Questions tagged [total-positivity]
For questions related to totally positive (or totally nonnegative) matrices, and related topics such as total positivity in a more general Lie-theoretic setting. (Not related to "totally positive integers" in the number-theoretic sense.)
32 questions
7
votes
2
answers
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Unimodality / Totally non-negative properties of the sequence of the number of non-zero minors of a matrix
Consider a field $\mathbb{K}$ and a matrix $A \in \mathbb{M}_n(\mathbb{K})$. Let's define for each $0\le k \le n$ , the number $N_k$ defined as the number of non-zero minors of size k of A. For k=0 , ...
- 73
0
votes
0
answers
119
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Invertibility of a semi-infinite lower triangular Toeplitz matrix
Let $\left(\dots, 0, 0, a_0, a_1, a_2, \dots \right)$ be a totally positive (TP) sequence. Is its corresponding Toeplitz matrix
$$A = \begin{bmatrix}
a_0 & \cdots & \cdots & \...
- 7
0
votes
0
answers
61
views
Divided differences of TP kernels yield distinct, real-rooted polynomials
Let $\{w_j:~1\le j\le N\}$ be a set of non-zero real numbers with $\sum_{j} \frac{1}{|w_j|}<\infty$. We define a polynomial $P(\xi,z)=\sum_{k=0}^{N-1}f_s(\xi)z^{s}$, where $f_s(\xi)$ is a real ...
- 7
5
votes
0
answers
239
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Totally positivity and the open Toda chain
The open Toda chain is the coupled system of differential equations
\begin{equation}
(\dagger) \quad
\begin{array}{ll}
x_n'(t) &= \, y_n(t) \, - \, y_{n-1}(t) \\
y_n'(t) &= \, y_n(t) \big( x_{...
- 2,257
0
votes
0
answers
69
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Is the non-central Beta distribution $\mathrm{STP}_{\infty}$?
$\newcommand\TP{\mathrm{TP}}\newcommand\STP{\mathrm{STP}}\newcommand\SVR{\mathrm{SVR}}$This excerpt is from the book Testing Statistical Hypotheses by Lehmann and Romano.
A family of distributions ...
- 55
0
votes
1
answer
169
views
Totally positive distributions of order 2 but not order 3
This excerpt is from the book "Testing Statistical Hypotheses" by Lehmann and Romano.
A family of distributions with probability densities $p_{\theta}(x)$,
$\theta$ and $x$ real-valued and ...
- 55
4
votes
1
answer
179
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Are the minors of this Hadamard product Schur positive?
Let $h_i (x)$ denote the complete symmetric function of degree $i$ in some set of variables $x = (x_1 , x_2 , \dots)$. Then the minors of the Toeplitz matrix $T (x) = \left(h_{i-j} (x) \right)_{i,j}$ ...
- 563
6
votes
1
answer
911
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Total positivity, log-concavity and Pólya frequency
I am not familiar with the definition of total positivity. I am not sure about the link between log-concavity and total positivity.
In a paper On Variation-Diminishing Integral Operators of the ...
- 403
4
votes
0
answers
194
views
The Grassmann twist-map, an associated semi-group action, and RSK
Let me begin by setting some notation: Let $\mathrm{Mat}_{k,n}(\Bbb{R})$ denote the vector space of all $k \times n$
real-valued matrices. Given $g \in \mathrm{Mat}_{k,n}(\Bbb{R})$ and two (ordered) ...
- 2,257
4
votes
1
answer
155
views
Total positivity tests: optimal in the number of minors vs. the computational cost
A real matrix is called totally positive if all of its minors are positive. Of course, to check that a given matrix is totally positive it is not necessary to check all the minors: for example, it ...
- 3,530
1
vote
0
answers
112
views
Embedding of co-oriented subspaces into positive Grassmannian
$\def\R{\mathbb{R}}$Let $P_1$, $P_2$, $P_3$ be three $m$-dimensional subspaces in $\R^n$. With a slight abuse of notation they will also denote the ortho-projectors on the respective subspaces. We ...
- 181
3
votes
0
answers
191
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Positivity of sequences
Totally positive sequences $\lbrace a_n\rbrace_{n\in\mathbb{Z}}$ are defined as those such that the Töplitz matrix $A_{ij}=a_{i-j}$ is totally positive (all its minors are non-negative). An ...
2
votes
0
answers
187
views
Total positivity of order 2 of generalized absolute value density or likelihood ratio order of "shifted" generalized absolute value
If $f$ is the Lebesgue density of a real valued symmetric random variable $X$ (symmetric means $X \overset{d}{=} -X$) then for fixed $u > 0$
$$f^*(v,u) := f(-u -v) + f(-u+v)$$
is the density of $\...
- 131
5
votes
0
answers
329
views
Infinite tridiagonal matrices and a special class of totally positive sequences
Let $\Bbb{y} = \big(y_1, y_2, y_3, \dots \big)$ be an infinite sequence of positive real numbers such that following $\Bbb{N} \times \Bbb{N}$ tridiagonal matrix
\begin{equation}
T(\Bbb{y}) := \,
\...
