Timeline for Bounding weight multiplicities by number of certain Coxeter elements
Current License: CC BY-SA 3.0
11 events
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| Jul 27, 2017 at 18:08 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
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| Jul 21, 2017 at 15:52 | comment | added | Jingren Chi | @JimHumphreys I have edited according to your suggestions. Thanks again! | |
| Jul 21, 2017 at 13:51 | comment | added | Jim Humphreys | P.S. The history of Coxeter's work in geometry makes it tricky to adapt terminology to Lie theory, so it may be simpler to work just with root systems and specify your conjectured lower bound as (say) $2^{n-1}$ when $n$ is the Lie rank. There is no need to mention "Coxeter elements", which are geometrically indistinguishable and form a single conjugacy class in a finite group $W$ generated by euclidean reflections. For finite dimensional representations of semisimple algebraic groups it's essential to fix a simple system of roots to get a dominant Weyl chamber. | |
| Jul 20, 2017 at 17:47 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
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| Jul 20, 2017 at 15:21 | comment | added | Jim Humphreys | @HCCH: To clarify further, there is just one definition of "Coxeter element". Usually a Coxeter system includes specification of the set of simple reflections, but in fact it doesn't matter whether you use another such generating set since all of these are conjugate. (This isn't emphasized in the literature, but I think an elementary argument takes care of it.) The crucial fact is that all Coxeter elements are conjugate, and thus the number of them is just the order of $W$ divided by the order $h$ of the centralizer of one such element. | |
| Jul 20, 2017 at 5:13 | comment | added | Jingren Chi | Thanks for your clarification and references. I apologize for the confusion of terminology. The number in my lower bound only counts those Coxeter elements that can be written as products of a fixed set of simple reflections, so it's much smaller than the number of all Coxeter elements, for $S_n$, it's $2^{n-2}$ vs $(n-1)!$. | |
| Jul 19, 2017 at 18:07 | comment | added | John Shareshian | Sorry, I didn't read the comments above. Everything is clear after reading them. | |
| Jul 19, 2017 at 17:15 | comment | added | Hugh Thomas | Note that the OP is using a different definition of Coxeter element from the one in this answer. | |
| Jul 19, 2017 at 15:52 | comment | added | John Shareshian | Are you sure that all conjugates of a given Coxeter element are Coxeter elements themselves? This seems false in type $A_3$. Maybe I misunderstand your definition. | |
| S Jul 19, 2017 at 15:41 | history | answered | Jim Humphreys | CC BY-SA 3.0 | |
| S Jul 19, 2017 at 15:41 | history | made wiki | Post Made Community Wiki by Jim Humphreys |