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.)

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11
votes
1answer
324 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. ...
7
votes
0answers
206 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 ...
17
votes
1answer
622 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 ...
2
votes
0answers
92 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 ...
13
votes
4answers
2k 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)...
4
votes
1answer
202 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....
11
votes
1answer
171 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 ...
3
votes
0answers
188 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 ...
5
votes
2answers
341 views

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
votes
0answers
156 views

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 ...
4
votes
3answers
2k views

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 ...
2
votes
1answer
407 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 ...
70
votes
4answers
5k views

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 ...
67
votes
2answers
8k 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 ...
5
votes
1answer
454 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 ...
9
votes
3answers
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 &...