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

Source abstract

We define a family of algebras $\mathbb{Q}[\mathbf{x}_n]/I_{n,a,b}$ whose Hilbert series, when $n\le a+b\le n+2$, $a>0$, and $0<b<n$, enumerate certain positroid cells by codimension. Equivalently, these series enumerate certain affine permutations by length. We determine the graded $\mathfrak{S}_n$-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 $\mathrm{grFrob}(\mathbb{Q}[\mathbf{x}_{a+b-1}]/I_{a+b-1,a,b};q)=\operatorname{rev}_q(ω((Θ_{e_{a-1}}Θ_{e_{b-1}}\nabla e_1)\mid_{t=0}))$, where $\operatorname{rev}_q$ reverses the $q$-coefficients and the specialization $(Θ_{e_{a-1}}Θ_{e_{b-1}}\nabla e_1)\mid_{t=0}$ appears in the Theta conjecture. For $a,b\ge1$, we also show that $e_a^\perp\mathrm{grFrob}(\mathbb{Q}[\mathbf{x}_{a+b}]/I_{a+b,a,b};q)=q^{\binom{a}{2}}\operatorname{rev}_q(ω((Δ_{e_{a-1}}^\prime e_b)\mid_{t=0}))$, where $Δ_{e_{k-1}}^\prime e_n$ 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 $(\mathbb{P}^{k-1})^n$ whose rational Chow ring is isomorphic to $\mathbb{Q}[\mathbf{x}_n]/I_{n,k,n-k}$. This variety admits a stratification induced by certain positroids. Its strata need not be affine, and their codimension generating function equals $\mathrm{Hilb}(\mathbb{Q}[\mathbf{x}_n]/I_{n,k,n-k};q)$.