Timeline for Dimension of roots of irreducible Schur polynomial on unit circle

7 events

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Jan 16, 2018 at 14:13	history	edited	M. Hosseini		edited tags
Sep 8, 2017 at 15:18	comment	added	David E Speyer		I don't understand your last remark, but I also see that I missed the hypothesis that $\mathrm{GCD}(\lambda_i+n-i)=1$, so this rules out taking $\lambda_i=n-i$.
Sep 8, 2017 at 6:47	comment	added	M. Hosseini		$s_{(n-1)(n-2) \cdots 210}(z_1, \ldots,z_n) =1$. I'd like to know the dimension of distinct roots i.e. $\theta_1\neq\ldots\neq\theta_n$ and $\in [0,2\pi)^n$.
Sep 8, 2017 at 0:57	comment	added	David E Speyer		Also, $s_{(n-1)(n-2) \cdots 210}(z_1, \ldots,z_n) = \prod_{1 \leq i < j \leq n} (z_i+z_j)$, so in that case $V$ is codimension $1$.
Sep 8, 2017 at 0:57	comment	added	David E Speyer		I find your formula for $V_i$ confusing but, in any case, there are some other $1$-dimensional subsets of $V$ to consider. For any $1 \leq i < j \leq n$ and any $(k_1, \ldots, k_n) \in \mathbb{Z}^n$, we can consider the set of all vectors of the form $(\theta, \theta, \ldots, \theta) + \tfrac{2 \pi}{(\lambda_i+n-i) - (\lambda_j+n-i))}(k_1, \ldots, k_n)$, and the $i$-th and $j$-th rows of the generalized Vandermonde determinant will be proportional.
Sep 7, 2017 at 17:20	review	First posts
Sep 7, 2017 at 17:37
Sep 7, 2017 at 17:18	history	asked	M. Hosseini	CC BY-SA 3.0