42 problems

Let $N\in\mathbb{N}$ and $y\in\Gamma_N$. Let $\eta\in X_{N,y}(\mathbb{Z})$ be an integral point. For $\widehat{w}\in\Sigma W^0$ and $g\in G$, define the action $g\cdot\widehat{w}$…

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…

Fix a regular block of $U_\chi(\mathfrak{g})$, with $\chi$ nilpotent, and let $\mathcal{S}$ be a complete set of nonisomorphic simple modules in this block. If $\chi$ corresponds t…

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

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…

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

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 $N\geq N_0$…

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 $n$:

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

Paper Boat zone conjecture. For every affine Weyl group there is a partition $\mathcal{Z}$ of $(\Lambda^\vee)^+$ with at most $3^n$ parts such that, for every $Z\in\mathcal{Z}$ and…

Sign conjecture. In Theorem, one has

Leading-matrix conjecture. There exists an ordering of the basis $d401_{d706}$ of $M_{d706}$ such that the leading matrices of the distinguished involutions satisfy

Let $t_\lambda$ be the affine Weyl group element associated with the dominant coweight $\lambda$, and let $f$ denote the finite Weyl group element used to define the dominant lower…

Let $(\lambda_1,w_1)$ and $(\lambda_2,w_2)$ be elements of the affine Weyl group written in semidirect-product form, where $\lambda_1$ and $\lambda_2$ lie in the root lattice and…

Let $f: \mathbb{C}^n \to \mathbb{C}^n$ be post-critically finite. An affine Weyl iterated-monodromy characterization conjecture asserts that if the iterated monodromy group of $f$…

Let $\mathfrak{g}$ be a simple Lie algebra, let $P_{\ell}$ be the set of level-$\ell$ dominant weights, let $W_{\ell}$ be the affine Weyl group, and let $V(\lambda)$ denote the irr…

Isomorphism conjecture. The map $\varphi$ is an isomorphism, equivalently

Let $W$ be the affine Weyl group, let $x\in W$, and let $[w,\mathbf{r}]\in\frak{P}$. Let $\calI(x)^+_{w,\mathbf{r}}$ be the corresponding space and let $\frak{c}_{w,\mathbf{r}}$ be…

Let $n$ be the rank parameter of the type $B$ MultiTASEP, and write $\langle a,b\rangle$ for the two-point correlation of entries in the last two positions, with barred symbols den…

Let $Y$ be a conical symplectic singularity, let $\lambda,\lambda'\in\mathfrak{h}_Y^*$, and let $\overline{\operatorname{HC}}(\mathcal{A}_{\lambda'},\mathcal{A}_{\lambda})$ denote…

Ngô's compatibility conjecture. The following diagram is commutative: