M(9,2,3) - Revision history

http://wiki.manchester.ac.uk/blocks/index.php?action=history&feed=atom&title=M%289%2C2%2C3%29 M(9,2,3) - Revision history 2026-05-13T12:18:08Z Revision history for this page on the wiki MediaWiki 1.30.1 http://wiki.manchester.ac.uk/blocks/index.php?title=M(9,2,3)&diff=627&oldid=prev Charles Eaton at 07:43, 23 October 2018 2018-10-23T07:43:22Z

<p></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr style="vertical-align: top;" lang="en-GB"> <td colspan="2" style="background-color: white; color:black; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 07:43, 23 October 2018</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l15" >Line 15:</td> <td colspan="2" class="diff-lineno">Line 15:</td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|defect-morita-inv? = Yes</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|defect-morita-inv? = Yes</div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|inertial-morita-inv? = Yes</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|inertial-morita-inv? = Yes</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>|O-morita? = <del class="diffchange diffchange-inline">Yes</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>|O-morita? = <ins class="diffchange diffchange-inline">No</ins></div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|O-morita = <math>\mathcal{O} ((C_3 \times C_3):C_2)</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|O-morita = <math>\mathcal{O} ((C_3 \times C_3):C_2)</math></div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|decomp = <math>\left( \begin{array}{cc}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|decomp = <math>\left( \begin{array}{cc}</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l31" >Line 31:</td> <td colspan="2" class="diff-lineno">Line 31:</td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|sourcereps =</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|sourcereps =</div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|k-derived-known? = No</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|k-derived-known? = No</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>|k-derived = <del class="diffchange diffchange-inline">none known</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>|k-derived =  </div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|O-derived-known? = No</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|O-derived-known? = No</div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|coveringblocks =  </div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|coveringblocks =  </div></td></tr> <tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l47" >Line 47:</td> <td colspan="2" class="diff-lineno">Line 47:</td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>== Other notatable representatives ==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>== Other notatable representatives ==</div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">== Covering blocks and covered blocks ==</del></div></td><td colspan="2"> </td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>== Projective indecomposable modules ==</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>== Projective indecomposable modules ==</div></td></tr>  </table>

Charles Eaton http://wiki.manchester.ac.uk/blocks/index.php?title=M(9,2,3)&diff=624&oldid=prev Charles Eaton at 07:23, 23 October 2018 2018-10-23T07:23:04Z

<p></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr style="vertical-align: top;" lang="en-GB"> <td colspan="2" style="background-color: white; color:black; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 07:23, 23 October 2018</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l5" >Line 5:</td> <td colspan="2" class="diff-lineno">Line 5:</td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|defect = [[C3xC3|<math>C_3 \times C_3</math>]]</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|defect = [[C3xC3|<math>C_3 \times C_3</math>]]</div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|inertialquotients = <math>C_2</math></div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|inertialquotients = <math>C_2</math></div></td></tr> <tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>|k(B) = <del class="diffchange diffchange-inline">9</del></div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>|k(B) = <ins class="diffchange diffchange-inline">6</ins></div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|l(B) = 2</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|l(B) = 2</div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|k-morita-frob = 1  </div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>|k-morita-frob = 1  </div></td></tr>  </table>

Charles Eaton http://wiki.manchester.ac.uk/blocks/index.php?title=M(9,2,3)&diff=622&oldid=prev Charles Eaton: Created page with "{{blockbox |title = M(9,2,3) - <math>k((C_3 \times C_3):C_2)</math> |image = M(9,2,3)quiver.png |representative = <math>k((C_3 \times C_3):C_2)</math> |defect = C3xC3|<mat..." 2018-10-22T20:20:30Z

<p>Created page with "{{blockbox |title = M(9,2,3) - <math>k((C_3 \times C_3):C_2)</math> |image = M(9,2,3)quiver.png |representative = <math>k((C_3 \times C_3):C_2)</math> |defect = C3xC3|<mat..."</p> <p><b>New page</b></p><div>{{blockbox<br /> |title = M(9,2,3) - <math>k((C_3 \times C_3):C_2)</math> <br /> |image = M(9,2,3)quiver.png<br /> |representative = <math>k((C_3 \times C_3):C_2)</math><br /> |defect = [[C3xC3|<math>C_3 \times C_3</math>]]<br /> |inertialquotients = <math>C_2</math><br /> |k(B) = 9<br /> |l(B) = 2<br /> |k-morita-frob = 1 <br /> |Pic-k= <br /> |cartan = <math>\left( \begin{array}{cc}<br /> 5 & 4 \\<br /> 4 & 5 \\<br /> \end{array} \right)</math><br /> |defect-morita-inv? = Yes<br /> |inertial-morita-inv? = Yes<br /> |O-morita? = Yes<br /> |O-morita = <math>\mathcal{O} ((C_3 \times C_3):C_2)</math><br /> |decomp = <math>\left( \begin{array}{cc}<br /> 1 & 0 \\<br /> 0 & 1 \\<br /> 1 & 1 \\<br /> 1 & 1 \\<br /> 1 & 1 \\<br /> 1 & 1 \\<br /> \end{array}\right)</math><br /> |O-morita-frob = 1<br /> |Pic-O = <math>C_2</math><ref>Proposition 4.3 of [[References|[BKL18]]]</ref><br /> |PIgroup = <br /> |source? = No<br /> |sourcereps =<br /> |k-derived-known? = No<br /> |k-derived = none known<br /> |O-derived-known? = No<br /> |coveringblocks = <br /> |coveredblocks = <br /> |pcoveringblocks = <br /> }}<br /> <br /> The representative <math>k((C_3 \times C_3):C_2)</math> is given by <math>C_3 \times C_3</math> acted on by an element inverting those of <math>C_3 \times C_3</math>, i.e., it is the group SmallGroup(18,4).<br /> <br /> == Basic algebra ==<br /> <br /> '''Quiver:''' a:<1,2>, b:<1,2>, c:<2,1>, d:<2,1><br /> <br /> '''Relations w.r.t. <math>k</math>:''' <math>ad=bc</math>, <math>cb=da</math>, <math>aca=bdb=0</math>, <math>cac=dbd=0</math><br /> <br /> == Other notatable representatives ==<br /> <br /> == Covering blocks and covered blocks ==<br /> <br /> == Projective indecomposable modules ==<br /> <br /> Labelling the unique simple <math>B</math>-module by <math>1,2</math>, the projective indecomposable modules have Loewy structure as follows:<br /> <br /> <math>\begin{array}{cc}<br /> \begin{array}{ccccc}<br /> & & 1 & & \\<br /> & 2 & & 2 & \\<br /> 1 & & 1 & & 1 \\<br /> & 2 & & 2 & \\<br /> & & 1 & & \\<br /> \end{array}, &<br /> \begin{array}{ccccc}<br /> & & 2 & & \\<br /> & 1 & & 1 & \\<br /> 2 & & 2 & & 2 \\<br /> & 1 & & 1 & \\<br /> & & 2 & & \\ <br /> \end{array} \\<br /> \end{array}<br /> </math><br /> <br /> == Irreducible characters ==<br /> <br /> All irreducible characters have height zero.<br /> <br /> [[C3xC3|Back to <math>C_3 \times C_3</math>]]<br /> <br /> == Notes ==<br /> <references /></div>

Charles Eaton