20
votes
Accepted
Biographical information on Anne Marie Whitney
There is more under her married name, Anne Calloway (October 2, 1921, December 27, 2008). Here is a photograph. Anne Whitney married her graduate school class mate Jean Calloway, who himself became a ...
- 188k
13
votes
Vandermonde matrix is totally positive
Total positivity: tests and parametrizations by Fomin and Zelevinsky
According to this reference (bottom of page 5), Fekete proved that a sufficient condition for total positivity is that all solid ...
- 2,175
12
votes
What is the amplituhedron?
There is now an AMS Notices article whose title is the same as the title of this question (and which therefore may be enlightening to anyone on this page): http://www.ams.org/journals/notices/201802/...
- 24.2k
12
votes
Accepted
Total positivity of $q$-Pascal matrix?
Yes. Consider the set $V$ of points with integer coordinates as vertices of a weighted directed acyclic graph. Namely, for any $(i,j)\in V$, the edge from $(i,j)$ to $(i+1,j)$ has weight 1, the edge ...
- 109k
11
votes
Accepted
Vandermonde matrix is totally positive
Totally Non-Negative Matrices Johnson and Fallat, Princeton University Press. Chapters 0 and 1.
Allan Pinkus (2009), Totally Positive Matrices, Cambridge University Press, ISBN 9780521194082
Perhaps ...
- 22.8k
9
votes
Vandermonde matrix is totally positive
The minors of a Vandermonde matrix are known as alternants. Their positivity follows from the fact that they are products of the Vandermonde determinant of the variables involved with a Schur function ...
- 33.8k
8
votes
Accepted
Total positivity, log-concavity and Pólya frequency
$\newcommand{\R}{\mathbb R}$For a positive integer $r$, a measurable function $f\colon\R\to\R$ is called a Pólya frequency function of order $r$ (abbreviated as PF$_r$) if the matrix $(f(x_i-y_j))_{i,...
- 128k
8
votes
Accepted
$q$-analogs of total positivity
This is a wonderful question but unfortunately I don't think that there is a definite answer in the literature just yet. Let's look at two somewhat recent lines of research in this direction:
1) A. ...
- 85.6k
7
votes
Accepted
Map from Bruhat stratification to Catalan stratification for the space of totally nonnegative upper-triangular matrices
The Catalan strata are unions of Bruhat strata, and the resulting map from permutations to Dyck paths is indeed given by taking the left-to-right maxima.
There is a way to parametrize totally ...
- 85.6k
6
votes
Vandermonde matrix is totally positive
V. Prasolov's book Problems and Theorems in Linear Algebra contains it as Theorem 1.2.12.2.
Russian text in pdf Alas, this very useful book seems to be not translated into English. The previous, ...
- 109k
5
votes
Accepted
Total positivity tests: optimal in the number of minors vs. the computational cost
There are at least a few places looking at complexity for total positivity counting arithmetic operations. I don't know if people have looked at trying to find smaller minors. These use initial minors ...
- 7,875
5
votes
Decorated permutations and subset permutations
Here is a direct proof for the formula via generating functions.
Using the formula for derrangement numbers, we have $$\begin{split}
F(n,k)=&\binom nk\cdot !(n-k)\\
=&\frac{n!}{k!}\cdot [x^{n-...
- 34.3k
4
votes
Accepted
Big cells in a Grassmann and permutations
Positroid cells in $Gr(k,n)$ are indexed by many objects we often want to go between. The big cell will be given by the bounded affine permutation $i \mapsto i+k$. See Postnikov's original preprint (...
- 7,875
4
votes
An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes
In this recent preprint of Postnikov https://arxiv.org/abs/1806.05307 (which is for an upcoming ICM talk in Rio) he explains a beautiful direct connection between the geometry of the positive ...
- 24.2k
3
votes
Decorated permutations and subset permutations
Max's answer is neat in using $e^{2x}\frac{e^{-x}}{1-x}=\frac{e^x}{1-x}$. We offer one approach without generating functions. To this end, let $f_n:=\sum_{k=0}^n\frac1{k!}$ and $g_n:=\sum_{k=0}^n\frac{...
- 43.2k
2
votes
Accepted
Are the minors of this Hadamard product Schur positive?
The answer to this question is no, with a quick counterexample being given by the determinant of the matrix
$$\begin{pmatrix}
h_{2} (x) h_{2} (y) & h_{3} (x) h_{3} (y) & h_{4} (x) h_{4} (y)\\\...
- 553
1
vote
$q$-analogs of total positivity
Here's an example of $\Bbb{R}[[q]]$-total positivity arising from specializing an instance of Schur-positivity related to the Grassmannian:
Given an ordered $k$-subset $I = \{i_1 < \dots < i_k \}...
- 2,137
1
vote
Accepted
An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes
From 4gravitons' [Post on the Amplitudes 2017 Conference] 1: "Between the two of them, Nima (Arkani-Hamed) and Yuntao (Bai) covered an interesting development, tying the Amplituhedron together ...
- 10.5k
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