128 problems

Berenstein–Kirillov conjecture. The relations

Let $D^\lambda$ be the simple module indexed by a multipartition $\lambda$, and call a standard tableau of shape $\lambda$ a Dipper–James–Murphy tableau if every standard tableau w…

Type C K-Stanley expansion conjecture.

Let $\lambda/\mu$ be a skew shape, let $m$ be a positive integer, and let $k$ be a positive integer. Write $m\lambda/m\mu$ for the skew shape obtained by multiplying every part of…

Let $\omega$ be the permutation governing the switching algorithm, and let $\mathcal{O}\in\operatorname{Orb}(\omega)$ be one of its orbits. A genomic tableau is a tableau counted b…

Parabolic Hall–Littlewood polynomials are the Catalan functions associated with concatenations of empty root ideals and a partition weight. Their Schur expansion has coefficients r…

The stronger factor-sequence conjecture. Under these hypotheses, $(\dots,X',Y',\dots)$ is also a factor sequence for $\gamma$.

The tableau interpretation conjecture. The coefficient $c_\mu(r)$ equals the number of sequences of tableaux satisfying the conditions specified by the quiver-coefficient construct…

The paper studies the polynomials $K_{\lambda;R}(q)$, which are $q$-analogues of Littlewood–Richardson coefficients, and the set of $R$-catabolizable tableaux of shape $\lambda$. F…

Let $R\trianglerighteq R'$ be a covering relation in the dominance order on sequences of rectangles. Let $\theta_R^{R'}:LRT(\lambda;R)\to LRT(\lambda;R')$ be the embedding obtained…

Let $R$ be a sequence of rectangles, and let $CT(\lambda;R)$ and $CCT(\lambda^t;R^t)$ denote the row- and column-catabolizable tableaux defined in the source. Catabolizable transpo…

Let $R$ be a sequence of rectangles such that $R^t$ is dominant. Let $b_R$ be the map from rigged configurations to tableaux, and let $CCT(\lambda;R)$ denote the set of $R$-column-…

Let $R$ be a sequence of rectangles, let $\widetilde{R}$ and $\widetilde{\lambda}$ be the complementary dual data defined in the source, and let $\operatorname{dual}$ be the corres…

Let $\operatorname{ev}$ be evacuation on column-strict tableaux, restricting to a bijection $LRT(\lambda;R)\to LRT(\lambda;\operatorname{rev}(R))$. Let $\theta$ complement each rig…

Let $R$ be a sequence of rectangles and let $u$ be a permutation of the rectangle indices. The bijection $u_R:LRT(\lambda;R)\to LRT(\lambda;uR)$ is defined by composing the adjacen…

Let $R$ be a sequence of rectangles. Let $b_R:RC(\lambda^t;R^t)\to CST(\lambda,\gamma(R))$ be the restriction of the inverse rigged-configuration bijection to the indicated subset,…

Let $R$ be a dominant sequence of rectangles. Let $LRT(\lambda;R)$ be the corresponding Littlewood–Richardson tableaux, let $RC(\lambda^t;R^t)$ be the rigged configurations, and le…

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

Generalized charge conjecture. For $R$ dominant,

Let $\mathfrak{g}$ be of type $C_n$, let $\lambda/\mu$ be a skew diagram, and let $\chi_{\lambda/\mu,a}$ be the associated Jacobi–Trudi determinant for $a\in\mathbb{C}$. Let…

Fix positive integers $m,n$ and set $\delta_n=(n-1,n-2,\ldots,1,0)$. Let $E_{n,k}$ be defined by … and let $D_n^{(m),\lambda}(z;q)$ denote the generalized tableau sum associated wi…

Fix positive integers $m,n$ and set $\delta_n=(n-1,n-2,\ldots,1,0)$. For a partition $\lambda\subseteq m\delta_n$, let $T$ be a semistandard tableau of shape…

The refined shuffle conjecture. For $1\leq k\leq n$,

Fix $n$ and let $\delta_n=(n-1,n-2,\ldots,1,0)$. For a partition $\lambda\subseteq\delta_n$, let $T$ range over semistandard tableaux of skew shape $(\lambda+(1^n))/\lambda$, and l…

Let $\lambda$ be a partition with $\ell$ parts and fixed parity for each part. Let $G_\lambda(x/x)$ be the specialized symmetric function defined by substituting $2p_{2i-1}$ for…