87 problems

Alternating-sign conjecture. For every $u,v,w\in W$,

Canonical-basis characterization conjecture.

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…

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 $\mathfrak g$ 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…

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 $\mathrm{SL}(E)$.…

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

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…

Sign conjecture. The coefficient of $[w]$ in the expansion of $\mathcal{C}[p]$ is a Laurent polynomial in $\tau_1,\ldots,\tau_n,y$ whose terms have sign $(-1)^{\ell(p)-\ell(w)}$.

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 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…

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…

Let $S_n$ be the symmetric group, let ${\cal G}_w(x_1,\ldots,x_n)$ be the Grothendieck polynomial associated with $w\in S_n$, and let $G_i^{A_{n-1}}$ be the elements defined by…

A permutation array is an element $P$ of $[n]^d$ satisfying the permutation-array conditions, and $X^o_P$ denotes the corresponding permutation array variety, consisting of tuples…

Path-reversal conjecture. In these Littelmann path models, $\widetilde{\Theta}$ is given by reversing Lakshmibai–Seshadri paths and translating the initial points of the reversed p…

Let $X$ be any flag variety, and let $\mathcal M_{0,n}(X,\beta)$ be the open part of the moduli space of stable maps, parametrizing maps without boundary degenerations. Let the ind…

Let $G$ be a semisimple algebraic group, let $P$ be a parabolic subgroup, and let $X=G/P$ be a general flag variety. For nonnegative integers $n$ and curve class $\beta$, write…

Let $(W,S)$ be a crystallographic finite Coxeter system, let $qBE(W,S)$ be the associated quantum algebra, and let ${\bf R}[q_s][\tilde{\theta}_s\mid s\in S]$ denote the subalgebra…

Quantum K-theory Chevalley conjecture. For every $w\in W$,

Knutson's conjecture. The integral

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

Let $X = \operatorname{\mathbf{Flag}}(p_1|q_1, \dots, p_t|q_t;\;V)$ be a flag supervariety of type $\tau$, and let $Y$ be the variety associated to $X$ by the construction in the p…