Timeline for Computing a determinant involving roots of unity
Current License: CC BY-SA 3.0
23 events
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| Oct 19, 2017 at 16:32 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Jun 25, 2017 at 20:43 | comment | added | GH from MO | @deterroot: The exponent of $i$ in Schur's article is $(d^2-d)/2$ as opposed to $(3d-2)(d-1)/2$ in my response. These two quantities are congruent to each other mod $4$ if and only if $d$ is odd (which is the case Schur is considering). For $d$ even, the exponent in Schur's style would be $1+(d^2-d)/2$. | |
| Jun 25, 2017 at 20:34 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Jun 25, 2017 at 20:08 | comment | added | GH from MO | @deterroot: It seems to me that Schur's derivation of the Gauss sum requires $d$ to be odd. Specifically, the easily known absolute value of the Gauss sum ($=\mathrm{tr}\,\Phi(d)$) and its structure as a sum of eigenvalues of $\Phi(d)$ determines the Gauss sum up to a $4$th root of unity. Using the known value of $\det\Phi(d)$, we can narrow the number of possibilities to $2$. Then, for odd $d$ there is a trick modulo $4$ that pins down the Gauss sum uniquely (see the bottom of page 210 in Landau: Elementary number theory). For even $d$, I don't see a way to generalize this trick. | |
| Jun 25, 2017 at 19:54 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Jun 25, 2017 at 2:01 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Jun 24, 2017 at 17:53 | comment | added | GH from MO | @deterroot: You are welcome. I don't have time to look at Schur's paper now, but I guess it is restricted to odd values of $n$ only, because this is the classical case of the Gauss sum. BTW I plan to work out (when I get some time) the general Gauss sum (i.e. the trace of $\Phi(d)$) using the above formula for $\det\Phi(d)$ (which will be used to choose the correct sign for the Gauss sum). I might add it here when I am done. | |
| Jun 24, 2017 at 13:59 | comment | added | deterroot | The paper by Schur, formula (7) at the end of p. 149 seems to give a different exponent for i, why is that so? | |
| Jun 24, 2017 at 12:52 | vote | accept | deterroot | ||
| Jun 24, 2017 at 5:03 | comment | added | Zach Teitler | Great! I'm glad to learn of this. Thank you Alexey and GH. | |
| Jun 24, 2017 at 4:20 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Jun 24, 2017 at 4:17 | comment | added | GH from MO | @AlexeyUstinov: Great, thanks again! I will update my Added sections accordingly. | |
| Jun 24, 2017 at 4:15 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Jun 24, 2017 at 4:13 | comment | added | Alexey Ustinov | Original Schur's article is here eudml.org/doc/59098 | |
| Jun 24, 2017 at 4:09 | comment | added | GH from MO | @ZachTeitler: See also my added sections for some history. | |
| Jun 24, 2017 at 4:08 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Jun 24, 2017 at 3:49 | comment | added | GH from MO | @AlexeyUstinov: Thanks for your valuable remark, which led to the "Added" section in my response. | |
| Jun 24, 2017 at 3:47 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Jun 24, 2017 at 3:08 | comment | added | Alexey Ustinov | $\Phi$ is known as Schur Matrix mathworld.wolfram.com/SchurMatrix.html | |
| Jun 24, 2017 at 3:06 | comment | added | GH from MO | @ZachTeitler: Very interesting, thank you. I was not aware of your paper. | |
| Jun 24, 2017 at 2:25 | comment | added | Zach Teitler | May I mention that essentially the same calculation is in section 3.1 of arxiv.org/abs/1304.7202. | |
| Jun 23, 2017 at 23:22 | history | edited | GH from MO | CC BY-SA 3.0 |
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| Jun 23, 2017 at 23:00 | history | answered | GH from MO | CC BY-SA 3.0 |