Timeline for Families of Hessenberg varieties for $GL_n$
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Aug 20, 2018 at 19:25 | comment | added | Cheng-Chiang Tsai | On the other hand, it is possible to generalize the method/result of Brosnan and Chow in the following sense: the (sum of shifted) local systems on the regular semisimple locus as well as the stalks of the corresponding perverse sheaves are now combinatorially known. Thus one can compare the stalks given by the full support components with Tymoczko's result on the Betti numbers of the fibers. They agree iff the full support assertion is true. This definitely can be verified (for a not-so-large fixed n) by a program, though I am probably not going to write one soon... | |
| Aug 20, 2018 at 19:24 | comment | added | Cheng-Chiang Tsai | Examples on $GL_2$ and $GL_3$ suggest me that the naive way to use the dimension bound is not enough - not saying there couldn't be some smarter way. | |
| Aug 20, 2018 at 17:49 | comment | added | Alexander Woo | Would an affirmative answer imply that the locus where the fibers (i.e. the Hessenberg variety for $\gamma$) are big is small? Dimensions can be calculated from the combinatorics in Tymoczko's paper (and there are papers that have done these calculations in various cases). | |
| Aug 20, 2018 at 4:11 | comment | added | Cheng-Chiang Tsai | And of course, their result works for the usual Hessenberg varieties for $GL_n$, not necessarily for $\mathfrak{h}$ in the "Hessenberg form"! For my interest I can reduce something about affine Springer fiber that I want to compute to Hessenberg varieties in this form (among with other things), but of course the question can be asked in general. | |
| Aug 20, 2018 at 3:53 | comment | added | Cheng-Chiang Tsai | Thank you! I haven't read the paper before and I really appreciate the direction. If I just read it correctly, I think part of their main result, Theorem 127, answers my (second) question affirmatively on the regular locus, i.e. every component whose support intersects nontrivially with the regular locus has full support. I can't think of a way to generalize their method for now. I will add comment if I find anything. | |
| Aug 20, 2018 at 0:33 | comment | added | Alexander Woo | I'm not sure an answer is there, but look at the (unique) paper of Patrick Brosnan and Timothy Chow. | |
| Aug 19, 2018 at 21:07 | history | asked | Cheng-Chiang Tsai | CC BY-SA 4.0 |