11 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…

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…

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

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…

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…

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…

Hilbert coefficient bounds conjecture. For $i=0,1,\ldots,n-s$,