Monomial stability of Frobenius images

TL;DR

This work develops a general, basis-aware framework for representation stability of FI-modules by analyzing stability in symmetric-function bases. It proves that stabilizing Schur coefficients of homogeneous symmetric functions is equivalent to stabilizing monomial coefficients and supplies explicit transfer criteria between bases, enabling practical stability analysis from monomial expansions. The authors then apply these results to key combinatorial objects: diagonal and multi-graded diagonal coinvariant algebras, and Garsia-Haiman modules, obtaining new stable ranges and refining known stability phenomena via monomial formulas (e.g., Kostka numbers, special rim-hooks, and Haglund–Haiman–Loehr constructions). The approach yields concrete stable ranges for bi-graded components and clarifies how stability transfers across bases, offering a unified route to derive representation stability from monomial data and to compute stable ranges from monomial expansions.

Abstract

We study representation stability in the sense of Church, Ellenberg, and Farb \cite{FI-module} through the lens of symmetric function theory and the different symmetric function bases. We show that a sequence, $(F_n)_n$, where $F_n$ is a homogeneous symmetric function of degree $n$, has stabilizing Schur coefficients if and only if it has stabilizing monomial coefficients. More generally, we develop a framework for checking when stabilizing coefficients transfer from one symmetric function basis to another. We also see how one may compute representation stable ranges from the monomial expansions of the $F_n$.\parspace As applications, we reprove and refine the representation stability of diagonal coinvariant algebras, $DR_n$. We also observe new representation stability phenomena of the Garsia-Haiman modules. This establishes certain stability properties of the modified Macdonald polynomials, $\tilde{H}_{μ^{(n)}}[X;q,t]$ and the modified $q,t$-Kostka numbers, $\tilde{K}_{μ^{(n)},ν[n]}(q,t)$, for arbitrary sequences of partitions with $μ^{(n)}\vdash n$ and $μ^{(n)}\subseteq μ^{(n+1)}$.

Figures (7)

• Figure 1: A brick tableau of shape $\lambda=(5,3^2)$ and content $\mu=(3,2^3,1^2)$. The weight is $1\cdot 3\cdot 2=6$.
• Figure 2: An example of special rim hook tableau of shape $\lambda=(5,3,3)$ and content $\mu=(6,4,1)$, with the special rim hooks highlighted in red,blue, and green. The sign is $(-1)^2=1$.
• Figure 3: An application of $\varphi_{11}$ to an element of $L^{(11)}$ with $\lambda[11]=(6,4,1)$ and $\mu[11]=(5,3,3)$. The sign of both special rim hook tableau is $-1$.
• Figure 4: A labeled Dyck path, $P$, with reading word $\sigma(P)=543612$, which is a $\alpha$-shuffle for $\alpha=(2,1,1,2)$ as well as all refinements of this composition. The area of $P$ is 3 and $\text{dinv}(P)=6$ with the following label pairs contributing diagonal inversions: $(2,6),(1,6),(3,4),(4,5),(3,5),(1,3)$. Hence $P$ contributes to the monomial coefficients of $m_{(2^2,1^2)},\,m_{(2,1^3)},\,m_{(1^4)}$ in $[DR_6]_{(5,3)}$.
• Figure 5: The labeled Dyck path obtained from the one in Fig \ref{['labeled Dyck path']} after applying $\psi_6$.

Theorems & Definitions (54)

• Definition 2.0.1
• Lemma 2.1.1
• proof
• Remark 2.1.1
• Corollary 2.1.1
• proof
• Lemma 2.1.2
• proof
• Corollary 2.2.1
• proof