14 problems

Let $\lambda$ and $\mu$ be partitions, and let $K_{\lambda,\mu}(t)$ be the Kostka–Foulkes polynomial defined by the expansion … where $s_{\lambda}$ is a Schur polynomial and…

Lecouvey's conjecture.

Catabolizable-tableau conjecture. If $R$ is dominant, then

Generalized charge conjecture. For $R$ dominant,

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…

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…

Type-C identities.

Type $C_n$ charge conjecture. The identity $$ holds when $chin^C$ is replaced by $mathrm{ch}{Cn}$, for all partitions $lambda$ and $mu$.

Type $C_n$ cyclage-statistic conjecture. This statistic yields an analogue of Lascoux–Schützenberger's theorem.

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…

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