229 problems

Andrews–El Bachraoui positivity conjecture. For every positive integer $k$, the series $F_{k,1}(q)$ is positive.

Andrews–Merca's conjecture. The coefficient of $q^m$ in this series is non-negative.

Positivity conjecture. For all integers $k\ge1$, $d\ge2$, and $n\ge0$, the Laurent polynomial $\mathscr{L}_{k,n}^{d}(z)$ has positive coefficients. This conjecture is presented as…

Beltrametti–Sommese conjecture. If $D-K_X$ is nef, then $D$ is ample.

Let $\mathcal{A}(X,\mathbf{h},\Lambda,\widetilde{B})$ be a generalized quantum cluster algebra with initial extended cluster $\{X_1,\ldots,X_m\}$. A Laurent polynomial in this clus…

Let $m\in\mathcal{M}_+$, and let $L_q(m)$ be the corresponding canonical-basis element of the quantum Grothendieck ring. Call $L_q(m)$ real if, for every…

Hankel total positivity conjecture. The sequences of polynomials

Let $M$ be a matroid, and let $P_M(t)\in\mathbb{Z}[t]$ be its Kazhdan–Lusztig polynomial, characterized by the recursive conditions in the paper. Kazhdan–Lusztig positivity conject…

Let $Q$ be a quiver, let $\mathbf{r}\in\mathbb{Z}_{\geq0}^{Q_{0}}$ be a rank vector, and let $\alpha\geq1$. Let $A_{(Q,\alpha),\mathbf{r}}$ be the associated Kac polynomial for the…

Let $\lambda$ be a partition and let $\theta^{(\alpha)}_\mu(\lambda)$ be the Jack-character coefficient defined by the paper, with $b:=\alpha-1$. Let $m_1(\mu)$ denote the number o…

Let $n\geq 1$. For partitions $\pi,\mu,\nu\vdash n$, define the coefficient $f^\pi_{\mu,\nu}(\alpha)$ by … where $e_\pi$ denotes the elementary symmetric function. Monomial positiv…

Let $\beta$ be a formal variable, let $u,v,w\in S_\infty$, and let $\mathfrak{G}_w(\mathbf{x};\mathbf{y})$ denote the double Grothendieck polynomial. Define the coefficients…

Let $m\in\mathbb Z_{>0}$, let $\mathcal C\subseteq\mathsf{RPD}_{m,n}$ be a $\sqrt{\mathfrak{gl}_n}$-crystal, and let $w\in S_n$. Write … A polynomial is Lascoux positive if it is a…

Let $P_n(x)$ be the Boros–Moll polynomials, whose coefficients form a polynomial sequence in $n$. A polynomial sequence is infinitely $q$-log-convex if every iterate under the oper…

Let $(W,S)$ be a Coxeter system, and let $P_{y,w}$ denote its Kazhdan–Lusztig polynomials for $y,w\in W$, with $\leq$ the Bruhat order. Property B. The polynomials $P_{y,w}$ are de…

Let $X_{w,2}(z)$ be the depth-two extremal quasimodular form of weight $w$, and define … A quasimodular form is completely positive when all coefficients in the relevant complete-p…

Fix $n\geq 1$. For partitions $\pi,\mu,\nu\vdash n$, define $d^\pi_{\mu,\nu}(\alpha)$ by … where $m_\nu$ denotes the monomial symmetric function in the alphabet $\boldsymbol{q}$. M…

Let $c^\rho_{\tau, u}(\alpha)$ be the coefficients defined by … Here $b:=\alpha-1$ is the shifted Jack parameter. Goulden–Jackson's conjecture. The coefficients satisfy…

Let $n$ be a positive integer, let $w\in S_n$ be a permutation, let $\zeta$ be a weak composition, and let $\boldsymbol{x}=(x_1,\ldots,x_n)$. Write…

Let $m^{(i)}[p,s]$ be a Kirillov–Reshetikhin monomial, and let $F_q(m^{(i)}[p,s])$ and $L_q(m^{(i)}[p,s])$ denote the associated KR-polynomial and canonical-basis element. KR-polyn…

Nonnegativity conjecture. For all $u,v,w$, $c_{uv}^w(y;z)$ is a polynomial in the differences $y_i-z_j$ with nonnegative integer coefficients.

Weak Campana–Peternell conjecture. If the tangent bundle of a Fano manifold is nef, then it is also big.

Polynomial-positive decomposition conjecture. For each permutation $w\in S_n$, there exist codominant permutations $w_1,\ldots,w_k\in S_n$ such that…

Let $\Delta$ be an ideal triangulation of $\frak{S}$ and let $\mathbb{I}^q_\Delta$ be the proposed quantum duality map. Quantum Laurent coefficient positivity conjecture. For every…

Let $X$ be a projective manifold and let $L$ be an ample line bundle on $X$. For a very general point $x\in X$, write … the maximal value of the Seshadri constant of $L$ on $X$. La…