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Given a representation , there is an action of \( \operatorname {\mathrm {GL}}(n_1)\times \operatorname {\mathrm {GL}}(s)\) on \({X= \operatorname {\mathrm {P}}( \operatorname {\mathrm {Hom}}(\mathbb {C}^{n_1},\mathbb {C}^{n_2}))}\) via \((g_1,g_2)\cdot [f]=[\rho (g_2)\circ f\circ g_1^{-1}]\) . We explicitly prove that there is a correspondence between semistable points \(x\in X\) and semistable points in the Grassmanian of \(n_1\) -planes in \(\mathbb {C}^{n_2}\) . This is done by computing the Hilbert-Mumford weight in the two different scenarios.

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Computations on the Hilbert-Mumford Criterion for Grassmanians

  • 会议论文

  • 发表日期：

    2025年

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  • Cesare Goretti

摘要

Given a representation , there is an action of \( \operatorname {\mathrm {GL}}(n_1)\times \operatorname {\mathrm {GL}}(s)\) on \({X= \operatorname {\mathrm {P}}( \operatorname {\mathrm {Hom}}(\mathbb {C}^{n_1},\mathbb {C}^{n_2}))}\) via \((g_1,g_2)\cdot [f]=[\rho (g_2)\circ f\circ g_1^{-1}]\) . We explicitly prove that there is a correspondence between semistable points \(x\in X\) and semistable points in the Grassmanian of \(n_1\) -planes in \(\mathbb {C}^{n_2}\) . This is done by computing the Hilbert-Mumford weight in the two different scenarios.