197 problems

Let $\mathrm{GL}(n)/P$ be a partial flag variety, and let its cohomology ring have structure constants defined with respect to the relevant Schubert basis. A puzzle is a tiling of…

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…

Kac–Moody extension conjecture. The analogues of the equations giving the Chevalley formulae for $\lambda$ and $-\lambda$ hold for $W$.

Let $G$ be a complex reductive group, let $B$ be a Borel subgroup, and let $W$ be the Weyl group. For $u,w\in W$ with $u\leq w$, write the equivariant Chern–Schwartz–MacPherson cla…

Let $T$ be a torus, let ${\mathcal F}$ be its flag manifold, and let $W$ be the Weyl group. For the opposite Schubert varieties $X^u$ and $X^v$, write their equivariant structure-s…

Let $\operatorname{Gr}(k,n)$ be a Grassmannian, let $(\lambda^1,\dotsc,\lambda^r)$ be a Schubert problem, and let $F^1_{\bullet},\dotsc,F^r_{\bullet}$ be flags secant to a rational…

Let $\lambda$, $\mu$, and $\nu$ be partitions whose Young diagrams fit in a $p$ by $n-p$ rectangle, and suppose the corresponding Littlewood–Richardson number is positive. For thre…

Let $\mathsf{M}$ be a matroid of rank $k$ on a ground set of size $n$, and let $\eta$ be a partition whose Young diagram is contained in the $k\times(n-k)$ rectangle. The Chow clas…

Let $k$ be a positive integer, let $q$ denote the parameter used in the $k$-Schur functions, and let $H_*(\operatorname{Gr}_{\mathrm{aff}})$ be the homology ring of the affine Gras…

Let $\omega$ be the permutation governing the switching algorithm, and let $\mathcal{O}\in\operatorname{Orb}(\omega)$ be one of its orbits. A genomic tableau is a tableau counted b…

Cosmall-root characterization conjecture. Assume that $R$ is simply laced and let $\alpha\in R^+\setminus R^{+}_P$. Then $\alpha$ is $P$-cosmall if and only if

KL-Schubert positivity conjecture. The evaluation $({\mathfrak S}_v)_w$, for any $w\leq v$, can be expressed as a possibly infinite sum of monomials in the $y_{-\alpha}$, where…

Let $D$ be a finite diagram contained in $[k] \times [n-k]$. Let $X_D$ be the associated diagram variety in $\operatorname{Gr}(k,n)$, with cohomology class…

Let $\ell_1,\dotsc,\ell_4$ be lines tangent to a twisted cubic at real points. Shapiro conjecture. The two solutions to the corresponding problem of four lines are real. This is th…

Let $S_n$ be the symmetric group, let $w_0$ be its longest element, and let $\widetilde{\operatorname{S}}_u$ denote the quantum Schubert polynomial indexed by $u\in S_n$. Consider…

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 $A_n$ be the vector space spanned by the classical Schubert polynomials, and let $\mathfrak{S}^q_w$ be the quantum Schubert polynomial representing the Schubert class $sigma_w$…

Let $r=(r_{ij})$ be rank conditions and let $c_\mu(r)$ be the corresponding coefficient. A factor sequence for a tableau diagram with $n$ rows is a sequence of tableaux…

Let $X=G/B$ be the flag variety of a complex semisimple group, with $B\supset T$ a Borel subgroup and maximal torus. Let $H_T^*(X)$ be the $T$-equivariant cohomology, viewed as a f…

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…

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 $Fl(n)$ be the flag variety, let $d_i$ and $d_{i+1}$ be consecutive elements in the specified degeneration order, and let an equivariant puzzle variety be the subvariety of…

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 $X=Gr_k(\mathbb{C}^n)$, and let $\sigma_{\lambda}$ and $\sigma_{\mu}$ be Schubert classes indexed by partitions contained in the $k\times(n-k)$ rectangle. Write … where…

Let the Lagrangian Grassmannian be the homogeneous space considered in the paper, and let the naive inequalities be the necessary inequalities obtained from Proposition. A collecti…