157 problems

Let $A = \left(\genfrac{\langle}{\rangle}{0pt}{}{n}{k}\right)_{n,k\geq 0}$ be the Eulerian triangle, whose entries count permutations on $[n]$ by descents. Brenti's conjecture. The…

Canonical-basis characterization conjecture.

Hankel total positivity conjecture. The sequences of polynomials

Let $\bm M_d$ be the matrix in the McMullen correspondence, of size $\left(\left\lfloor\frac{d}{2}+1\right\rfloor\right)\times(d+1)$. A matrix is totally non-negative if every mino…

Sokal's functional-digraph conjecture. The following assertions hold: (i) $[\psi_{n,k}]_{n,k}$ is totally positive; (ii) $(\bm{\psi}_n(y))_{n\geq0}$ is coefficientwise Hankel-total…

Let $P_n(x;\bm{\mu})$ be the row-generating polynomials of the Graham–Knuth–Patashnik recurrence, with parameters $\bm{\mu}=(\alpha,\beta,\gamma,\alpha',\beta',\gamma')$ treated as…

Let $A$) be a totally nonnegative $n\times n$ matrix, meaning that all its minors are nonnegative, and let $\lambda\vdash n$ be a partition. For a class function $\phi^{\lambda}$ o…

Let $A$ be a totally positive element of $GL_n$, and let $R$ be a ratio of products of minors as above. In a weighted planar-network parametrization, call $R=p/q$ subtraction free…

Let $S_n$ be the symmetric group, let $0<k<n$ be an integer, let $v in Sn$ avoid the pattern $12cdots(k+1)$, and let $operatorname{Imm}v X$ denote the corresponding Kazhdan–Lusztig…

Let $\operatorname{Gr}(k,n)$ denote the Grassmannian of $k$-planes in $\mathbb{R}^n$, and let $\operatorname{Gr}_{\geq 0}(k,n)$ be its totally nonnegative part, consisting of the p…

For a sequence $\{a_k\}_{k\geq0}$, define the operator $\mathcal{L}$ by … A Pólya frequency (PF) sequence is a sequence whose associated generating polynomial has the relevant tota…

Let $M$ be an $n\times n$ matrix with entries in $\mathbb R$ that is totally positive, meaning that all minors of $M$ are positive. For a commutation class of double wiring diagram…

Williams' conjecture. The cell decomposition of $(G/P)_{\geq 0}$ is a regular CW complex: the closure of each cell is homeomorphic to a closed ball and the boundary of each cell is…

Let $M$ be a generic square matrix, and let a double wiring diagram be arbitrary. The chamber-minor Laurent positivity conjecture. Every minor of $M$ can be written as a Laurent po…

Let $D_1,D_2,D_3$ be positive definite matrices, and let $M$ be the matrix whose entries are the coefficients of $y$ and $z$ in … A minor means the determinant of any finite square…

Let $f(x,y)=\sum f_i(x)y^i\in\mathbf{P}^{\mathrm{pos}}_2$, and form the upper-triangular coefficient matrix … A minor means the determinant of any finite square submatrix. The tota…

Let $f\in\mathbf{P}^{\mathrm{pos}}$ and define the Hankel matrix from its derivatives by … For a fixed size $d$, write $\det H((f(x),f'(x),f^{(2)}(x),\dots);d)$ for the correspondi…

Let $M_d$ be the transfer matrix associated with the McMullen correspondence, with entries … A matrix is totally nonnegative if every minor is nonnegative. Bjbfrner's conjecture. A…

Regularity conjecture. The CW-complex formed by the cells $S_\mathcal{M}^\mathrm{tnn}$ is regular. The closure of each cell is homeomorphic to a closed ball of appropriate dimensio…

Let $G$ be a simple algebraic group, let $W$ be its Weyl group, and let $\flag\times\flag$ be the double flag variety with positive part $\pos{(\flag\times\flag)}$. For…

Cube conjecture. The cell-decomposed space $Y_{ge 0}$ is homeomorphic to an $(n-1)$-dimensional cube.

Let $G/P$ be an arbitrary flag variety, and consider its totally nonnegative part together with the cell decomposition induced by total positivity. Regular CW ball conjecture. The…

Let $G$ be a reductive algebraic group, let $P$ be a parabolic subgroup, and let $\mathcal Y_P$, $\mathcal X_P$, and $X_P$ be the varieties associated with the Peterson isomorphism…

Let $\alpha^1,\alpha^2,\ldots,\alpha^s\in\binom{[n]}{k}$ satisfy … and let $g_2,g_3,\ldots,g_s$ be totally positive matrices. Given any real flag $F_\bullet^1$, define…

Let $G$ be a reductive group, and let a double Bruhat cell be the intersection of a Bruhat cell for one Borel subgroup with a Bruhat cell for the opposite Borel subgroup. Consider…