20 problems

For each $n$, let ${\mathcal B}_n$ be the recursively constructed set of monomials in ${\rm DQ}_n$, using the rules … when $i+j<n-1$, together with the diagonal rule … for…

Let $(W,S)$ be a finite Coxeter system, let $BE(W,S)$ be the associated algebra, and let ${\bf R}[\theta_s\mid s\in S]$ denote the subalgebra generated by its Dunkl elements. The c…

Exterior-degree conjecture. The exterior degree $j$ part of $SR_{SL_n(\mathbb{F}_q)}$ is Brauer-isomorphic to

Super Artin basis conjecture. The set ${\mathcal S}{\mathcal A}_{m,n}$ is a basis for $\operatorname{SRG}_{m,n}$.

For $n\geq 0$, define … where $S^o[n,k]$ are the ordered type $A$ $q$-Stirling polynomials. Palindromicity and sign-pattern conjecture. The polynomial $F_n(q)$ is palindromic after…

Let $\operatorname{SRB}_n$ be the type $B$ super coinvariant algebra, defined by … Here $S_B^o[n,k]$ denotes the ordered type $B$ $q$-Stirling number. Swanson–Wallach conjecture. W…

The type $A$ superdiagonal coinvariant algebra is a triply graded algebra whose Hilbert series records the ordered $q$-Stirling numbers $S^o[n,k]$. Zabrocki's conjecture. The tri-g…

For $1\leq k\leq n$, let $\delta_{n,k}\in\Omega_n$ be the superspace Vandermonde, and let $V_{n,k}$ be the smallest linear subspace of $\Omega_n$ containing $\delta_{n,k}$ and clos…

Let $d_j:\Omega_n\to\Omega_n$ be defined for $j\geq1$ by … and let $\delta_n\in\mathbb{C}[x_1,\dots,x_n]$ be the Vandermonde determinant. Let $SR_n$ be the superspace coinvariant a…

Let $\mathfrak{g}$ be a semisimple Lie algebra with Weyl group $W$, let $\mathfrak{u}_0(\mathfrak{g})$ be the corresponding small quantum group, let…

Let $\mathfrak{u}_0(\mathfrak{sl}_m)$ be the small quantum group at the relevant root of unity, let $\mathbf{z}(\mathfrak{u}_0(\mathfrak{sl}_m))$ denote its center, and let…

Let $\mathcal{E}_n=\sum_{\mu\vdash n}\mathcal{C}_\mu\otimes s_\mu(\bm{z})$ be the pure bosonic multivariate coinvariant characteristic, where…

Let $n$ be a positive integer, let $k$ and $j$ index the commuting and anticommuting variable sets, and let $\mathcal{E}_n(\bm{q};\bm{z})$ be the generic Frobenius characteristic o…

Let $a$, $b$, and $c$ be non-negative integers. Consider the homogeneous component of order $n$ in $\operatorname{SDR}_n$ with $x$-degree $a$, $y$-degree $b$, and $z$-degree $c$. H…

Let $n\geq1$. For each ordered set partition $\Pi$ of $[n]$, let $\mathcal{B}_\Pi$ be the set of monomials constructed from the associated permutation, Grassmann monomial, and sche…

Let $n\geq1$. For an ordered set partition $\Pi$ of $[n]$, let $|\Pi|$ be its number of blocks, let $\tau(\Pi)$ be its associated permutation, and let $w_\Pi(c)$ be the schedule nu…

Let $S_n$ act on the super-polynomial algebra $\bk[\bm{x}_n,\bm{\theta}_n]$ in commuting variables $x_1,\ldots,x_n$ and anticommuting variables $\theta_1,\ldots,\theta_n$ by permut…

Let ${\mathbf{x}}_n=(x_1,\dots,x_n)$ and ${\mathbf{y}}_n=(y_1,\dots,y_n)$ be two sets of variables, and let $DR_{n,k}$ be the bigraded $S_n$-module quotient by the ideal generated…

Let $n,k,r$ be parameters, let $R_{n,k,r}$ be the graded $S_n$-module quotient defined from $I_{n,k,r}$, and let … be the ideal obtained by replacing the homogeneous symmetric func…

The center–diagonal coinvariant algebra conjecture. In type $A$, there exists an $S_m$-action on $\mathsf z_0$ extending the action on $\mathrm{C}_m$, commuting with the…