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Higher Specht polynomials under the diagonal action

Maria Gillespie

Abstract

We introduce higher Specht polynomials - analogs of Specht polynomials in higher degrees - in two sets of variables $x_1,\ldots,x_n$ and $y_1,\ldots,y_n$ under the diagonal action of the symmetric group $S_n$. This generalizes the classical Specht polynomial construction in one set of variables, as well as the higher Specht basis for the coinvariant ring $R_n$ due to Ariki, Terasoma, and Yamada, which has the advantage of respecting the decomposition into irreducibles. As our main application of the general theory, we provide a higher Specht basis for the hook shape Garsia--Haiman modules. In the process, we obtain a new formula for their doubly graded Frobenius series in terms of new generalized cocharge statistics on tableaux.

Higher Specht polynomials under the diagonal action

Abstract

We introduce higher Specht polynomials - analogs of Specht polynomials in higher degrees - in two sets of variables and under the diagonal action of the symmetric group . This generalizes the classical Specht polynomial construction in one set of variables, as well as the higher Specht basis for the coinvariant ring due to Ariki, Terasoma, and Yamada, which has the advantage of respecting the decomposition into irreducibles. As our main application of the general theory, we provide a higher Specht basis for the hook shape Garsia--Haiman modules. In the process, we obtain a new formula for their doubly graded Frobenius series in terms of new generalized cocharge statistics on tableaux.
Paper Structure (14 sections, 20 theorems, 62 equations, 4 figures)

This paper contains 14 sections, 20 theorems, 62 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Definitions and background
  4. Tableaux and statistics
  5. A weight-preserving bijection
  6. Symmetric functions and graded Frobenius series
  7. Diagonal coinvariants and Garsia--Haiman modules
  8. Young symmetrizers and one-variable higher Specht polynomials
  9. General constructions in two sets of variables
  10. Conditions for independence
  11. A higher Specht basis for the hook-shape Garsia--Haiman modules
  12. Independence in the full polynomial ring
  13. Independence in
  14. Future directions and observations

Key Result

Theorem 1.4

The polynomials $F_T^S$ form a basis for the one-variable coinvariant ring $R_n=\mathbb{C}[x_1,\ldots,x_n]/(e_1,\ldots,e_n)$, where $e_i$ is the $i$-th elementary symmetric polynomial.

Figures (4)

  • Figure 1: A tableau $T\in \mathrm{Tab}(4,3,1)$ in French notation, and the Specht polynomial $F_T$.
  • Figure 2: From left to right: A semistandard Young tableau, a standard Young tableau, and a standard general tableau.
  • Figure 3: A tableau $S$ at left, and its values of $\mathrm{ccTab}_\mu(S)$ and the negated labels from $\mathrm{ccTab}'_\mu(S)$ superimposed at right. Here $n=16$ and $k=9$, so $n-k+1=8$.
  • Figure 4: A set of higher Specht polynomials for $\mathrm{DR}_3$.

Theorems & Definitions (70)

  • Definition 1.3
  • Theorem 1.4: ATY
  • Definition 1.5
  • Theorem 1.6
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Example 2.6
  • ...and 60 more