- 2,257
11
votes
1
answer
387
views
Map from Bruhat stratification to Catalan stratification for the space of totally nonnegative upper-triangular matrices
$\DeclareMathOperator\SL{SL}$This question came up in a class "Total Positivity and Cluster Algebras" being taught by Chris Fraser.
Let $N^+$ denote the space of uni-upper-triangular ...
- 25.9k
2
votes
0
answers
330
views
Finding the decorated permutation of a non-reduced plabic graph
This is a question about Postnikov's theory of positroids and plabic graphs. The short version is
If we have an non-reduced plabic graph $G$, how do we look at the alternating strands and read off ...
- 161k
11
votes
2
answers
688
views
$q$-analogs of total positivity
A real matrix $M$ is called totally positive if all of its minors are positive; these matrices have been extensively studied, and there are generalizations to other Lie types, for example by Lusztig.
...
- 2,783
7
votes
0
answers
323
views
Cyclic shift acting on finite Grassmannian
The (twisted) cyclic shift $(v_1,v_2,\ldots,v_n) \mapsto (v_2,v_3,\ldots,v_n,(-1)^{k-1}v_1)$ acting on the Grassmannian $\mathrm{Gr}(\mathbb{C};k,n)$ of $k$-planes in $\mathbb{C}^n$ is an important ...
- 25.9k
18
votes
1
answer
799
views
Biographical information on Anne Marie Whitney
I am looking for information on the mathematician Anne Marie
Whitney. She wrote a number of significant papers related to total positivity with her thesis adviser Isaac Schoenberg. All I could find on ...
- 53.6k
2
votes
0
answers
155
views
What is the symmetry group of the totally nonnegative Grassmannian $Gr_{tnn}(k,n)$?
What is the symmetry group of the totally nonnegative Grassmannian $Gr_{tnn}(k,n)$? [The latter consists of those elements of the Grassmannian that can be represented by $k \times n$-matrices all of ...
15
votes
4
answers
3k
views
Vandermonde matrix is totally positive
A totally positive matrix $M\in \mathcal{M}_{n\times m}(\mathbb R)$ is such that all of its minors of all sizes are positive. It is true that any Vandermonde matrix (with well-ordered positive entries)...
- 5,562
4
votes
1
answer
342
views
Big cells in a Grassmann and permutations
In the lecture notes, it is said that (Theorem 3.1.3) the set of positroid cells in $Gr(k,n)$ are in one to one correspondence with the set of bounded affine permutations of type $(k,n)$. In Example 4....
- 6,351
12
votes
1
answer
339
views
Total positivity of $q$-Pascal matrix?
A matrix of real numbers is called totally positive if all its minors are non-negative. A well-known example is the Pascal matrix $(\binom{i}{j})$.
Is it true that the minors of the $q$-Pascal matrix ...
- 5,832
3
votes
0
answers
249
views
Definition of loop amplituhedrons
In the paper The Amplituhedron
, Nima Arkani-Hamed and Jaroslav Trnka introduced the geometric object amplituhedron. It is defined as follows (see also the lecture notes).
Let $Z$ be a $(k+m)\times ...
- 6,351
7
votes
2
answers
4k
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Decorated permutations and subset permutations
Decorated permutations are defined as permutations where the fix-points come in two colors (say $\overline{\cdot}$ and $\underline{\cdot}$). For example, the 16 decorated permutations of length 3 are
$...
- 3,331
3
votes
0
answers
200
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Spectra of certain totally positive matrices
Let $S$ be the set of $3 \times 3$ matrices $A$ satisfying the following conditions:
All minors are $>0$ (i.e., $A$ is a strictly totally positive matrix);
all principal minors are $>1$, except ...
- 2,509
7
votes
3
answers
2k
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An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes
What are the roles that the classic number arrays-- Eulerian, Narayana--play in the application of totally non-negative Grassmannians, or amplituhedrons, to string / twistor scattering theory?
(This ...
- 11.1k
3
votes
1
answer
999
views
Positroids and Totally Nonnegative Complex Grassmanian
Recently I begin working on matroids, in particular to a generalization of oriented matroids to the complex case.
I found on arxiv the following interesting articles:
1)Alexander Postnikov: Total ...
- 741
76
votes
4
answers
6k
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What is the amplituhedron?
The paper ”Scattering Amplitudes and the Positive Grassmannian” by Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, and Jaroslav Trnka, introduces ...
- 25.3k
74
votes
2
answers
9k
views
The amplituhedron minus the physics
Is it possible to appreciate the geometric/polytopal properties of the amplituhedron without delving into the physics that gave rise to it?
All the descriptions I've so far encountered assume ...
- 153k
5
votes
1
answer
632
views
On totally nonnegative Grassmannian
I was reading Postnikov's paper [TOTAL POSITIVITY, GRASSMANNiANS, AND NETWORKS][1] when I came across the definition of the totally nonnegative Grassmannian $Gr_{kn}^{tnn} \subset Gr_{kn}$ as the ...
- 519
9
votes
3
answers
4k
views
Eigenvectors of a certain big upper triangular matrix
I'm looking at this matrix:
$$
\begin{pmatrix}
1 & 1/2 & 1/8 & 1/48 & 1/384 & \dots \\
0 & 1/2 & 1/4 & 1/16 & 1/96 & \dots \\
0 & 0 & 1/8 & 1/16 &...
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