72 problems

For $1\leq i<j\leq n$, let $\mathcal{S}_{ij}$ be the Schubert variety in ${\rm Gr}(2,n)$, let $\mathcal{S}_{ij}^2$ be its square image, let $P$ be the associated symmetric matrix,…

Unimodality and log-concavity conjecture for Mather polynomials. The polynomial $M_w$ has strictly positive coefficients and is unimodal. If in addition…

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…

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…

Let $I$ be a $k$-subset of ${1,\ldots,n}$, and let $\mathit{CS}_I$ denote the conormal Schubert variety, equivalently the unique component of $\mu^{-1}(\mathcal O_{k,n})$ satisfyin…

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…

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…

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…

Flat-locus dimension conjecture. A tuple $f\in R^{\iota}_{\bf d}$ is in the flat locus of $\pi$ if and only if

Let $\mathcal{S}_{ij}$ be a Schubert variety in ${\rm Gr}(2,n)$. GD-degree conjecture. If $1<j-i<n-1$, then the Schubert variety $\mathcal{S}_{ij}$ has GD degree $2$. The paper rep…

Let $1\leq i<j\leq n$, and let $\mathcal{S}_{ij}$ be the Schubert variety in ${\rm Gr}(2,n)$. ED-degree conjecture. If $i<j-1$, then … The formula was verified in the paper for…

Let $1\leq i<j\leq n$ with $j<n$, and let $\mathcal{S}_{ij}^2$ be the square image of the Schubert variety $\mathcal{S}_{ij}$. Degree recursion conjecture. The degrees satisfy … an…

Richmond–Slofstra's conjecture. The element $w$ has a Grassmannian BP decomposition.

Billey–Fan–Losonczy's conjecture. The interval $[e,w]$ admits a Lehmer code.

Let $D'$ be a Go-diagram in a partition $\lambda$, and let $c\prec c'$ be a crossing and uncrossing pair in $D'$ with $D'(c)=\circ$. Let $D$ be obtained by replacing the stones at…

Let $D'$ be a Go-diagram in a partition $\lambda$, and let $c\prec c'$ be a crossing and uncrossing pair in $D'$ with $D'(c)=\circ$. Let $D$ be obtained by replacing the stones at…

Let $w\in S_n$, let $P_{\mathrm{id},w}(q)$ be the Kazhdan–Lusztig polynomial, and let $\operatorname{ms}(w)$ denote the set of permutations indexing the irreducible components of t…

Indecomposable-root conjecture. The root $\check\eta_w(k)$ is indecomposable in $\check R^+(w)$. In particular, the set