7 problems

Let $F$ be the set of the following 13 partitions: … Define … … For $1\leq a\leq3$, let $\mathsf L_a$ be the set of partitions that do not begin with any $\boldsymbol c\in I_a$ and…

Let $\Phi^{\Delta}_{\Lambda,\Lambda}(\alpha,\beta,c_{\emptyset};z)$ be a ramified irregular vertex operator with expansion coefficients $c_{\lambda}^{(m)}$, and let…

Let $p,q$ be positive integers, let … and let $\chi_{p,q}$ be the singular vector of level $pq$ in $M_{\Delta_{p,q}}$. A ramified irregular vertex operator is called singular when…

Let $r$ be a positive integer and let $M^{[r]}_{\Lambda}$ be an irregular Verma module with $\Lambda_{2r-1}\neq 0$ and $\Lambda_{2r}=0$. A ramified irregular vertex operator has an…

Let $\widetilde\Phi(z):\mathcal F_{\vec u}\to\mathcal F_{\vec v}$ be the crystal-limit vertex operator, with $\vec u=(u_1,u_2)$ and $\vec v=(v_1,v_2)$, and let…

Let $\mathbf{u}=(u_1,\ldots,u_m)$ and $\mathbf{v}=(v_1,\ldots,v_m)$, and let $\Phi(w)=\Phi_{\mathbf{u}}^{\mathbf{v}}(w):\mathcal{F}_{\mathbf{u}}\to\mathcal{F}_{\mathbf{v}}$ be the…

Eigenvalue conjecture. The eigenvalues of $\big[\eta(z;p q^{-1}t)\big]_1$ on $\mathcal{F}$ are given by