29 problems

Let $n$) and $k$ satisfy $0\le k<n$, and let $\Delta'_{e_{n-k-1}}$ be the Delta operator on modified Macdonald polynomials. Let a module of super-diagonal coinvariants denote the m…

Haiman's q,t-Catalan evaluation conjecture. The specialization at $q=t=1$ equals the classical Catalan number:

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

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

Consider the algebra obtained from the Orlik–Solomon algebra by introducing two generators $x_{ij}$ and $y_{ij}$ for each pair $1\leq i\ne j\leq n$ and imposing the analogous relat…

Bergeron–Shapiro level-one conjecture. The function $b$ satisfies

Bergeron–Shapiro conjecture. There exists an integer-valued function $b=b(l)$ such that the number of admissible pairs $(k,l)$ satisfying

Haglund–Loehr conjecture. The coefficient of $q^at^b$ in $R_n(q,t)$ equals

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 $TDR_n$ be the quotient of the polynomial-superpolynomial ring in two sets of commuting variables and two sets of anti-commuting variables by the ideal generated by the invaria…

Let $C_n(q,t)$ be the rational function introduced as the $q,t$-Catalan number, let $D_n[X;q,t]$ denote the diagonal coinvariant alternant, and let $e_n$ be the $n$-th elementary s…

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 $\mathbb{F}$ be a field and let $\operatorname{coinv}^{\,}_{n}(\mathbb{C}^{3})$ denote the diagonal coinvariant algebra for the symmetric group on $n$ variables, obtained by qu…

Let $R_n$ be the diagonal coinvariant algebra in $n$ pairs of variables, let $e_n$ be the $n$th elementary symmetric function, and let $\nabla$ be the operator diagonal in the modi…

The Shuffle Conjecture. The following equation holds:

Let $\mathcal{DR}_{3,n}^{m}$ be the $m$-extended diagonal coinvariant space in three sets of variables, with its natural action of the symmetric group $\mathfrak{S}_n$. Let…

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 $\mathsf{DR}_n$ be the diagonal coinvariant ring for the diagonal action of $S_n$ on $\mathbb{C}[x_1,\ldots,x_n,y_1,\ldots,y_n]$, and let $\mathsf{PF}_n$ be the permutation rep…