11 problems

Let $\lambda$ be a $k$-bounded partition, let $\lambda^{\omega_k}$ be its $k$-conjugate, and define $\omega_t f=(\omega f)|_{t\to1/t}$. The $t$-involution conjecture. For every $k$…

For $k$-bounded partitions $\lambda$, $\mu$, and $\nu$, write the product of $k$-Schur functions as … where $c_{\lambda\mu}^{\nu}$ are the ordinary Littlewood–Richardson coefficien…

Let $\lambda$ be a $k$-bounded partition, meaning that its largest part is at most $k$. Let $\lambda^{\omega_k}$ denote its $k$-conjugate, and let $\omega$ be the usual involution…

Haiman–Hugland's conjecture. The symmetric function

Interval conjecture. For every $4\lambda\in\mathcal{P}_k4$ and $1\le t\le k$, there exists $4\mu\in\mathcal{P}_k4$ such that

Nonnegativity conjecture. For every $4\lambda\in\mathcal{P}_k4$ and every $\mu$, one has $a_{P,\lambda,\mu}\ge 0$.

For $k>0$, let $\mathcal P^k$ be the set of partitions with first part at most $k$, let $s_\lambda^{(k)}[X;t]$ be the $k$-Schur function with parameter $t$, and let…

Let $k$ be a positive integer, let $u$ be a $k$-bounded partition, and let $R( u)$ be its residue table. The theory also involves $k$-bounded partitions and $k$-Schur functions, i…

The branching-path conjecture. For all $\lambda\in\mathcal C^{k+1}$ and $\mu\in\mathcal C^k$,

Charged branching-polynomial conjecture. For all $\lambda\in\mathcal{C}^{k+1}$ and $\mu\in\mathcal{C}^k$,

The k-Schur filtration conjecture. There are polynomials $\tilde b_{\mu\lambda}^{(k\to k')}(t)\in\mathbb{Z}_{\geq 0}[t]$ such that