arXiv subjects

Raymond Chou

Publications and source records attributed to Raymond Chou.

5 recordsLinked to original sources

arXiv2026

Nonvanishing higher Specht polynomials and a construction for three row and hook shape Garsia--Procesi modules

Given a polynomial ring quotient $R=\mathbb{C}[x_1,\ldots,x_n]/I$ with an action of the symmetric group induced by permuting the variables, a higher Specht basis is a collection of bases for each irreducible $\mathfrak{S}_n$-module in its decomposition that mimics the behavior of the classical Specht polynomial construction in the lowest degrees. Higher Specht bases have now been constructed for the coinvariant ring, the full polynomial ring, the rings $R_{n,k}$ appearing in the $t=0$ Delta conjecture, the hook shape Garsia-Haiman modules, and the two-row Garsia-Procesi modules, and more. We establish a general theory for determining when a higher Specht polynomial is nonzero, and give a proof of a conjectural higher Specht basis for all Garsia-Procesi modules in the cases of three row shapes and hook shapes.

math.CO↗

arXiv2026

Torus Equivariant Cohomology for the $Δ$-Springer Fiber

We define a torus $U \subset T = (\mathbb{C}^\times)^K$ which acts on the $Δ$-Springer varieties $Y_{n,λ,s}$ defined by Griffin-Levinson-Woo and give a Borel-style presentation for the equivariant cohomology ring $H^*_U(Y_{n,λ,s})$. Our presentation arises from the orbit harmonics deformation technique, and uses methods of Chou-Matsumura-Rhoades and Abe-Horiguchi.

math.AG↗

arXiv2025

Representation Theoretic Bases for the $Δ$-Springer Module

We give a descent monomial basis of $Δ$-Springer modules $R_{n,λ,s}$, first defined by Griffin. Our construction simultaneously generalizes the descent basis for the Garsia-Procesi module $R_λ$ studied by Carlsson-Chou and Hanada, as well as the descent basis for the generalized coinvariant algebras $R_{n,k}$ studied by Haglund-Rhoades-Shimozono. This basis is deeply connected with a combinatorial object called battery-powered tableaux, introduced by Gillespie-Griffin. We highlight the representation theoretic properties of this monomial basis by using it to give a direct combinatorial proof of the graded Frobenius character of $R_{n,λ,s}$ in terms of battery-powered tableaux, a fact which has only the geometric proof of Gillespie-Griffin. We also conjecture a higher Specht basis of $R_{n,λ,s}$, generalizing the higher Specht basis of the coinvariant ring defined in Ariki-Terasoma-Yamada. This construction coincides with the Gillespie-Rhoades higher Specht basis for $R_{n,k}$. We give a proof for when $λ= (λ_1,λ_2)$ is a partition of two rows.

math.CO↗

arXiv2024

Equivariant cohomology of Grassmannian spanning lines

Given integers $n \geq k \geq d$, let $X_{n,k,d}$ be the moduli space of $n$-tuples of lines $(\ell_1, \dots, \ell_n)$ in $\mathbb{C}^k$ such that $\ell_1 + \cdots + \ell_n$ has dimension $d$. We give a quotient presentation of the torus-equivariant cohomology of $X_{n,k,d}$. The form of this presentation, and in particular the torus parameters appearing therein, will arise from the orbit harmonics method of combinatorial deformation theory.

math.CO↗

arXiv2024

A descent basis for the Garsia-Procesi module

We assign to each Young diagram $λ$ a subset $\mathcal{B}_{λ'}$ of the collection of Garsia-Stanton descent monomials, and prove that it determines a basis of the Garsia-Procesi module $R_λ$, whose graded character is the Hall-Littlewood polynomial $\tilde{H}_λ[X;t]$. This basis is a major index analogue of the basis $\mathcal{B}_λ\subset R_λ$ defined by certain recursions in due to Garsia and Procesi, in the same way that the descent basis is related to the Artin basis of the coinvariant algebra $R_n$, which in fact corresponds to the case when $λ=1^n$. By anti-symmetrizing a subset of this basis with respect to the corresponding Young subgroup under the Springer action, we obtain a basis in the parabolic case, as well as a corresponding formula for the expansion of $\tilde{H}_λ[X;t]$. Despite a similar appearance, it does not appear obvious how to connect these formulas appear to the specialization of the modified Macdonald formula of Haglund, Haiman and Loehr at $q=0$.

math.RT↗