Timeline for Representation of $\mathrm{AGL}(V)$ on the homology of the poset of affine subspaces of $V$
Current License: CC BY-SA 4.0
14 events
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| Nov 2, 2022 at 14:55 | comment | added | Benjamin Steinberg | Your answer was quite helpful. Thanks | |
| Nov 2, 2022 at 14:54 | comment | added | Benjamin Steinberg | For any geometric lattice it is a wedge of spheres of the same dimension (the dimension of the complex)by EL-shellability. Your count gives the number of spheres. | |
| Nov 2, 2022 at 13:57 | comment | added | Jesper M. Moller | Yes, something like what you say seems to follow from Siegel's study of the irreducible characters of the affine group. I assume that the homotopy complementation formula of Bj{\"o}rner and Walker or Bj{\"o}rner can be used determine the homotopy type of $A_n(q)$. | |
| Nov 1, 2022 at 16:43 | comment | added | Benjamin Steinberg | This irreducible rep of the affine group appears once as a constituent of the n-chains and carries the Steinberg rep of GL_n as a constituent. Moreover, if I understand the proof, then n-1-chains on this geometric lattice as an affine group rep is induced from the n-1-chains of the Tits building as a GL_n-rep and the unique copy of the Steinberg rep in this restriction is in the kernel of the boundary map on the Tits building. It follows that the Affine group module generated by this copy of the Steinberg representation is the unique irreducible of our desired dimension and is in the homology | |
| Nov 1, 2022 at 16:39 | comment | added | Benjamin Steinberg | It seems like the paper On the characters of the affine group over a field sciencedirect.com/science/article/pii/0021869392901337 by Siegel, who was a student of Solomon, says the affine group has a unique irreducible character of that dimension and it is a constituent in the module of n-1-chains. Moreover, if we restrict that character to the general linear group you get a character with the Steinberg representation as a constituent with multiplicity one. To be ctd | |
| Nov 1, 2022 at 13:01 | comment | added | Jesper M. Moller | Yes, I combined the two answers into one. | |
| Nov 1, 2022 at 12:59 | history | edited | Jesper M. Moller | CC BY-SA 4.0 |
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| Nov 1, 2022 at 12:57 | comment | added | Benjamin Steinberg | By the way would it not make more sense to combine your two answers into one? | |
| Nov 1, 2022 at 12:56 | comment | added | Benjamin Steinberg | Thanks. My understanding is Solomon is looking at that representation but I didn't know how to compute the dimension to see if Solomon was picking out a subrepresentation or the whole thing. I didn't know enough about this kind of geometric lattice stuff. All I could find was counting nonbroken circuits. The recursion is a good approach. Thanks. I'm glad this representation is irreducible | |
| Nov 1, 2022 at 12:54 | vote | accept | Benjamin Steinberg | ||
| Nov 1, 2022 at 12:53 | history | edited | Jesper M. Moller | CC BY-SA 4.0 |
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| Nov 1, 2022 at 12:42 | history | edited | Jesper M. Moller | CC BY-SA 4.0 |
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| Nov 1, 2022 at 12:37 | history | edited | Sam Hopkins | CC BY-SA 4.0 |
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| Nov 1, 2022 at 12:33 | history | answered | Jesper M. Moller | CC BY-SA 4.0 |