16 problems

Let $m,n$ be positive integers, let $L_{mn,n}^+$ denote the set of rational $m$-Dyck paths, and let $L_{n,n}^+$ denote the set of Dyck paths of size $n$. For any rational Dyck path…

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…

Euler-number specialization conjecture.

The $\Lambda\Sigma=\Delta$ conjecture. We have $\Lambda\Sigma=\Delta$.

Let $G=GL_n$, let $\gamma$ be the regular element used to define the affine Springer fiber $\Fl^{\gamma}_{\bI}$, let $\Lambda$ be the lattice acting on it, and let $P$ be the perve…

For $1\leq k\leq n$, let $\mathbb{V}_{n,k}$ be the smallest subspace of $\Omega_n[y_1,\dots,y_n]$ containing the superspace Vandermonde $\delta_{n,k}$, closed under…

Let $\Omega_n[y_1,\dots,y_n]=\mathbb{C}[x_1,\dots,x_n]\otimes\mathbb{C}[y_1,\dots,y_n]\otimes\bigwedge\{\theta_1,\dots,\theta_n\}$, with $S_n$ acting triply diagonally, and let … b…

Let $\mathsf{DR}_n$ be the doubly graded diagonal coinvariant ring, and let $\mathsf{PF}_n$ be the labeled classical parking functions. For each $P$, let $\mathsf{area}(P)$,…

Let $\mathsf{PF}_n$ be the set of labeled classical parking functions. For $P\in\mathsf{PF}_n$, let $\mathsf{area}(P)$ be its area statistic and let $\mathsf{dinv}(P)$ be its diago…

Let $\Phi$ be a root system, let $\mathcal{M}(\Phi)$ be its diagonal coinvariant ring, and define its bigraded Hilbert series by … Let $\mathcal{I}(\Phi^+)$ denote the order ideals…

Haglund–Haiman–Loehr–Remmel–Ulyanov conjecture. We have an identity

Haiman's kernel conjecture. The kernel of this surjection does not contain a copy of the trivial representation. For $m=1$, the source attributes the conjecture to Haiman; the sour…

Let $W$ be a well-generated complex reflection group, let $D\!R^{(m)}(W)$ be the space of generalized diagonal coinvariants, and let $\mathbf{e}_{\epsilon}$ denote the idempotent p…