126 problems

Let $V$ be a subspace with coordinate projections $\pi_{E \setminus i}$, and let $Y_V \subset \prod_{i=1}^N \mathbb{P}^{n_i}$ be the associated Schubert variety. Suppose that the s…

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

Billey–Crites conjecture. The variety $X_w$ is smooth if and only if $w$ avoids the patterns $3412$ and $4231$.

Let $G$ be a semisimple algebraic group with Borel subgroup $B$, Weyl group $W$, and dominant weights $\lambda,\mu\in P^+$. For $v,w\in W$, let $V_v(\lambda)$ and $V_w(\mu)$ be Dem…

For permutations $a,b,v,w$, let $P_{a,b}(q)$ and $P_{v,w}(q)$ denote Kazhdan–Lusztig polynomials, and let $[a,b]$ and $[v,w]$ be the corresponding Bruhat order intervals. Lusztig's…

Let $G$ be the group, let $X\in G$ be regular, let $w\in W$, and let $\mathcal{Y}_w(X)$ be the corresponding variety. Let $\Omega_w$ be the Schubert variety and let…

Let $G$ be the group under consideration and let $\lambda$ be a dominant coweight. Let $\overline{\mathcal{Y}^\lambda}\subset\operatorname{Gr}_G$ be the moduli scheme of principal…

New-variable sign conjecture. The coefficients of the $s_i,\delta$-monomials in the $\mathbf{[w]}$-expansion of the $\mC[p]$ classes have sign $(-1)^{\ell(w)}$.

Let $\mathcal{B}(X,h)$ be a regular nilpotent Hessenberg variety in Lie type $A$, and let the associated Ding variety be the Schubert variety whose Betti numbers agree with those o…

Catanese's conjecture. The variety $G/P$ should possess a full strongly exceptional poset together with a bijection between the elements of the poset and the Schubert varieties of…

Kazhdan–Lusztig singular-support conjecture. The singular support of $\sideset{^{\pi}}{_{X_w}}{\Bbb C}$ is irreducible.

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…

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 $\pi$ be a permutation, and let $L_\pi$ be the linear map associated with $\pi$ whose surjectivity is equivalent to the locally factorial property of the corresponding Schubert…

Let $X_v$ be a Schubert variety and let $e_u$ be a torus-fixed point. Write $\mathcal{I}_{\mathrm{not\ factorial}}$ for the order ideal of intervals describing pairs $(u,v)$ for wh…

Let $X_v$ be a Schubert variety and let $e_u$ be a torus-fixed point. Write $\mathcal{I}_{\mathrm{not\ Gorenstein}}$ for the order ideal of intervals describing pairs $(u,v)$ for w…

Let $X$ be the affine flag manifold, let $X_w$ be the Schubert variety indexed by $w\in W$, and let $\mu$ be a weight. Write $\mathscr{O}_X(\mu)$ and $\mathscr{O}_{X_w}(\mu)$ for t…

Schubert-cell crystal conjecture. For every $w\in W$,

Schubert-cell perfect-basis conjecture. For every $w\in W$, the coordinate algebra $\mathbb{C}[X_w^\vee]$ has a perfect basis.

Let $G(l,m)$ be a Grassmann variety and let its associated Grassmann code have higher weights. A Schubert union is a union of Schubert varieties in the Grassmann variety. Schubert-…

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…

Let $X_w$ be a Schubert variety, and let $X_v$ range over the Schubert varieties contained in its maximal singular locus. The maximal non-Gorenstein locus characterization proposes…

Fusion-product conjecture. In general, the structure of the cohomology groups as $\slt\otimes\mathbb C[t]$-modules can be described in terms of fusion products.

First-$n$-coefficients conjecture. If $(\Phi_+,w)$ is not a rationally smooth pair, then comparing the first $n$ coefficients and the last $n$ coefficients of $P_w(t)$ suffices to…

Let $w,\tau\in W/W_{P_d}$. Let $R^+$ and $R_{P_d}^+$ denote the positive roots of the ambient root system and of the parabolic subsystem, respectively. Define $S'_{w,\tau}$ to be t…