Timeline for Dual versions of "folding" symmetric ADE Dynkin diagrams?
Current License: CC BY-SA 4.0
9 events
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| Feb 17, 2023 at 12:55 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
http -> https (the question was bumped anyway)
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| Oct 9, 2022 at 15:36 | comment | added | Allen Knutson | Consider a slice (2-dimensional) of the subprincipal nilpotent inside the principal, and take its preimage under the Springer resolution; the fiber over the simple surface singularity will be an ADE Dynkin curve. But what if you start with a non-simply-laced group? Then the subprincipal nilpotent isn't simply connected, and so, you get an action of its pi_1 on the ADE Dynkin diagram. So a non-simply-laced algebra "tells you" how to obtain it from folding. Interestingly, not every possible folding arises this way -- the groups have favored ways they wish to be obtained. | |
| Oct 9, 2022 at 3:48 | answer | added | Kimyeong Lee | timeline score: 1 | |
| Nov 28, 2019 at 16:25 | answer | added | Sam Hopkins | timeline score: 7 | |
| Nov 5, 2012 at 3:08 | comment | added | Dima Pasechnik | geometers study these sorts of foldings - although for them the diagrams have slightly different meaning. IIRC, in the J.Tits' book on buildings of spherical type folding of diagrams is mentioned. The book books.google.com/books/about/… by A.Pasini has a chapter on foldings. | |
| Nov 4, 2012 at 23:22 | comment | added | Jim Humphreys | @Theo and Marty: I agree that there is a strong flavor of both roots and coroots in the ways these foldings arise, plus sometimes a clear rationale for taking fixed points in the Lie algebra (or group). Asking about origins of these ideas is just a step toward unifying them, which may or may not be feasible. My last two paragraphs indicate the kind of problem where I get stuck, even though the folding is somehow implicit in the eventual results on representation theory. | |
| Nov 4, 2012 at 19:28 | comment | added | Marty | Why look at roots instead of coroots? You can see the $F_4$ embedded in $E_6$, for example, by seeing each simple coroot of $F_4$ as a sum of one or two coroots from $E_6$. When one group embeds in another compatibly with max tori, e.g. $F_4$ in $E_6$, the map is covariant from coroot lattice of the small group to coroot lattice of the large group. | |
| Nov 4, 2012 at 17:05 | comment | added | Theo Johnson-Freyd | My guess is that there is a basic duality going on here. If memory serves, in the case of folding for Lie algebras, one recovers precisely the subalgebra fixed by the symmetry group (Z/2 most of the time, permutations of 3 things in the $D_4 \to G_2$ case). Could the "dual" cases correspond to times when you recover the cofixed points of the symmetry, rather than the fixed points? | |
| Nov 4, 2012 at 15:58 | history | asked | Jim Humphreys | CC BY-SA 3.0 |