86 problems

Let $h:[n]\to[n]$ be a Hessenberg function, let $\mathsf{x}\in\mathfrak{gl}_n(\mathbb{C})$, and let $\varphi_{\mathsf{x},h}$ be the one-parameter family constructed from the associ…

Panyushev's irreducibility conjecture. If the rank of the symmetric pair $(\mathfrak g,\mathfrak g_0)$ is greater than $1$, then $\mathfrak C(\mathfrak g_1)$ is irreducible.

Let $X$ and $Y$ be Dynkin diagrams of ADET type, with Cartan matrices $\mathcal{C}(X)$ and $\mathcal{C}(Y)$, ranks $r(X)$ and $r(Y)$, and Coxeter numbers $h(X)$ and $h(Y)$. Define…

Let $G$ be a real reductive group with Cartan decomposition $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$, and let $U(\mathfrak{g})$ and $C(\mathfrak{p})$ be the universal envelopi…

Let ${\mathfrak{g}}$ be a semisimple Lie algebra, let $Y({\mathfrak{g}})$ be its Yangian, and let $M_{\mathfrak{g}}$ be the De Concini–Procesi compactification of the regular part…

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, let $U_q(\mathfrak{g})$ be its quantised enveloping algebra, and let $W$ be the Weyl group of $\mathfrak{g}$. Consider the B…

Let $a_n$ be the number of triangulations of a fixed convex $n$-gon such that at least six triangles meet at each internal vertex. Let…

Let $\gog$ be a simple Lie algebra with Weyl group $W$, and let $\VV_{\la}$ and $\VV_{\mu}$ be irreducible representations with highest weights $\la$ and $\mu$. For a weight $\xi$,…

Let $\mathfrak g$ be a finite-dimensional Lie algebra, let $n$ be a positive integer, and let $x_1,\ldots,x_n\in\mathfrak g$. Define … For each $i$, let $\partial_{x_i}F_i$ be the…

Let $\alpha_{ij}=e_i-e_j$ be the roots of the type $A$ root system. An alcoved polytope is a polytope whose faces lie in hyperplanes $\langle\alpha_{ij},\tau\rangle=m$ for…

Casian–Kodama conjecture. The number of blow-ups is also given by the number of real roots of those Schur polynomials for a specific variable.

Degree–singularity–root conjecture. The degree $\eta(w_*)$ of $p(q)$ is the multiplicity $d$ at $p_o$, where $d$ is given by the minimal degree of the product of the $\tau$-functio…

Poisson vector bundle conjecture. (1) Every homogeneous Hermitian Poisson vector bundle over ${\mathfrak g}^*$ is Poisson-trivial. (2) There is a one-to-one correspondence between…

Irrational-angle counterexample conjecture. It is impossible to find a counterexample satisfying $\varphi_0\notin 2\pi\mathbb{Q}$.

Generalized Duistermaat–Heckman conjecture. An analogous correlation-function formula should hold for every compact group, with the triangular integration domain identified with…

For positive integers $n$ and $M$, define the symmetric function … Here, $\operatorname{Lie}_{n/d}[p_d]$ denotes the plethystic substitution of the power-sum symmetric function…

Let $r\geq 3$, let $\mathcal{C}(D_r)$ be the Cartan matrix of type $D_r$, and let $(q;q)_m=\prod_{j=1}^{m}(1-q^j)$. For $a\in\\{0,1\\}$, consider nonnegative integer vectors…

Let $V(\lambda)$ be a finite-dimensional representation of $\mathfrak{sl}_n$ equipped with an MV basis and a dual canonical basis, and write the change-of-basis matrix from the MV…

Let $V(\lambda)$ be a finite-dimensional representation of $\mathfrak{sl}_n$, with its Gelfand–Tsetlin basis, and consider bases obtained by lower unitriangular changes of basis fo…

Let $L_w$ be a simple highest weight module in the regular block of the BGG category $\mathcal{O}$ for $\mathfrak{sl}_n$. Kährström's conjecture. As $n$ tends to infinity, the answ…

Let $\mathfrak{g}$ be a reductive Lie algebra. Choose a well-chosen $\mathbb{Q}$-factorial terminalization … and let $X_{\mathfrak{g},\mathrm{Losev}}$ be the intermediate symplecti…

Let $G$ be a complex reductive group with Lie algebra $\mathfrak{g}$ and Weyl group $W$. Define the varieties $X_{\mathfrak{g},\mathrm{sgn}}$ and $X_{\mathfrak{g},\mathrm{diag}}$ b…

Let $G$ be a complex reductive group with Weyl group $W$, and let $X_{\mathfrak{g}}$ be one of the varieties attached to its reductive Lie algebra $\mathfrak{g}$. Consider the toru…

Theta–$\mathfrak{sl}_3$ identification conjecture. Up to conventions and normalizations,

Let $F$ be a local or finite field, let $\underline G$ be a split semisimple group with Lie algebra $\mathfrak g$, and let $\Omega\subset\mathfrak g$ be a conjugacy class with clos…