29 problems

Let $R_n^{(3,0)}$ denote the $(3,0)$-bosonic coinvariant ring. Haiman's conjecture. Its dimension is … This conjecture was made by Haiman in 1994 and remains open in general.

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…

Let $R_n^{(2,2)}$ be the corresponding four-graded coinvariant ring, let $e_r$ be the elementary symmetric function, let $\Theta_{e_r}$ be the Theta operator indexed by $e_r$, let…

Let $R_n^{(2,1)}$ denote the corresponding triply graded coinvariant ring, with Frobenius series in variables $q,t,u$, and let $e_r(\mathbf{z})$ be the elementary symmetric functio…

Let … Write $\operatorname{Frob}(R_n^{(1,2)};q;u,v)$ for its multigraded Frobenius series, and let $\ell(\mu)$ and $d_i(\mu)$ denote respectively the length of a partition $\mu$ an…

Let $G(m,1,n)$ be the complex reflection group of $n\times n$ monomial matrices whose nonzero entries are $m$th roots of unity. Let the operator theorem refer to the type $B$ opera…

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…

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…

Let $R_n^{(1,2)}$ be the $(1,2)$ bosonic-fermionic coinvariant ring, and let $B_n^{(1,2)}$ be the monomial set constructed by interpolating between the Kim–Rhoades basis and the su…

Let $R_{\mathfrak{B}_n}^{(1,2)}$ be the type $B$ $(1,2)$ coinvariant ring. Type $B$ dimension conjecture. … The conjecture is based on computations for $n\leq 5$. The source proves…

Let $R_{\mathfrak{B}_n}^{(1,2)}$ be the type $B$ $(1,2)$ coinvariant ring, and let $B_{\mathfrak{B}_n}^{(1,2)}$ be the explicitly constructed set of monomials in the source. Type…

Let $R_n^{(2,2)}$ be the $(2,2)$-bosonic-fermionic coinvariant ring, and let $\Theta_f$ and $\nabla$ denote the symmetric-function operators appearing in the conjectural formula. T…

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

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

Fields Institute conjecture. The doubly graded ${\mathfrak S}_n$-structure of $SR_n$ is given by

Fields Institute conjecture. The dimension of $SR_n$ is the ordered Bell number, and, as an ungraded ${\mathfrak S}_n$-module, $SR_n$ is the permutation representation on ordered s…

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 $W$ be a real reflection group, let $R_W$ be its diagonal coinvariant ring, let $n$ be the rank of $W$, and let $h$ be its Coxeter number. Haiman–Gordon's conjecture. The ring…

Let $W$ be a real reflection group of rank $n$, let $R$ be the polynomial ring in two sets of $n$ variables, and let $R_W$ be its quotient by the ideal generated by positive-degree…

Let $\mathcal{F}_n\in\Lambda(\mathbf{x})\otimes\Lambda(\mathbf{q})$ be the stable character associated with fermionic multidiagonal coinvariants, and let…

Let $FDR(n;k)$ be the fermionic multidiagonal coinvariant ring, graded by multidegrees $\boldsymbol{i}=(i_1,dots,i_k)\in\mathbb{Z}_{\geq 0}^k$, and write…