83 problems

Homogeneous-limit conjecture. For $\alpha\in Q_+$, $f_\alpha$ remains finite at $s_1=\cdots=s_n=1$ if and only if $\alpha\in C(\Delta)$. Otherwise it diverges in this homogeneous l…

Weyl-antisymmetry conjecture. This polynomial satisfies

Let $B_n$ be the Weyl group of type $B_n$. Let $\Lambda_w(B_n)$ be its left-invariant differential algebra, and let $\Lambda_{quar}(B_n)$ be the quotient of the quadratic algebra…

Let $W$ be a Weyl group, let ${\cal C}$ be its set of reflections, and let $\Lambda_w$ be the left-invariant differential algebra associated with this differential structure. Let…

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

Shapovalov-preserving Weyl group representation conjecture. There exists a representation of the Weyl group $W$ in $U$ such that each generator $s_i$ is mapped to $\tilde s_i$. The…

Orbit-product decomposition conjecture. Under these hypotheses,

Multiplicity-one conjecture. Under these hypotheses, the multiplicity of $O(w\lambda+\mu)$ in $O(\lambda)\otimes O(\mu)$ is $1$.

Let $G$ be a reductive group with Langlands dual group $G^\vee$, and let $W$ be the Weyl group. Let $X$ be a smooth spherical $G$-variety such that $T^*X$ is hyperspherical and has…

Let $G$ be a connected complex reductive group with Weyl group $W$, let $\psi\in\operatorname{Irr}(W)$ be inflated from a quotient … and let $\alpha_{\psi,G}^{z}$ be the Laurent po…

Let $\mathcal X_{|\mathcal A_n|}$ be the cluster variety associated with the $\mathcal A_n$-quiver, let $O(\sqrt{\mathcal X_{|\mathcal A_n|}})$ be the ring of universal Laurent pol…

The CII maximal-involution conjecture.

The maximal fixed-point-free involution conjecture.

Let $W=W_n^+$ be the type D Weyl group, let $w_0$ be its longest element, and let … For a strict partition $\lambda=(\lambda_1>\dots>\lambda_m>0)$ and a part $a$, write…

Let $\mathfrak{g}$ be a reductive Lie algebra with Weyl group $W$, let $\mathfrak{C}_W$ be the set of two-sided cells in $W$, and let $\mathbf{a}:\mathfrak{C}_W\to\mathbb{N}$ be Lu…

Let $G$ be a complex reductive group with Cartan subalgebra $\mathfrak{t}$, Weyl group $W$, and $A=\mathbb{C}[\mathfrak{t}\oplus\mathfrak{t}^*]^{\mathrm{sgn}}$. Set…

Let $(\mathbf {U},\mathbf{B},\gamma)$ be a commutative triple, and suppose that the action of $W_\Sigma$ on $\mathfrak{B}(\gamma)$ is transitive and that one stabiliser $W_J$ is a…

Flip Combinatorial Invariance conjecture. Flip Combinatorial Invariance holds for flipclasses of Weyl type.

Minuscule or cominuscule periodicity conjecture. Let $t\in I$. Then $\PD(x_t)$ is finite if and only if $t\in I_{\mathrm{MC}}$. Moreover, if $t\in I_{\mathrm{MC}}$, then the follow…

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

Rank log-concavity conjecture. The interval $[u,v]$ is rank log-concave.

Let $\mathfrak{g}$ be a simple Lie algebra with Weyl group $W$. Let $\mathcal{O}$ be the unique minimal solvable nilpotent orbit of $\mathfrak{g}$. For each Lusztig family $F$, let…

Let $\mathfrak{g}$ be a simple Lie algebra with Weyl group $W$, reflection representation $V$, and Spaltenstein map $d$ on nilpotent orbits. Let $\Psi$ be the map from genuine irre…

Let $\mathfrak{g}$ be a simple Lie algebra with Weyl group $W$, Lie group $G$, nilpotent cone $\mathcal{N}$, and set of nilpotent orbits $G\backslash\mathcal{N}$. Let…

Let $\mathfrak{h}^{\mathrm{reg}}$ be the regular locus for the Cartan subalgebra and let the boundary Casimir connection descend to the quotient $\mathfrak{h}^{\mathrm{reg}}/W$, wh…