10 problems

Let $\mu=(\mu_1,\ldots,\mu_m)$ be a partition and let $r\geq m$. Define … Here $r\mu=(r\mu_1,\ldots,r\mu_m)$. Monotonicity conjecture. For consecutive values of $j$, the difference…

Cocharge-atom decomposition conjecture. For any partition $\lambda$, there exists a partition $f(\lambda)$ such that

Positivity conjecture. The coefficients in these Schur-basis expansions are polynomials with nonnegative integer coefficients.

Non-negativity conjecture. The coefficients of each polynomial $c^\nu_{\lambda^{(0)},\ldots,\lambda^{(r-1)}}(q)$ are non-negative. The polynomial $\tilde G_\lambda^{(r)}(X;q)$ is a…

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…

Let generalized Kostka polynomials be the polynomials studied in the paper, let ribbon tableaux carry their spin statistic, and let rigged configurations be the corresponding combi…

Let $\lambda$ be a partition of size $k$, and let $s_\lambda({\mathbf u})$ and $h_\lambda({\mathbf u})$ be the non-commutative Schur and homogeneous symmetric functions in the ribb…

The ribbon Schur operators $u_i$ add ribbons to partitions, and $s_\nu({\mathbf u})$ denotes the corresponding non-commutative Schur function. The notation $h_\lambda({\mathbf u})$…

Kazhdan–Lusztig immanant positivity conjecture. All Kazhdan–Lusztig immanants of $A_{\lambda/\mu,R}$ should be Schur positive.

Let $r \in \mathbb{N}$, let $n \in \mathbb{N}_0$, let $\nu$ be a partition of $rn$, and let $(a,b)$ be a partition of $n$. An $r$-ribbon tableau of shape $\nu$ and weight $(a,b)$ i…