17 problems

Let $R_n^{(1,3)}$ be the $(1,3)$-bosonic-fermionic coinvariant ring, let $\operatorname{Frob}(R_n^{(1,3)};1;1,1,1)$ denote its Frobenius series with all grading variables specializ…

Let $R_n^{(1,2)}$ be the $(1,2)$-bosonic-fermionic coinvariant ring, and let $\operatorname{Frob}(R_n^{(1,2)};q;u,v)$ denote its multigraded Frobenius series. The notation…

Let $R_n^{(2,2)}$ be the $(2,2)$-bosonic-fermionic coinvariant ring, and let $\operatorname{Frob}(R_n^{(2,2)};q,t;u,v)$ denote its multigraded Frobenius series. Here $\Theta_{e_k}$…

Let $\Omega_n=\mathbb{C}[x_1,\ldots,x_n]\otimes\bigwedge\langle\theta_1,\ldots,\theta_n\rangle$ be the superspace, and let $SR^B_n$ be its type $B$ superspace coinvariant ring. Wri…

Let $SR_n$ be the bigraded superspace coinvariant ring with its natural $\mathfrak{S}_n$-action, and let $\operatorname{grFrob}(SR_n;q,z)$ denote its bigraded Frobenius characteris…

Let $\Omega_n$ be the superspace ring with diagonal $\mathfrak{S}_n$-action, let $SR_n=\Omega_n/SI_n$ be its superspace coinvariant ring, and let $\mathbb{F}[\mathcal{OP}_n]$ be th…

For fixed $n$ and nonnegative integers $k,j$, let $R_n^{(k,j)}$ be the corresponding multigraded coinvariant ring, and write … For $0\leq k'\leq k$ and $0\leq j'\leq j$, restrictio…

Let $R_n^{(1,3)}$ be the $(1,3)$-bosonic-fermionic coinvariant ring, let $\operatorname{Frob}(R_n^{(1,3)};q;u,v,w)$ denote its Frobenius series, and let $s_{(1^n)}$ denote the Schu…

Let $R_{\mathfrak{B}_n}^{(1,1)}$ be the type $B$ $(1,1)$ coinvariant ring, and let $B_{\mathfrak{B}_n}^{(1,1)}$ be the type $B$ super-Artin set. Type $B$ super-Artin conjecture. ……

Let $B_n^{(1,2)}$ be the proposed monomial basis of $R_n^{(1,2)}$. For $b\in B_n^{(1,2)}$, write $\deg_x(b)$, $\deg_\theta(b)$, and $\deg_\xi(b)$ for its degrees, and let…

Quaternionic Haiman conjecture. There is a $(g+1)^n$-dimensional quotient ring of the zero-fiber ring of $W$.

Hard Lefschetz conjecture. For every $J\subseteq[n]$, the quotient ring $A_J$ satisfies the Hard Lefschetz property.

J-Artin monomial conjecture. For every subset $J\subseteq[n]$, the $J$-Artin monomials descend to a basis of

D'Adderio–Iraci–Vanden Wyngaerd conjecture. The space $(TDR_n)_{i,j}$ vanishes whenever $i+j\geq n$. If $i+j<n$, then

Let $\bell$ and $m$ be positive integers with $bell ygeq 2$ and $m$ dividing $bell$, and let $G(bell,m,n)$ denote the corresponding monomial group. Let $R{G(bell,m,n)}$ be its diag…

Let $\Omega_n$ be the superspace algebra with commuting variables $x_1,\ldots,x_n$ and anticommuting variables $\theta_1,\ldots,\theta_n$, acted on by the symmetric group…

Higher Specht basis conjecture. The set of polynomials