229 problems

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 $\mathcal{GT}(\lambda,\sigma)$ be the generalized Gelfand–Tsetlin polytope associated with a partition $\lambda$ and a permutation $\sigma$, and let…

Let $b,c$ be positive integers, and let $X[a_1,a_2]$ denote the quantum greedy element of the rank $2$ quantum cluster algebra $\mathcal{A}_v(b,c)$ associated with…

Positivity conjecture. Then $K_Y+B_Y$ is $\mathbb{Q}$-Cartier and the moduli part $M_Y$ is $\mathbb{Q}$-free: for some $\nu\in\mathbb{N}$, $\mathcal O_Y(\nu M_Y)$ is generated by g…

For two shifted shapes $\lambda$ and $\mu$, let $\lambda\cup\mu$ be the partition obtained by rearranging all parts of $\lambda$ and $\mu$ in weakly decreasing order, and write … D…

Let $\lambda/\mu$ and $\nu/\rho$ be skew shifted shapes. For partitions, define componentwise operations … and extend them to skew shifted shapes by…

Let $P_n$ be the algebra containing the element $f$, let $\mathrm{Pfaf}'_D$ denote the diagram pfaffinants, and let $A_{\lambda/\mu}$ be a generalized $Q$-Jacobi-Trudi matrix assoc…

Positivity conjecture. For all ${\underline\alpha}$, $c_{\text{SM}}({\mathbb{S}}({\underline\alpha})^\circ)\in A_*({\mathbb{S}}({\underline\alpha}))$ is represented by an effective…

Nonnegative product conjecture.

Positivity conjecture. For any object $M$ and any exceptional object $K$ of ${\mathcal C}$, the integer $r(M,K)$ is nonnegative.

Let $\mathcal{A}$ be a cluster algebra of rank $n$ with a seed whose cluster and frozen variables are $x_1,\dots,x_m$. By the Laurent phenomenon, every cluster variable is a Lauren…

Let $A$ be a projective manifold, and let ${\mathcal L} \subset \bigwedge^r \Omega^j_A$ be a line bundle. A curve $C$ on $A$ is movable if its deformations fill up $A$. The positiv…

Let $(W,S)$ be a crystallographic Coxeter system, let $qBE(W,S)$ be the associated quantum algebra, and let $\tilde{P}_w\in {\bf S}(V^{\ast})\otimes {\bf R}[q_s]$ be the polynomial…

Let $(W,S)$ be a crystallographic Coxeter system. Let $BE(W,S)$ be the associated algebra, let $X_w$ be the Bernstein–Gelfand–Gelfand polynomial indexed by $w\in W$, and let…

Let $P^{(p,p')}$ be the parameter set used for the affine weights, let $P_{+}^{r}$ denote the dominant weights at level $r$, and let $X_{\eta,\Lambda,\Lambda'}(q)$ be the configura…

Let $f:Y\to X$ be a surjective morphism from a smooth projective variety to a smooth projective variety of dimension $n$, and suppose that $f$ is smooth over $X_0=X\setminus B$ for…

For $\mathbf{b}=(b_1,\dots,b_n)\in\mathbb{Z}_{>0}^n$, let $\mathfrak{X}_n(\mathbf{b})$ denote the $\mathbf{b}$-parking-function polytope, the convex hull of all $\mathbf{b}$-parkin…

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…

Positivity conjectures. The Laurent polynomials $Q_{x,w}\in\mathbb{Z}[v,v^{-1}]$ have non-negative coefficients for all $x\in W_{\mathrm{aff}}$ and $w\in W_{\mathrm{aff}}^+$, and t…

Let $\operatorname{Par}_n$ denote the set of partitions with at most $n$ parts. A $(0,1)$-substitution map is an operator $\varphi$ on…

Let $G$ be a connected complex reductive group with Weyl group $W$, let $\psi\in\operatorname{Irr}(W)$ be inflated from a quotient … and let $\alpha_{\psi,G}^{z}$ be the Laurent po…

Consider the $\imath$quantum group of type ${\rm AIII}_{2r}$ and its dual canonical basis. Type-${\rm AIII}_{2r}$ positivity conjecture. The positivity results for the dual canonic…

Let $\widetilde{{\mathbf U}}^\imath$ be a right coideal subalgebra of $\widetilde{{\mathbf U}}$, with embedding $\iota$, and let $\mathcal{B}^\imath$ and $\mathcal{B}$ be the respe…

Homogeneous positivity conjecture. The polynomial $\rho(\{y\},\beta)$ has non-negative coefficients as a homogeneous polynomial in $y_i$ and $\beta$, and it has degree $7$ in…

Let $\phi$ be the displayed map from the five auxiliary variables into the specified linear space, and write $\{m,n,r\}=(m_1,m_2,m_3,n_1,n_2,n_4,r_1,r_2)$. Stanley positivity conje…