Timeline for How is the free modular lattice on 3 generators related to 8-dimensional space?
Current License: CC BY-SA 3.0
8 events
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| Sep 20, 2015 at 14:29 | history | edited | Hugh Thomas | CC BY-SA 3.0 |
point out horrible errors
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| Sep 20, 2015 at 3:12 | comment | added | Hugh Thomas | Going in the more-and-more-trivial direction, it also holds for the star quiver with 1 external vertex (which is $A_2$) (i.e., we get the free bounded modular lattice on one generator) and with 0 external vertices ($A_1$), where we get the free bounded modular lattice on zero generators. | |
| Sep 20, 2015 at 2:46 | comment | added | Hugh Thomas | Well, for $D_3=A_3$, the corresponding lattice of torsion-free classes is the free bounded modular lattice on two elements, which is encouraging. (Note, though, that the connection to the root poset, and the dimension of the Lie algebra, both disappear.) | |
| Sep 20, 2015 at 2:34 | comment | added | John Baez | +Hugh Thomas - Thanks for your new improved answer! I'm not quite sure what is left to understand. For now I guess we can summarize by saying the lattice of torsion-free classes in the category of injective representations of the $D_4$ quiver is the free modular lattice on 3 generators. I guess one could try to generalize this a bit: is the lattice of torsion-free classes in the category of injective representations of the $\widetilde{D}_4$ quiver the free modular lattice on 4 generators? ($\widetilde{D}_4$ has infinitely many indecomposable representations but it's still 'tame'.) | |
| Sep 20, 2015 at 2:07 | history | edited | Hugh Thomas | CC BY-SA 3.0 |
added stuff
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| Sep 20, 2015 at 0:15 | comment | added | John Baez | Thanks. Apparently Dedekind was in the latter camp, since he says the free modular lattice on 3 generators had 28 elements. Apparently the person who drew the picture (I'm afraid I forget who) was in the former camp. Getting this straight should help figure out the puzzle... along with Hugh's nice remark about the top part being the poset of positive roots! | |
| Sep 19, 2015 at 22:00 | comment | added | Todd Trimble | Perhaps a bit of culture clash. Some authors say a lattice should (by definition) admit all finite meets and joins; others say it should admit binary meets and joins, and say "bounded lattice" if they mean to include the empty meet and empty join. It's my impression that many lattice theorists (who call themselves such) are in the latter camp. | |
| Sep 19, 2015 at 21:37 | history | answered | Hugh Thomas | CC BY-SA 3.0 |