162 problems

Let $(\mathbf {U},\mathbf{B},\gamma)$ be a commutative triple. Let $W_\Sigma$ be the restricted Weyl group acting on the set $\mathfrak{B}(\gamma)$ of relevant $M$-types, let…

Let $W$ be a Coxeter group. The Calogero–Moser cells are the equivalence classes in $W$ defined using the $\mathbb{C}^*$-fixed points of the Calogero–Moser space, while Lusztig's t…

Dilks–Petersen–Stembridge's conjecture. For any irreducible Weyl group, the affine Eulerian polynomial $\widetilde{W}(x)$ has only real zeros.

For even $n$, let $\omega_0$ be the cluster DT-transformation corresponding to the longest element of the birational Weyl group. A maximal green sequence is a mutation sequence wit…

Let $W$ be the Weyl group associated with a singularity of type $D_n$ or $E_n$. For some proper smooth surface $X/K$, consider an integral model of $X$ whose special fiber has a si…

Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$, fix a Cartan subalgebra $\mathfrak{h}$, and let $\Phi$ be the associated root system with Weyl group $W$. Choose posit…

Let $G$ be a reductive algebraic group with Weyl group $W$, let $p$ denote the characteristic of the ground field, and let Springer representations mean the representations of $W$…

Lehrer–Solomon's conjecture. The cohomology groups $H^m(\mathscr{M}_{\mathcal{W}},\mathbb{C})$ decompose into a sum of representations induced from one-dimensional representations…

Eisenstein conjecture. Each Weyl group multiple Dirichlet series appears as the first Fourier-Whittaker coefficient in an Eisenstein series on the metaplectic $n$-fold cover of the…

Reuman's conjecture. If $x$ lies in the shrunken Weyl chambers, then $X_x(b)\ne\emptyset$ if and only if

Type $A_n$ measure conjecture. The probability measure induced on $W$ by this random choice agrees with the probability measure given by $x_q$. This proposes that the construction…

Let $G$ be a connected, semisimple, simply connected group defined over a finite field of $q$ elements, and let $F$ be an $F$-split Frobenius automorphism of $G$. The Weyl group of…

Let $G$ be a reductive, connected, simply connected group of Lie type with Frobenius automorphism $F$, Weyl group $W$, rank $r$, and defining-field size $q$. For a semisimple orbit…

Gelfand–Tsetlin p-part conjecture. The $p$-part constructed from $f(\mathbf{x};\ell)$ should coincide with the $p$-part arising from the Gelfand–Tsetlin-pattern construction.

Let $X^{{\mathcal P},{\mathcal Q}}_{c,d}$ be a generalized Steinberg variety, and let $e_c\in\mathbb{Q}W_P$ and $e_d\in\mathbb{Q}W_Q$ be primitive idempotents associated with the c…

A generalized Steinberg variety $X^{{\mathcal P},{\mathcal Q}}_{c,d}$ is associated with nilpotent adjoint orbits in $\operatorname{Lie}(P/U_P)$ and $\operatorname{Lie}(Q/U_Q)$, de…

Let ${\cal BE}(G_2)$ be the bracket algebra of type $G_2$, and let $\Theta_1,\Theta_2$ be the corresponding RSM-elements; let $\phi_1$ and $\phi_2$ denote the two polynomial expres…

Let $YB(B_n)$ be the Yang–Baxter algebra generated by the elements $h_{ij}(x)$, $g_{ij}(x)$, and $h_j(y)$, and let $\Theta_j^{B_n}(x,y)$, $1\leq j\leq n$, be the associated commuti…

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 $X$ be either $A_{n-1}$ or $D_n$, and let $\theta_1^X,\ldots,\theta_n^X$ be the corresponding elements of the quadratic algebra $\Lambda_{quad}(X)$. Write $\epsilon_k$ for the…

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…