A family of algebras that connects positroids, the Theta conjecture, the Delta conjecture, and Eulerian quasisymmetric functions

Abstract: We define a family of algebras $\mathbb{Q}[\mathbf{x}<em>n]/I</em>{n,a,b}$ whose Hilbert series, when $n\le a+b\le n+2$, $a&gt;0$, and $0<b<n$, enumerate certain positroid cells by codimension. Equivalently, these series enumerate certain affine permutations by length. We determine the graded $\mathfrak{S}<em>n$-module structure of these algebras by using exact sequences to derive an invertible recursion for their graded Frobenius characteristics when $a+b\ge n$ and then passing to the limit $a\rightarrow\infty$. In particular, the graded Frobenius characteristic $$\mathrm{grFrob}(\mathbb{Q}[\mathbf{x}</em>{a+b-1}]/I_{a+b-1,a,b};q)=\operatorname{rev}<em>q(ω((Θ</em>{e_{a-1}}Θ<em>{e</em>{b-1}}\nabla e_1)\mid_{t=0}))$$, where $\operatorname{rev}<em>q$ reverses the $q$-coefficients and the specialization $$(Θ</em>{e_{a-1}}Θ<em>{e</em>{b-1}}\nabla e_1)\mid_{t=0}$$ appears in the Theta conjecture. For $a,b\ge1$, we also show that $$e_a^{\perp\mathrm{grFrob}(\mathbb{Q}[\mathbf{x}<em>{a+b}]/I</em>{a+b,a,b};q)=q^{{\binom{a}{2}}\operatorname{rev}<em>q(ω((Δ</em>{e_{a-1}}^\prime}} e_b)\mid_{t=0}))$$, where $Δ<em>{e</em>{k-1}}^\prime e_n$ occurs in the Delta conjecture. In the three cases $a+b-n=0,1,2$, the ungraded Frobenius characteristics agree with three families of Eulerian quasisymmetric functions, respectively. We also construct an open subvariety of $(\mathbb{P}<sup>{k-1})<sup>n$ whose rational Chow ring is isomorphic to $\mathbb{Q}[\mathbf{x}<em>n]/I</em>{n,k,n-k}$. This variety admits a stratification induced by certain positroids. Its strata need not be affine, and their codimension generating function equals $$\mathrm{Hilb}(\mathbb{Q}[\mathbf{x}<em>n]/I</em>{n,k,n-k};q)$$.