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Higher Specht bases for generalizations of the coinvariant ring

Maria Gillespie, Brendon Rhoades

Abstract

The classical coinvariant ring $R_n$ is defined as the quotient of a polynomial ring in $n$ variables by the positive-degree $S_n$-invariants. It has a known basis that respects the decomposition of $R_n$ into irreducible $S_n$-modules, consisting of the higher specht polynomials due to Ariki, Terasoma, and Yamada. We provide an extension of the higher Specht basis to the generalized coinvariant rings $R_{n,k}$. We also give a conjectured higher Specht basis for the Garsia-Procesi modules $R_μ$, and provide a proof of the conjecture in the case of two-row partition shapes $μ$. We then combine these results to give a higher Specht basis for an infinite subfamily of the modules $R_{n,k,μ}$ recently defined by Griffin, which are a common generalization of $R_{n,k}$ and $R_μ$.

Higher Specht bases for generalizations of the coinvariant ring

Abstract

The classical coinvariant ring is defined as the quotient of a polynomial ring in variables by the positive-degree -invariants. It has a known basis that respects the decomposition of into irreducible -modules, consisting of the higher specht polynomials due to Ariki, Terasoma, and Yamada. We provide an extension of the higher Specht basis to the generalized coinvariant rings . We also give a conjectured higher Specht basis for the Garsia-Procesi modules , and provide a proof of the conjecture in the case of two-row partition shapes . We then combine these results to give a higher Specht basis for an infinite subfamily of the modules recently defined by Griffin, which are a common generalization of and .

Paper Structure

This paper contains 14 sections, 16 theorems, 100 equations, 4 figures.

Table of Contents

  1. Introduction and Background
  2. Higher Specht polynomials for the coinvariant ring
  3. The rings $R_{n,k}$
  4. The rings $R_\mu$
  5. The rings $R_{n,k,\mu}$
  6. Main results
  7. Outline
  8. Acknowledgments
  9. Higher Specht bases for $R_{n,k}$
  10. Higher Specht bases for $R_\mu$
  11. Semistandard higher Specht modules in ${\mathbb {Q}}[{\mathbf {x}}_n]$
  12. Independence in $R_\mu$ for two-row shapes
  13. Beyond two-row shapes
  14. A higher Specht basis for $R_{n,k,(n-1)}$

Key Result

Theorem 1.7

Let $k \leq n$ be positive integers. Consider the set of polynomials where $T, S \in {\mathrm {SYT}}(n)$ have the same shape and $(i_1, i_2, \dots, i_{n-k})$ is a tuple of $n-k$ nonnegative integers whose sum is $< k - {\mathrm {des}}(S)$. The set ${\mathcal{B}}_{n,k}$ descends to a higher Specht basis for $R_{n,k}$.

Figures (4)

  • Figure 1: A standard Young tableau $T$ of partition shape $\lambda=(3,3,1)$.
  • Figure 2: A standard Young tableau $S$ at left, with its cocharge labels shown at right.
  • Figure 3: A standard Young tableau $S$ (at left) and its destandardization $S'$ (middle). The tableau $R$ defined from $S'$ in the proof of Lemma \ref{['enumeration-lemma']} arising from the tuple $(i_1,\ldots,i_8)=(0,1,0,0,2,0,1,0)$ is shown at right.
  • Figure 4: The transition matrix that expresses the elements of $\mathcal{B}_{(3,3)}^{(2)}$ (the row labels) in terms of those of $\mathcal{C}_{(3,3)}^{(2)}=\mathcal{B}_{(3,2)}^{(2)}\cup x_6 \mathcal{B}_{(3,2)}^{(1)}$ (the column labels).

Theorems & Definitions (47)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Example 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Conjecture 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 37 more