Monomial stability of Frobenius images
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912395800084480 |
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| author | Borisov, Nikita |
| author_facet | Borisov, Nikita |
| contents | 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 $μ^{(n)}\subseteq μ^{(n+1)}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_04950 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Monomial stability of Frobenius images Borisov, Nikita Combinatorics Representation Theory 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 $μ^{(n)}\subseteq μ^{(n+1)}$. |
| title | Monomial stability of Frobenius images |
| topic | Combinatorics Representation Theory |
| url | https://arxiv.org/abs/2503.04950 |