183 problems

Let ${\mathcal{F}}l(\mathbb C^n)$ be the complete flag variety, and let $\operatorname{reg}(\iota_*{\mathcal{O}}_{{\mathcal{F}}l(\mathbb C^n)})$ denote the multigraded regularity r…

Canonical-basis characterization conjecture.

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 $E$ be a Euclidean space of dimension $d\geq 3$, and let $\nu$ be a totally strongly irreducible probability distribution supported on a discrete subgroup of $\mathrm{SL}(E)$.…

Let $S_n$ be the symmetric group, let $[ij]$ be the generators of the Fomin–Kirillov algebra, and let $\theta_i$ be the Dunkl elements … Let $\mathfrak{S}_w$ be the Schubert polyno…

Buch's conjecture. The structure constants $c_{u,v}^w$ have alternating signs for all flag varieties.

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…

Wahl's conjecture. The Gaussian map is surjective when $\mathcal L_1$ and $\mathcal L_2$ are ample.

Let $\mathfrak g$ be a Kac–Moody algebra, let $\Lambda_1,\dots,\Lambda_n$ be dominant integral weights, let $z_1,\dots,z_n$ be points, and let $\boldsymbol b$ be symmetric paramete…

Fargues–Rapoport's conjecture. There exists a group-theoretic condition which expresses precisely when

Kazhdan–Lusztig irreducibility conjecture. In type A, the characteristic cycle of the IC-sheaf $\mathcal{L}_Q$ is irreducible.

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

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 $X=G/B$ 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 $F=F(n_1,\ldots,n_l,n)$ be a partial flag manifold, and let $X\subset F$ be a generic complete-intersection Calabi–Yau variety. Let $P$ be the toric degeneration of $F$, let…

Let $F=F(n_1,\ldots,n_l,n)$ be a partial flag manifold. Let $M_q$ be the affine varieties defined by the roof and box equations, with parameters specialized to $(q,1,1,\ldots,1)$,…

Uniqueness conjecture. The WDVV equation defines ${\cal F}(t)$ uniquely among potentials satisfying: (1) the normalization

Let $G=SL_{n+1}$ over $\mathbb C$. Degenerate the pair $(G/B,-K_{G/B})$ to the toric pair $(X_{\Delta},-K_{X_{\Delta}})$, let $\Delta^*$ be the dual polytope, and consider the subf…

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 $G$ 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…

Let $G$ be a simple Lie group with Borel subgroup $B$, let $G/B$ be its flag variety, let $P$ be the weight lattice, and let $R$ be the coefficient ring used in the quantum deforma…

The isomorphism conjecture. The algebra homomorphism $\eta$ is an isomorphism. The statement appears as a conjecture in the source, but the supplied material gives no resolution ev…

Let $Fl_n$ be the flag variety of type $A$, let $\widetilde{{\mathcal{E}}}_n$ be the algebra over $\mathbb{Z}[q_1,\ldots,q_{n-1}]$ obtained from ${\mathcal{E}}_n$ by replacing rela…

Equivariant mirror conjecture. A full set of solutions to the $T$-equivariant quantum differential equations of $G/P$ is given by these integrals. This is the $T$-equivariant exten…

Let $G/P$ be a partial flag variety, let $Z_P^{\mathfrak h^\vee}$ be the pullback of the mirror family to $(\mathfrak h^\vee)^{W_P}$, and let $\omega$ and $\mathcal F_P$ be respect…