77 problems

Geometric polynomial conjecture. For any affine Weyl group, the map $q_a$ is a geometric polynomial.

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

Let $W$ be a finite Weyl group of type $A$, and let $\widetilde{W}$ be its corresponding affine Weyl group. Consider the reduced random walk on the alcoves of $\widetilde{W}$ obtai…

Let ${\mathcal O}$ be a nilpotent orbit, let $a({\mathcal O})=\frac12\operatorname{codim}_{\mathcal N}{\mathcal O}$, and let $j:{\mathcal O}\to {\mathcal N}$ be the inclusion of th…

For each generalized Cartan matrix of affine type $A=(a_{ij})_{i,j\in I}$, let $\tau_i$ be the tau-functions in the Weyl-group representation described above. Strong regularity con…

Vertex-wise admissibility conjecture. Admissibility has a vertex-by-vertex characterization; equivalently,

Let ${\underline c}_f\subset W_f(\zeta)$ be a two-sided cell whose corresponding family is not exceptional. Let ${\mathcal G}_{{\underline c}_f}$ be Lusztig's associated finite gro…

Let $G$ be one of the groups $SL_2$, $SL_3$, or $Sp_4$. Let $b=\epsilon^\nu$, and let $\ell(b)$ denote the length of the translation in $\widetilde W$ determined by $\nu$. Eventual…

Canonical left $\tilde{S}$-region conjecture. The ideal $J(\lambda)$ is completely prime.

Lusztig's matrix-algebra conjecture. There is a bijection

Weight-space and dimension conjecture. The weight spaces of simple $A$-modules are one dimensional. Consequently, the dimension of a simple module is the multinomial coefficient

Let $V_C=\left(\mathbb{Z}/(2n+1)\mathbb{Z}\right)^n$ and let $\mathcal{C}_n$ be the subset defined in the paper from the affine type C residue conditions. Affine type C difference-…

For an integer $n\geq1$, define … where the latter two conditions hold for $1\leq i<j\leq n$. The Euclidean form is $\sum_{i=1}^n x_i^2$. Affine type C universality conjecture. For…

Refined universality conjecture. Assume $n\geq6$. Then $Q_{\boldsymbol{s}}(t_1,\ldots,t_n)$ is universal on $\mathcal{O}_{\boldsymbol{s}}$. Computational evidence supports the clai…

Let $A_{n-1}^{(1)}$ be the affine Dynkin diagram of type $A$, let $\Lambda_0,\ldots,\Lambda_{n-1}$ be its fundamental weights, and let $\sL_{\Lambda_0+\cdots+\Lambda_{\ell-1}}$ den…

Let ${\rm Adm}^{\mathcal G}(\{\mu\})$ be the admissible set and let $|\Delta|$ be the divisor map from this poset to the face poset ${\mathscr F}(\sigma_{{\mathcal G},\mu})$. For…

Birational affine Weyl group action conjecture. The quotient set of such matrices admits a birational action of the affine Weyl group $W(E_8^{(1)})$, with

Let $i\geq 2$ and let $\mathbf{N}_{\lambda}^{i+1}$ be an element of the $(i+1)$-st pre-canonical basis, expanded in the $i$-th pre-canonical basis $\mathcal{N}^i$. Libedinsky's pos…

Piecewise-translation conjecture. There exists $g\in G$ such that

Weak combinatorial invariance conjecture. If $x,x',y,y'\in W_+$ satisfy that $[x,y]$ and $[x',y']$ are full and poset-isomorphic, and there exists $g_0\in G$ with…

Stabilization conjecture. For any dominant coweight $\lambda\in\Lambda^\vee$ and dominant $x,y\in W$, there exists an integer $N_0=N_0(x,y,\lambda)$ such that for every $N\geq N_0$…

Piecewise-isometric invariance conjecture. Whenever two Bruhat intervals are isomorphic, there exists an isomorphism realized by a piecewise isometry.

Let $d_1,\dots,d_\ell$ be the eigenvalues of the operator $\nabla^\eta E$, equivalently the eigenvalues of its representation matrix $A$ in the basis…

Type-$A_n$ polynomiality conjecture. The cardinality of the lower Bruhat interval is a polynomial in the coordinates $\lambda_1,\dots,\lambda_n$ of degree $n$:

Deep Paper Boat size conjecture. For any type, the size of a Paper Boat associated with a coweight sufficiently deep inside the dominant chamber equals