Geometric invariant theory and stretched Kostka quasi-polynomials
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| Format: | Preprint |
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2024
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| _version_ | 1866917914322403328 |
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| author | Besson, Marc Jeralds, Sam Kiers, Joshua |
| author_facet | Besson, Marc Jeralds, Sam Kiers, Joshua |
| contents | For $G$ a semisimple, simply-connected complex algebraic group and two dominant integral weights $λ, μ$, we consider the dimensions of weight spaces $V_λ(μ)$ of weight $μ$ in the irreducible, finite-dimensional highest weight $λ$ representation. For natural numbers $N$, the function $N \mapsto \dim V_{Nλ}(Nμ)$ is a quasi-polynomial in $N$, the stretched Kostka quasi-polynomial. Using methods of geometric invariant theory (GIT), we realize the degree of this quasi-polynomial as the dimension of a certain GIT quotient. As a result, we resolve a conjecture of Gao and Gao on an explicit formula for this degree. We also discuss periods of this quasi-polynomial determined by the GIT approach, and give computational evidence supporting a geometric determination of the minimal period. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_01651 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometric invariant theory and stretched Kostka quasi-polynomials Besson, Marc Jeralds, Sam Kiers, Joshua Representation Theory Algebraic Geometry 22E46, 17B10, 14L24, 14M15 For $G$ a semisimple, simply-connected complex algebraic group and two dominant integral weights $λ, μ$, we consider the dimensions of weight spaces $V_λ(μ)$ of weight $μ$ in the irreducible, finite-dimensional highest weight $λ$ representation. For natural numbers $N$, the function $N \mapsto \dim V_{Nλ}(Nμ)$ is a quasi-polynomial in $N$, the stretched Kostka quasi-polynomial. Using methods of geometric invariant theory (GIT), we realize the degree of this quasi-polynomial as the dimension of a certain GIT quotient. As a result, we resolve a conjecture of Gao and Gao on an explicit formula for this degree. We also discuss periods of this quasi-polynomial determined by the GIT approach, and give computational evidence supporting a geometric determination of the minimal period. |
| title | Geometric invariant theory and stretched Kostka quasi-polynomials |
| topic | Representation Theory Algebraic Geometry 22E46, 17B10, 14L24, 14M15 |
| url | https://arxiv.org/abs/2412.01651 |