Proceedings of the 44th Symposium on Ring Theory and ...
Proceedings of the 44th Symposium on Ring Theory and ...
Proceedings of the 44th Symposium on Ring Theory and ...
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❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ ✂ ❁ ✴<br />
1 3 ⊕ S ✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴✴<br />
1 ⊕ ✂ ✂ 2<br />
1<br />
2<br />
✂ ✂ ❁ ❄ ❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄❄<br />
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⊕ ✂ ✂ 2<br />
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1 3 ⊕ S 1 ⊕ S 3<br />
2<br />
✂ ✂ ❁ ✬<br />
1 3 ⊕ 2 ❁ ✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬ <br />
3 ⊕ S 3<br />
1❁ 3 ✂ ❉ ❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉❉<br />
♦♦♦♦♦♦♦♦♦♦♦♦♦♦ ✂ ⊕ ✚<br />
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S 1 ⊕ S 3<br />
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2<br />
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❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖<br />
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3 ⊕ S 2 ⊕ ✂ ✂ 2<br />
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2 ⊕ ✂ ✂ 3 ⊕ S 2<br />
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Figure 4. Exchange graph <str<strong>on</strong>g>of</str<strong>on</strong>g> Z/2Z-stable maximal rigid objects<br />
where N is a maximal unipotent subgroup <str<strong>on</strong>g>of</str<strong>on</strong>g> a Kac-Moody group, N − its opposite<br />
unipotent group, B − <str<strong>on</strong>g>the</str<strong>on</strong>g> corresp<strong>on</strong>ding Borel subgroup, <strong>and</strong> w is an element <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
corresp<strong>on</strong>ding Weyl group. In particular, if N is <str<strong>on</strong>g>of</str<strong>on</strong>g> Lie type <strong>and</strong> w is <str<strong>on</strong>g>the</str<strong>on</strong>g> l<strong>on</strong>gest<br />
element, <str<strong>on</strong>g>the</str<strong>on</strong>g>n N(w) = N.<br />
• Partial flag varieties corresp<strong>on</strong>ding to classical Lie groups.<br />
These results were obtained in [5] <strong>and</strong> [6] for <str<strong>on</strong>g>the</str<strong>on</strong>g> simply-laced cases <strong>and</strong> in [2] for <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
n<strong>on</strong> simply-laced cases.<br />
It permits for example to prove in <str<strong>on</strong>g>the</str<strong>on</strong>g>se cases that all <str<strong>on</strong>g>the</str<strong>on</strong>g> cluster m<strong>on</strong>omials (products<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> elements <str<strong>on</strong>g>of</str<strong>on</strong>g> a same extended cluster) are linearly independent (result which is now<br />
generalized but was new at that time) <strong>and</strong> o<str<strong>on</strong>g>the</str<strong>on</strong>g>r more specific results (for example <str<strong>on</strong>g>the</str<strong>on</strong>g><br />
–41–