21 problems

Let $V$ be a rational vertex operator algebra, and let $W_{1},\dots,W_{r}$ be a complete set of representatives of equivalence classes of irreducible $V$-modules. Here $U(V)$ is th…

Let $({\mathcal W},{\mathcal S})$ be a Coxeter system with length function $l$, and let ${\mathscr B}(x)^{[y]}\hookrightarrow {\mathscr B}(x)^y$ be the inclusion of the costalk int…

Let $({\mathcal W},{\mathcal S})$ be a Coxeter system with length function $l$, and let ${\mathscr B}(x)^y$ be the stalk of the sheaf ${\mathscr B}(x)$ at $y\in{\mathcal W}$. For a…

Let $R=k[x_1,\dots,x_n]$ be a standard graded polynomial ring over a field, and let $N$ be a finitely generated graded torsion $R$-module. Set $c=n-\dim N$ and, for a minimal grade…

For each positive integer $n$, let $I(n)$ and $J(n)$ be the ideals in the polynomial ring ${\mathbb C}[{\mathbf x}_{\binom{[n]}{2}}]$ on variables indexed by the two-element subset…

Closure conjecture. The class of RSK-standard modules remains closed under the sign involution.

For each index $t$, let $d_{k,t}$ be the number of weight-$k$ generators of the free module $J_{*,E_8,t}^{\operatorname{w},W(E_8)}$ of $W(E_8)$-invariant weak Jacobi forms. Stabili…

Let $S=k[x_1,\ldots,x_n]$ be a polynomial ring over a field $k$, and let $M$ be a finitely generated graded $S$-module of finite length. Write … for the total Betti number. Total R…

Diagonal irreducibility conjecture. Any irreducible $\mathbb{Z}$-graded $\widetilde{\mathcal{H}}$-module is diagonal.

Let $\mathbb{K}$ be a field, let $S=\mathbb{K}[x_1,\dots,x_n]$, and let $M$ be a finitely generated $\mathbb{Z}^n$-graded $S$-module. A Stanley decomposition of $M$ is a finite dir…

Stanley's conjecture. Every finitely generated $\mathbb{Z}^n$-graded $S$-module satisfies the Stanley inequality.

Stanley's conjecture. One always has

Generalized Eulerian duality conjecture. The shifted module $^\sharp M^\dagger(+n)$ is a left generalized Eulerian $A_n(K)$-module.

Puthenpurakal–Singh conjecture. For every $\nu\geq 0$, $\operatorname{Ext}^{\nu}_{A_n(K)}(M,N)$ is concentrated in degree zero; equivalently,

Let $\bD$ be the divided power algebra over $\bk$, let $R$ be the relevant grading ring, and let $\cC_n$ be the category of finitely generated $\bk$-modules supported on…

Stanley's conjecture. Every finitely generated $\mathbb{Z}^n$-graded $S$-module satisfies Stanley's inequality. This conjecture was recently disproved, so the asserted inequality d…

Let $R=\mathds{K}[X_1,\ldots,X_n]$ be a polynomial ring in an arbitrary positive number of variables, let $\mathfrak m=(X_1,\ldots,X_n)$ be its homogeneous maximal ideal, and let…

Extension conjecture. Theorem also holds for arbitrary weight vector $\mathbf{u}\in\mathbb{Z}^n$.

Let $S$ be a polynomial ring, and consider the rational cone of Betti tables of Cohen–Macaulay modules. A ray of this cone is a pure-resolution ray when it corresponds to the Betti…

Let $L$ be the Virasoro-like algebra considered in the paper, and let $V$ be a uniformly bounded ${\bf Z}$-graded irreducible $L$-module, meaning that the dimensions of its homogen…

Let $S=K[x_1,\ldots,x_n]$ and let $M$ be a finitely generated ${\mathbb Z}^n$-graded $S$-module. A Stanley decomposition of $M$ is a finite direct sum … Here $m_iK[Z_i]$ denotes a…