Timeline for Characterizing zeros of Schur functions over $\mathbb{R^n}$ or $\mathbb{C^n}$
Current License: CC BY-SA 4.0
30 events
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| Feb 17 at 16:47 | history | edited | YCor | CC BY-SA 4.0 |
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| Aug 31, 2011 at 14:30 | history | edited | Ahmed Roman | CC BY-SA 3.0 |
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| Aug 31, 2011 at 14:15 | comment | added | Ahmed Roman | If it does, then I am done. But if it does not how should one proceed? | |
| Aug 31, 2011 at 14:12 | comment | added | Ahmed Roman | But the sequence $m_i$ is arbitrary, so you don't know about the elements, but yes some sequence might have $1,2,\cdots ,n$ | |
| Aug 31, 2011 at 14:05 | history | edited | Ahmed Roman | CC BY-SA 3.0 |
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| Aug 31, 2011 at 14:02 | comment | added | Emmanuel Briand | I have seen your edit but there is still a technical problem: your $\Gamma$ seems to include the sequence $m=(1,2,\ldots,n-1)$, whose corresponding Schur function is $1$. Trivially the corresponding $Z_s$ is empty. Maybe the correct question is different. Then probably the correct answer will be obtained by considering the fact that the Schur polynomials are a linear basis for the symmetric polynomials. | |
| Aug 31, 2011 at 13:48 | history | edited | Ahmed Roman | CC BY-SA 3.0 |
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| Aug 31, 2011 at 13:01 | comment | added | Ahmed Roman | I am sorry, may be I am not being very clear. But I will attempt to fix that. | |
| Aug 30, 2011 at 21:17 | comment | added | Emmanuel Briand | Is this question really about Schur polynomials ? If I understand correctly, $Z_{d,n}$ is the set of common zeros of all Schur polynomials of degree $d$ (in $n$ variables). Then $Z_{d,n}$ is also the set of common zeros of all symmetric polynomials that are homogeneous of degree $d$ (since the Schur polynomials of degree $d$ are a basis for this space). Then we may choose to work with an easier basis, such as the products of elementary symmetric polynomials. For instance, if $k$ is less than or equal to $n$ and divides $d$, then $Z_{d,n}$ is contained in the zero locus of $e_k$. | |
| Aug 30, 2011 at 15:48 | comment | added | Stephen | Ahmed, I don't understand the definition of $Z_{d,n}$. Is it the set of common zeros of all Schur functions of degree $d$ in $n$ variables? Nor do I understand the assumption that follows: what is "the" Schur function whose zeros are assumed to generate a free abelian group of rank equal to the number of variables? It seems to me that this can basically never hold for a single polynomial (assuming $n>1$ and you mean zeros in $\CC^n$, anyway). | |
| Aug 30, 2011 at 15:37 | history | edited | Ahmed Roman | CC BY-SA 3.0 |
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| Aug 30, 2011 at 3:04 | comment | added | Ahmed Roman | I added a paragraph to answer your question at the end. I hope my answer/question is clear. | |
| Aug 30, 2011 at 2:59 | history | edited | Ahmed Roman | CC BY-SA 3.0 |
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| Aug 30, 2011 at 2:53 | history | edited | Ahmed Roman | CC BY-SA 3.0 |
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| Aug 29, 2011 at 22:28 | comment | added | Emmanuel Briand | Like Thierry, I do not understand well the kind of answer you expect. The condition of non-invertibility of a generalized Vandermonde matrix is already expressed as an equation. Are you looking for a geometrical study of the corresponding algebraic variety ? (Such a study is addressed in the following draft: www2.math.su.se/~shapiro/Articles/SML.pdf) Of its real points ? Are you interested in sufficient conditions for invertibility (like: all $x_i$ positive and pairwise distinct that you give) ? | |
| Aug 29, 2011 at 21:26 | history | edited | Ahmed Roman | CC BY-SA 3.0 |
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| Aug 29, 2011 at 11:54 | history | edited | Ahmed Roman | CC BY-SA 3.0 |
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| Aug 29, 2011 at 11:45 | history | edited | Ahmed Roman | CC BY-SA 3.0 |
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| Aug 29, 2011 at 11:40 | history | edited | Ahmed Roman | CC BY-SA 3.0 |
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| Aug 29, 2011 at 11:34 | history | edited | Ahmed Roman | CC BY-SA 3.0 |
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| Aug 29, 2011 at 11:12 | history | edited | Ahmed Roman | CC BY-SA 3.0 |
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| Aug 29, 2011 at 8:51 | comment | added | Ahmed Roman | I hope that clarifies things a bit. | |
| Aug 29, 2011 at 8:51 | history | edited | Ahmed Roman | CC BY-SA 3.0 |
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| Aug 27, 2011 at 14:10 | comment | added | Thierry Zell | By the way, this question mathoverflow.net/questions/60938/… discussed Vandermonde determinants and Schur functions, from a slightly different angle. The answers were pretty detailed, so maybe there is something there for you. | |
| Aug 27, 2011 at 14:06 | comment | added | Thierry Zell | I don't get your questions: I am not sure what kind of results you want to know about the zeros. As for the group question, you point out yourself that the symmetric group acts on the zeros, I am not sure what else you could be looking for. I see that you've been editing your question quite a bit, and your editing improved it, but there is still room for clarification. | |
| Aug 27, 2011 at 10:42 | history | edited | Ahmed Roman | CC BY-SA 3.0 |
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| Aug 21, 2011 at 22:45 | history | edited | Ahmed Roman | CC BY-SA 3.0 |
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| Aug 20, 2011 at 4:30 | history | edited | Ahmed Roman | CC BY-SA 3.0 |
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| Aug 20, 2011 at 4:04 | history | edited | Ahmed Roman | CC BY-SA 3.0 |
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| Aug 20, 2011 at 3:58 | history | asked | Ahmed Roman | CC BY-SA 3.0 |