6 problems

Let $n$ and $l$ be positive integers, and let $X^{(l)}{ u}(rho)$, $X^{(l)'}{ u}(rho)$, $F^{(l)}{ u}(q)$, and $F^{(l)'}{ u}(q)$ denote the restricted one-dimensional sums and fermio…

Let $G$ be one of the classical groups and let $\mu^{(1)},...,\mu^{(r)}$ and $\lambda$ index the corresponding $q$-analogue $\frak{D}_{\mu^{(1)},...,\mu^{(r)}}^{\lambda,G}(q)$ of a…

Let $B_\mu=B_{\mu_1}\otimes\cdots\otimes B_{\mu_m}$ be an affine crystal, and let $X$ denote the associated one-dimensional sum, obtained by grading highest-weight vertices in the…

A stable one-dimensional sum is the large-rank limit of the graded tensor-product multiplicities associated with a nonexceptional family of affine algebras. The four stable types a…

Let $B$ be a tensor product of crystals, let $\boldsymbol{L}$ record its multiplicities, and let $\bar{X}_{\lambda,B}(q)$ denote the one-dimensional sum for weight $\lambda$. Let…

Schilling–Shimozono conjecture. For $\diamondsuit\in\{(1),(2),(1,1)\}$,