24 problems

Let $h\colon [k]\rightarrow [k]$ be a Hessenberg function, let $S$ be a regular semisimple operator on $\mathbb C^k$, and let $\mathfrak S_k$ act on $H^*(\operatorname{Hess}(S,h))$…

Let $k$ be a positive integer, let $h\colon [k]\rightarrow [k]$ be a Hessenberg function, let $S$ be a regular semisimple operator on $\mathbb C^k$, and let $\mathfrak S_k$ act on…

Let $k$ be a positive integer, let $h\colon [k]\rightarrow [k]$ be a Hessenberg function, and let $G_h$ be the graph associated with $h$. For a proper coloring $\kappa$ of $G_h$, w…

Let $G=\operatorname{GL}_n(\mathbb{C})$, let $\mathbf{s}\in G^{\mathrm{rs}}$ be regular semisimple, and let $w,w'\in S_n$ be smooth permutations associated, via the natural combina…

Let $G$ be a reductive group with Lie algebra $\mathfrak{g}$, let $\mathfrak{g}^{\mathrm{rs}}$ denote its regular semisimple locus, and let $X(s)$ and $Y(s)$ be the associated Fano…

Smoothness conjecture. The following statements are equivalent: $\Omega_{w,h}$ is smooth; $\Gamma_{w,h}$ is a regular graph; and $w$ avoids all the patterns $\overline{2143}$,…

Let $\mathbf{m}\colon[n]\to[n]$ be a Hessenberg function, let $G_\mathbf{m}$ be its indifference graph, and let $g_k(G_\mathbf{m};x,q)$ be the symmetric functions defined in the pa…

Coefficient formula conjecture. For every partition $\lambda=(n-j,j)$, the coefficient $c_\lambda$ in the equivariant Poincaré polynomial of $X_{h_2}$ is given by the corresponding…

Hessenberg variety homeomorphism conjecture. There exists a codominant permutation $w'$ such that

Let $w\in S_n$ be smooth, and let $X\in SL(n)$ be regular. For $u\leq w$, let $[\mathcal{Y}_u(X)]\in H^*(\mathcal{Y}_w(X))$ be the corresponding geometric class. Cohomology-basis c…

Let $G$ be the group, let $X\in G$ be regular, let $w\in W$, and let $\mathcal{Y}_w(X)$ be the corresponding variety. Let $\Omega_w$ be the Schubert variety and let…

Let $X$ be an element of the group under consideration, let $\underline{s}$ be a reduced word, and let $\mathcal{Y}_{\underline{s}}(X)$ be the associated variety. A variety is rati…

Let $h:[n]\to[n]$ be a Hessenberg function, meaning a nondecreasing function with $h(i)\geq i$ for all $i$. Let $\mathcal{H}ess(\mathsf{S},h)$ be the regular semisimple Hessenberg…

Erasing marks conjecture. The collection

Let $G$ be a connected reductive algebraic group over $\mathbb{C}$, let $M$ be a Hessenberg subspace, and let $\pi_M:G\times^B M\to\mathfrak{g}$ be the Hessenberg map. Let…

Let $h$ be a Hessenberg function and let $\operatorname{Hess}(N,h)$ be the associated regular nilpotent Hessenberg variety. For a Hessenberg function $h'\subset h$, write…

Let $h:[n]\to[n]$ be a Hessenberg function, and let $\Gamma_h$ be its associated graph. Let $m=m(\Gamma_h)$ be the number of parts of the partition $\lambda\vdash n$. Let…

Radicality conjecture. The ideal $I_{w,\mathcal{B}(S,H)}$ is radical for any semisimple Hessenberg variety $\mathcal{B}(S,H)$, and is therefore the patch ideal of…

Let $I\subset\Phi^+$ be a lower ideal. A subset $Y\subset I$ is of Weyl type if $calpha,\beta\in Y$ and $\alpha+\beta\in I$ imply $\alpha+\beta\in Y$, and if…

Let $(G,K)=(SL(N),SO(N))$ with $N$ odd, and let $\operatorname{Hess}(K/P,{\mathfrak g}_1,\Sigma)\to{\mathfrak g}_1$ be a family of Hessenberg varieties associated to a parabolic su…

Let ${\bf m}$ be a Hessenberg vector, and let $P({\bf m})$ be the natural unit interval order on $[n]$ defined by $i<_{P({\bf m})}j$ if $j>m_i$. Let ${\mathcal H}({\bf m})$ be the…

Let $P$ be a natural unit interval order on $[n]$. For each $j$, let $\operatorname{ch}H^{2j}({\mathcal H}(P))$ be the Frobenius characteristic of Tymoczko's representation on degr…

Let $P$ be a natural unit interval order on $[n]$. Tymoczko's representation of $\mathfrak S_n$ on the cohomology $H^*({\mathcal H}(P))$ is the representation under consideration.…

Presentation conjecture. The quotient