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.)
10 questions with no upvoted or accepted answers
7
votes
0
answers
276
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 ...
- 24.2k
5
votes
0
answers
287
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}) := \,
\...
- 2,137
3
votes
0
answers
153
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) ...
- 2,137
3
votes
0
answers
137
views
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 ...
3
votes
0
answers
226
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 ...
- 6,211
3
votes
0
answers
182
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 ...
- 2,479
2
votes
0
answers
122
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
2
votes
0
answers
287
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 ...
- 156k
2
votes
0
answers
132
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 ...
1
vote
0
answers
82
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 ...
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