5 problems

The diagonal formula conjecture. For all $k$ between $1$ and $n-1$,

Let $\rho=(a\,|\,b)$ be a hook shape, let $\mathcal{S}_\rho$ denote the corresponding graded Frobenius characteristic, and let $L_{\leq 2}$ retain the components indexed by partiti…

Let $k=2$ and let $\rho$ be a hook partition indexing a Schur function $s_\rho$. Write $\nabla$ for the nabla operator and let the associated graded Frobenius characteristic be the…

Let $\varepsilon_{n,j}(\mathbf{w};q)$ be the symmetric functions defined in the source, let $\nabla$ be the nabla operator, and let $S_\mu(\mathbf{w})$ denote the Schur function in…

Let $\lambda$ and $\mu$ be partitions of the same integer $n$. Write $\nabla_{\lambda\mu}(q,t)=\langle \nabla(S_\lambda(\mathbf{w})),S_\mu(\mathbf{w})\rangle$, and let $k$ be the n…