16 problems

Let $\lambda$ be a shifted shape and $\mu$ a content, and suppose that a statistic $d$ on marked shifted tableaux is used to express … where the sum is over marked shifted tableaux…

Let $L_{\lambda\mu}(q)$ denote the $Q$-Kostka polynomial, with $\lambda$ and $\mu$ partitions in the indexing convention of the paper. Positivity conjecture. The $Q$-Kostka polynom…

Let $G_{\lambda}[X;q]$ be the $q$-analog of the Schur $Q$-function indexed by a strict partition $\lambda$, and write the transition from the $G$-basis to the $Q$-basis as … The co…

Let $P$ and $Q$ range over the relevant shifted standard tableaux, and let $A_{P,Q}$ denote the corresponding entry of the matrix associated with modified jeu de taquin. Symmetry c…

Connectivity conjecture. For each strict partition $\lambda$ with $\ell(\lambda)\leq n$, the square-root queer crystal $\mathsf{SetDecTab}_n(\lambda)$ is connected and the tableau…

The dual $K$-theoretic Schur $P$- and $Q$-functions $g\hspace{-0.1mm}p_\lambda$ and $g\hspace{-0.1mm}q_\lambda$ are specified by a Cauchy identity involving the $K$-theoretic Schur…

Let $\beta$ be a parameter, let $\beta$-dependent skew functions $j\hspace{-0.2mm}p_{\nu}$ and $j\hspace{-0.2mm}q_{\nu}$ be indexed by strict partitions, and let…

Let $\beta$ be a parameter, let $\beta$-dependent skew functions $ghspace{-0.2mm}p{ u}$ and $ghspace{-0.2mm}q{ u}$ be indexed by strict partitions, and let $u$ be a strict partiti…

Ikeda–Naruse's basis conjecture. The set of all $G\hspace{-0.2mm}P_\lambda$'s, respectively all $G\hspace{-0.2mm}Q_\lambda$'s, is a basis for a ring.

Converse CDE conjecture for shifted diagrams. The interval $[\emptyset,\lambda]$ has the CDE property if and only if $\lambda$ is balanced or trapezoidal.

The combinatorial-model conjecture. The restriction of this map to $\overline{\mathcal{T}(\mu^t,\mu,s(\mathcal{T}_\lambda))}$ is a bijection onto…

Let $\lambda$ be a strict partition and let $\mu$ be a partition. Write $g_{\lambda\mu}$ for the coefficient in the Schur-function expansion of the Schur $Q$-function, and let…

Let $v$ and $w$ be involution words, and let $\overset{\textsf{O}}\sim$ denote the shifted Hecke equivalence relation. The shifted Hecke insertion algorithm assigns to each word…

Shifted skew-shape product conjecture. One has

A shifted dual equivalence is a family of involutions satisfying the stated local axioms for shifted tableaux and peak sets. The paper considers weak dual equivalence as the analog…

Ring-generation conjecture. $\mathcal{P}_{\lambda/\mu}$ belongs to the ring generated by the shifted plactic Schur $P$-functions.