116 problems

Functional equation conjecture. The exponential generating function satisfies

Positivity conjecture for motivic Chern classes. For every $u\leq w\in W$,

Alternation conjecture. The Euler characteristic is alternating:

Strong positivity conjecture. One has $c_{v,w}>0$ for every $v\le w$. The weaker non-negativity assertion is known, but strict positivity remains the stronger conjectural statement…

Equivariant positivity conjecture. The coefficient $c_{v;w}(\alpha)$ is a polynomial in the positive roots $\alpha$ with non-negative coefficients. Numerical evidence supports this…

Let $R_n$ be the ring defined in the surrounding construction, with binary operation $\star$. Associativity conjecture. $(R_n,\star)$ is an associative ring. The supplied context d…

Let $\widetilde{\cal G}_n^0$ be the quotient of $\widetilde{\cal G}_n$ by the relations $[ij]^2=0$. Let $\widetilde\theta_1,\ldots,\widetilde\theta_n$ be the Dunkl elements, and le…

Let $S_n$ be the symmetric group, let $\mathcal{E}_n^+$ be the positive cone generated by noncommutative monomials in the Bruhat generators $[ij]$, and let…

Let $m,p\geq 1$, let $\gamma$ be the real rational normal curve, and let $w_1,\ldots,w_n$ be Schubert conditions satisfying … For distinct real points $s_1,\ldots,s_n\in\gamma$, co…

Let $m,p\geq 1$, let $\gamma$ be the real rational normal curve, and let $w_1,\ldots,w_n$ be Schubert conditions on $p$-planes in $(m+p)$-space. For points…

Vakil's geometric Littlewood–Richardson conjecture. There exists a subset $U\subseteq S\times S\times\{d_0,\dots,d_{\binom n2}\}$ such that: $(\alpha,\beta,d_0)\in U$ for all…

Let $\alpha$, $\beta$, and $\gamma$ be strings indexing Schubert classes in the two-step flag variety, and let $c^{\gamma}_{\alpha,\beta}$ denote the corresponding two-step Littlew…

Let $X_1$ and $X_2$ be the two components arising when a puzzle variety breaks in the geometric Littlewood–Richardson degeneration, and let $X_3$ be the puzzle variety correspondin…

Let $0\le d<n$, let $t_1<t_2<\dots<t_{(n+1)(d-n)}$ be real numbers, and let $\mathbf r:\mathbb C\to\mathbb C^{d+1}$ be a rational normal curve with coordinates $r_i(t)=t^i$ for…

For integers $d$ and $0<k<m<d$, let $P_{k;d}(x)$ and $P_{m;d}(x)$ denote the truncated binomial polynomials used in the paper. Common-root conjecture. The polynomials $P_{k;d}(x)$…

The Schubert calculus conjecture. For generic $z=(z_1,\ldots,z_n)$, the intersection of Schubert varieties $\mathcal{I}_{\{{\bf w}\}}(z)$ is transversal. Here generic means that…

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 $(W,S)$ be a finite Coxeter system, let $BE(W,S)$ be the associated algebra, and let ${\bf R}[\theta_s\mid s\in S]$ denote the subalgebra generated by its Dunkl elements. The c…

Knutson's conjecture. The integral

Let $d>r$ be integers and let $calpha^1,\ldots,\alpha^n$ be ramification data for degree $d$ rational curves in $\mathbb{P}^r$. Let $s_1,\ldots,s_n$ be distinct points of…

Let $d\geq 1$ be an integer, and let $\lambda$, $\mu$, and $\nu$ be Young diagrams indexing Schubert classes in the Grassmannian. Denote by $\langle\lambda,\mu,\nu\rangle_d$ the co…

Let $X$ be the Grassmannian, and let $\sigma_{\lambda}$ and $\sigma_{\mu}$ be Schubert classes indexed by Young diagrams. Write $d_{\min}$ and $d_{\max}$ for the smallest and large…

Let ${\bf a}=1<a_1<a_2<\cdots<a_r<a_{r+1}=n$, and let $(\alpha^1,i_1),\ldots,(\alpha^s,i_s)$ be Grassmannian Schubert data for the partial flag manifold $\mathbb{F}\ell_{\bf a}$, w…

Let $V$ be a vector space with a bilinear form, and let $G:=\operatorname{Aut}(V,\langle\cdot,\cdot\rangle)$ be respectively a general linear, orthogonal, or symplectic group accor…

Let $s_1,\ldots,s_n$ be distinct real numbers, and let $\gamma(s)$ be a parameterization of a rational normal curve in $\mathbf{R}^{m+p}$ of degree $m+p-1$. For integers $k_i$, def…