Abstract
Let $k \leq n$ be nonnegative integers and let $\lambda$ be a partition of $k$. S. Griffin recently introduced a quotient $R_{n,\lambda}$ of the polynomial ring $\mathbb{Q}[x_1, \dots, x_n]$ in $n$ variables which simultaneously generalizes the Delta Conjecture coinvariant rings of Haglund-Rhoades-Shimozono and the cohomology rings of Springer fibers studied by Tanisaki and Garsia-Procesi. We describe the space $V_{n,\lambda}$ of harmonics attached to $R_{n,\lambda}$ and produce a harmonic basis of $R_{n,\lambda}$ indexed by certain ordered set partitions $\mathcal{OP}_{n,\lambda}$. The combinatorics of this basis is governed by a new extension of the Lehmer code of a permutation to $\mathcal{OP}_{n, \lambda}$.
Highlights
In his Ph.D. thesis [5], Sean Griffin introduced the following remarkable family of quotients of the polynomial ring Q[xn] := Q[x1, . . . , xn] in n variables
We describe a generating set for the harmonic space Vn,λ as a Q[xn]-module (Theorem 15) and give an explicit harmonic basis {δσ : σ ∈ OPn,λ} of Rn,λ indexed by ordered set partitions in OPn,λ (Theorem 16)
Observe that trailing zeros have no effect on Young diagrams, so this would be the Young diagram of the partition (4, 2, 1, 0, 0)
Summary
In his Ph.D. thesis [5], Sean Griffin introduced the following remarkable family of quotients of the polynomial ring Q[xn] := Q[x1, . . . , xn] in n variables. In order to motivate harmonic spaces and bases, we recall some technical issues that arise in the study of quotient rings. For a homogeneous ideal I ⊆ Q[xn], the harmonic space V of I is the graded subspace of Q[xn] given by. The harmonic space V permits the study of the quotient ring R without the computational issues inherent in cosets. Given k n and a partition λ = (λ1 · · · λs) of k, let OPn,λ be the collection of length s sequences σ = (B1 | · · · | Bs) of subsets of [n] such that. We describe a generating set for the harmonic space Vn,λ as a Q[xn]-module (Theorem 15) and give an explicit harmonic basis {δσ : σ ∈ OPn,λ} of Rn,λ indexed by ordered set partitions in OPn,λ (Theorem 16).
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