32 problems

ABF Boltzmann-weight conjecture. The connection matrices for four point functions having arbitrary numbers of screening operators should be expressible in terms of the Boltzmann we…

Let $z_1^{d_1}v_{m_1}\otimes z_2^{d_2}v_{m_2}\otimes\cdots$ be a semi-infinite tensor, and define its $q$-antisymmetrization by … Here $T_\sigma$ acts through the infinite Hecke al…

The integral vertex-operator conjecture. For $\alpha\in\mathbb Z$, the following vertex operators exist:

Let $U_q[\widehat{gl(2|2)}]$ be the quantum affine superalgebra. For arbitrary $\beta$, define … and, for $\alpha\in\mathbb Z$, … Let $\overline V(\lambda^0_\beta)$,…

The Fock-space identification. There is an identification of highest weight $U_q[\widehat{gl(2|2)}]$-modules

Naive-function-field conjecture. For arbitrary $Z$, one has

Let $\Lambda_{a\pm2s,a}(u)$ be the tail operator, let $g_{P+h}$ be the gauge factor, and let $\widehat{F}^{+}$ and $F(v_j)$ denote the half-current and its integral-kernel realizat…

Let $\widehat{\Phi}_{\varepsilon}(u)$ be the gauge-transformed vertex operators, with $\varepsilon\in\{-k,-k+2,\ldots,k\}$, and let $W^{(k,k)}$ denote the fused face weights. Commu…

Elementary scalar particle conjecture. The operator $t(\zeta)$ satisfies

Let $ca_R$ be the algebra equipped with the hierarchy generated by the commuting Hamiltonians $H^{(n)}$, where … An integrable system is said to correspond to this hierarchy when i…

Generalized dominant-weight spanning conjecture. For all $\gamma^{(i)}\in\mathbb{Z}^{\eta_i}$, the product

Convergence conjecture. Such a composition converges to a map

Elliptic vertex-operator commutation conjecture. These vertex operators satisfy the commutation relation given by the source equations with the elliptic $R$-matrix, the choices…

The elliptic-algebra module and vertex-operator conjecture. The elliptic algebra ${\cal A}_{q,p}(\widehat{sl_2})$ has natural analogs of the level-one modules ${\cal H}^{(i)}$,…

The XXZ model is a spin-$1/2$ quantum spin chain with Hamiltonian $H$ and exact symmetry … where $U'_q(\widehat{sl}_2)$ is the subalgebra of $U_q(\widehat{sl}_2)$ without the gradi…

Let the vertex operator algebras be the algebras introduced in the paper for the eight-vertex model and its associated representations. Let the bosonization procedure mean the cons…

Let $\Lambda(u_0)^{m-2k}_m{\,}^{n-2l}_n$ denote the operator appearing in the free-field representation, and let $X(u)$ and $\tilde Y(u)$ be the vertex operators used in that repre…

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 $c7_1,c7_2$ be simple $qq$-characters, and define their combinatorial fusion $c7_1*c7_2$ by retaining the products of monomials with maximal relative pairing. Suppose this fusi…

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 $r$ be a positive integer, and let $\Phi^{[r],\lambda}_{\Delta,\Lambda}(z):M^{[0]}_{\Delta}\to M^{[r]}_{\Lambda}$ be an irregular vertex operator of rank $r$, with its defining…

Let $\eta^{(i)}(z)$ be the relevant vertex operator and define its modes by … Let $P_\lambda(a_{-n}^{(i)};q,t)|u\rangle$ be a Macdonald-function vector, and let $\mu\subset\lambda$…

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…