Timeline for Dimensions of irreducible representations of $GL(n,F_q)$ are polynoms in q having roots ONLY at roots of unity and zero?
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Aug 14, 2018 at 6:17 | answer | added | Dr. Evil | timeline score: 5 | |
| Aug 17, 2017 at 19:59 | history | edited | Alexander Chervov | CC BY-SA 3.0 |
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| Aug 15, 2017 at 6:50 | comment | added | Alexander Chervov | @user148212 Yes, it is good catch with GL(2,2) ! Thank you ! Some years ago I was wondering about S_3 = GL(2,2) irreps: math.stackexchange.com/questions/182724/… | |
| Aug 15, 2017 at 1:50 | answer | added | user148212 | timeline score: 2 | |
| Aug 15, 2017 at 0:14 | comment | added | user148212 | @AlexanderChervov This refers to the principal series one: E.g. in your GL_2 example, q+1 is not an irrep dim if q=2; in GL_2(F_2) there are 1 one dim cusp irrep, 1 trivial one dim irrep, and 1 two dim Steinberg irrep. | |
| Aug 14, 2017 at 18:31 | comment | added | Alexander Chervov | @user148212 thank you for your comments, however i am not sure i understand second one | |
| Aug 14, 2017 at 17:46 | comment | added | user148212 | Maybe you need the assumption that q is big enough to avoid some "missing dim"; e.g. (q+1)(q^2+q+1)...(q^{n-1}+...+1) is the dim of some irreps only if q is big enough. | |
| Aug 14, 2017 at 17:24 | comment | added | user148212 | For a general finite gp of Lie type, the integer coefficient property may not be true; e.g. SL_2(F_q) has irreps of dim (q-1)/2 and (q+1)/2. | |
| Aug 14, 2017 at 17:00 | answer | added | Jim Humphreys | timeline score: 6 | |
| Aug 13, 2017 at 23:24 | comment | added | Richard Stanley | Note that if $\mathrm{GL}(n,F_q)$ has an irrep whose dimension is a polynomial in $q$, then the zeros of this polynomial are indeed roots of unity and zero. This is because the dimension of an irrep divides the order of the group, and if $f(q),g(q)\in \mathbb{Q}[q]$ have the property that $f(q)|g(q)$ for infinitely many integers $q$, then $f(q)|g(q)$ as polynomials. | |
| Aug 13, 2017 at 19:42 | history | edited | Alexander Chervov | CC BY-SA 3.0 |
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| Aug 13, 2017 at 19:36 | history | edited | Alexander Chervov | CC BY-SA 3.0 |
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| Aug 13, 2017 at 19:30 | history | asked | Alexander Chervov | CC BY-SA 3.0 |