8 problems

Let $R$ be a dominant sequence of rectangles, and let $R^t$ be obtained by transposing every rectangle in $R$. Let $R'$ be a dominant rearrangement of $R^t$, and let $m$ be the tot…

For a sequence of rectangles $R$, let $r_{i,j}(R)$ be the number of rectangles containing $(i,j)$, and define … Set $\widetilde{K}_{\lambda;R}(q)=q^{n(R)}K_{\lambda;R}(q^{-1})$. Co…

Let $R=(R_1,\ldots,R_p)$ be a dominant sequence of rectangular partitions, and let $\Lambda$ be the partition with empty $p$-core and $p$-quotient $(R_1,\ldots,R_p)$. The polynomia…

Type-C identities.

Let $\Lambda_k^A$ and $\Lambda_{n-k}^A$ be the $k$-th and $(n-k)$-th fundamental weights of $U_q(\mathfrak{sl}_n)$, and set … Let $K_{\Lambda_{2k},0}(q)$ be the type $C_n$ Kostka–F…

Let $\beta$ be a dominant weight for the root system $C_n$, let $B(\beta)$ be the corresponding crystal, and let … be its subset of tableaux of weight $\mu$. Let…

No broken diamond in hexagon conjecture. The paths $E_i$ and $E_{i+1}$ in the graph induced by the modified crystal operators do not contain any edges that give way to Stembridge E…

Principal-filtration conjecture. If $\mu\in\mathfrak X_{+,H}$ satisfies the vanishing conditions of Theorem, then