--hboxt--></table> <div class="p"><!----></div> <br /><br /><b>The Tables</b> <div class="p"><!----></div> <br /><br /> <table> <tr><td valign="top" align="right">***</td><td valign="top"> p.</td><td valign="top" align="right <td valign="top"><table border="0"><tr><td valign="top"></td><td width="558"> In the 13 modular table, change 12A 12A to 12A ∗. </td></tr></table><!--vbox--> </td><td valign="top"></td></tr> <tr><td valign="top" align="right">***</td><td valign="top"> p.</td><td valign="top" align="right </td><td valign="top"><table border="0"><tr><td valign="top"></td><td width="558"> Change element orders of 4C, 12C, 12D in 4.G.2<sub>3</sub> to 4, 12, 12. </td></tr></table><!--vbox--> </td><td valign="top"></td></tr> <tr><td valign="top" align="right">***</td><td valign="top"> p.</td><td valign="top" align="right </td><td valign="top"><table border="0"><tr><td valign="top"></td><td width="558"> Change class names 10A-B in G.2 to 10E-F. </td></tr></table><!--vbox--> </td><td valign="top"></td></tr> <tr><td valign="top" align="right">***</td><td valign="top"> p.</td><td valign="top" align="right </td><td valign="top"><table border="0"><tr><td valign="top"></td><td width="558"> In G.2, change 6C to 6D and 8B to 8C where applicable. </td></tr></table><!--vbox--> </td><td valign="top"></td></tr> <tr><td valign="top" align="right">***</td><td valign="top"> p.</td><td valign="top" align="right </td><td valign="top"><table border="0"><tr><td valign="top"></td><td width="558"> In G.4, insert fusion signs joining characters 3 and 4, 6 and 7, 9 to 12, 13 and 14. </td></tr></table><!--vbox--> </td><td valign="top"></td></tr> <tr><td valign="top" align="right">***</td><td valign="top"> p.</td><td valign="top" align="right </td><td valign="top"><table border="0"><tr><td valign="top"></td><td width="558"> Change D∗40 to D∗&#tyle='color: green'>8727;5. </td></tr></table><!--vbox--> </td><td valign="top"></td></tr> <tr><td valign="top" align="right">***</td><td valign="top"> p.</td><td valign="top" align="right </td><td valign="top"><table border="0"><tr><td valign="top"></td><td width="558"> Change class name 12J in G.2<sub>3</sub> to 12K. </td></tr></table><!--vbox--> </td><td valign="top"></td></tr> <tr><td valign="top" align="right">C</td><td valign="top"> p.</td><td valign="top" align="right"> 1 </td><td valign="top"><table border="0"><tr><td valign="top"></td><td width="558"> Swap the two fusion markers for φ<sub>27</sub>-φ<sub>30</sub> in 2.G.2<sub>1</sub>. </td></tr></table><!--vbox--> </td><td valign="top"></td></tr> <tr><td valign="top" align="right">C</td><td valign="top"> p.</td><td valign="top" align="right"> 1 </td><td valign="top"><table border="0"><tr><td valign="top"></td><td width="558"> Change chis to phis. </td></tr></table><!--vbox--> </td><td valign="top"></td></tr> <tr><td valign="top" align="right">C</td><td valign="top"> p.</td><td valign="top" align="right"> 1 </td><td valign="top"><table border="0"><tr><td valign="top"></td><td width="558"> Rewrite y′63∗&#le='color: green'>8727;− z21∗10&17 to y′63∗∗+ z7. (Both expressions are equal.) </td></tr></table><!--vbox--> </td><td valign="top"></td></tr> <tr><td valign="top" align="right">***</td><td valign="top"> </td><td valign="top" align="right"> </ </td><td valign="top"><table border="0"><tr><td valign="top"></td><td width="558"> In the third and fourth degree 511 rows of the 73 modular table, exchange the indicators °6 and &#>176;2 in G.3. </td></tr></table><!--vbox--> </td><td valign="top"></td></tr> <tr><td valign="top" align="right">***</td><td valign="top"> p.</td><td valign="top" align="right </td><td valign="top"><table border="0"><tr><td valign="top"></td><td width="558"> In the 7 modular table of G.2<sub>1</sub>, change the indicator of the second degree 910 character from ++ to ++2. In the 13 modular table of G.2<sub>3</sub>, change the indicator of the second degree 91 character from °° </td></tr></table><!--vbox--> </td><td valign="top"></td></tr> <tr><td valign="top" align="right">***</td><td valign="top"> p.</td><td valign="top" align="right </td><td valign="top"><table border="0"><tr><td valign="top"></td><td width="558"> Change the value of the first degree 584 character on the class 5EF from ∗ to 2+b5&# </td></tr></table><!--vbox--> </td><td valign="top"></td></tr> <tr><td valign="top" align="right">***</td><td valign="top"> p.</td><td valign="top" align="right </td><td valign="top"><table border="0"><tr><td valign="top"></td><td width="558"> Swap the character values in G and 3.G on the classes 17A and 17B, and the character values in G.2 on the classes 34A and 34B. </td></tr></table><!--vbox--> </td><td valign="top"></td></tr></td></tr></table><!--hboxt--></table> <div class="p"><!----></div> <br /><br /><b>Appendix 1</b> <div class="p"><!----></div> <br /><br /> <table> <tr><td valign="top" align="right">C</td><td valign="top"> p.</td><td valign="top" align="right"> 2 <td valign="top"><table border="0"><tr><td valign="top"></td><td width="558"> Add the entry w41∗3 | X<sup> </td></tr></table><!--vbox--> </td><td valign="top"></td></tr> <tr><td valign="top" align="right">***</td><td valign="top"> p.</td><td valign="top" align="right </td><td valign="top"><table border="0"><tr><td valign="top"></td><td width="558"> The entries for b29 and b29∗ must be as follows. b29 | X + 2 | C<sub>2</sub>, b29&# </td></tr></table><!--vbox--> </td><td valign="top"></td></tr> <tr><td valign="top" align="right">***</td><td valign="top"> p.</td><td valign="top" align="right </td><td valign="top"><table border="0"><tr><td valign="top"></td><td width="558"> Replace y9∗∗ by y9&#style='color: green'>8727;2. </td></tr></table><!--vbox--> </td><td valign="top"></td></tr></td></tr></table><!--hboxt--></table> <div class="p"><!----></div> <br /><br /><b>Appendix 2</b> <div class="p"><!----></div> <br /><br /> <table> <tr><td valign="top" align="right">C</td><td valign="top"> p.</td><td valign="top" align="right"> 3 <td valign="top"><table border="0"><tr><td valign="top"></td><td width="558"> Remove the last change for U<sub>3</sub>(8), which suggests to add a note to the map. Note that the isoclines of 3.G.3<sub>1</sub> are in fact isomorphic. </td></tr></table><!--vbox--> </td><td valign="top"></td></tr></td></tr></table><!--hboxt--></table> <div class="p"><!----></div> <br /><br />The number of changes of each type is as follows: 14 *** 5 C. <div class="p"><!----></div> <br /> File created on 18 April 2000<br> Last modified: 17 October 2024 by <a href="mailto:sam@math.rwth-aachen.de">Thomas Breuer</a> <div class="p"><!----></div> <br /><br /><hr /><small>File translated from T<sub><font size="-1">E</font></sub>X by <a href="http://hutchinson.belmont.ma.us/tth/"> T<sub><font size="-1">T</font></sub>H</a>, version 3.59.<br />On 17 Oct 2024, 22:49.</small> </html>
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