A family of algebras that connects positroids, the Theta conjecture, the Delta conjecture, and Eulerian quasisymmetric functions
Abstract
We define a family of algebras ℚ[xn]/In,a,b whose Hilbert series, when n≤ a+b≤ 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 Sn-module structure of these algebras by using exact sequences to derive an invertible recursion for their graded Frobenius characteristics when a+b≥ n and then passing to the limit a→∞. In particular, the graded Frobenius characteristic grFrob(ℚ[xa+b−1]/Ia+b−1,a,b;q)=revq(ω((Θea−1Θeb−1∇ e1)|t=0)), where revq reverses the q-coefficients and the specialization (Θea−1Θeb−1∇ e1)|t=0 appears in the Theta conjecture. For a,b≥1, we also show that ea⊥grFrob(ℚ[xa+b]/Ia+b,a,b;q)=qbinoma2revq(ω((Δea−1′ eb)|t=0)), where Δek−1′ en 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 (ℙk−1)n whose rational Chow ring is isomorphic to ℚ[xn]/In,k,n−k. This variety admits a stratification induced by certain positroids. Its strata need not be affine, and their codimension generating function equals Hilb(ℚ[xn]/In,k,n−k;q).
Related
- Bicolored Eulerian quasisymmetric functions, affine lengths, and Theta operators
- The universal property of the decomposition space of quasisymmetric functions
- Chromatic word-quasisymmetric functions of matroids
- Ehrhart positivity for paving positroids
- Keller's conjecture for split finite-dimensional algebras
BibTeX
@misc{zhu2026family,
title = {{A family of algebras that connects positroids, the Theta conjecture, the Delta conjecture, and Eulerian quasisymmetric functions}},
author = {Hai Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21894},
eprint = {2610.21894},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21894},
}
RIS
TY - UNPB
TI - A family of algebras that connects positroids, the Theta conjecture, the Delta conjecture, and Eulerian quasisymmetric functions
AU - Hai Zhu
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21894
UR - https://arxiv.org/abs/2610.21894
AB - We define a family of algebras ℚ[x_n]/I_n,a,b whose Hilbert series, when n≤ a+b≤ 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 S_n-module structure of these algebras by using exact sequences to derive an invertible recursion for their graded Frobenius characteristics when a+b≥ n and then passing to the limit a→∞. In particular, the graded Frobenius characteristic grFrob(ℚ[x_a+b−1]/I_a+b−1,a,b;q)=rev_q(ω((Θ_e_a−1Θ_e_b−1∇ e_1)|_t=0)), where rev_q reverses the q-coefficients and the specialization (Θ_e_a−1Θ_e_b−1∇ e_1)|_t=0 appears in the Theta conjecture. For a,b≥1, we also show that e_a^⊥grFrob(ℚ[x_a+b]/I_a+b,a,b;q)=q^binoma2rev_q(ω((Δ_e_a−1^′ e_b)|_t=0)), where Δ_e_k−1^′ 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 (ℙ^k−1)^n whose rational Chow ring is isomorphic to ℚ[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 Hilb(ℚ[x_n]/I_n,k,n−k;q).
ER -