Towards a standard monomial theory for exceptional groups

Lakshmibai, V.; Rajeswari, K. N.

doi:10.1090/conm/088/999999

Cited by 7 publications

(5 citation statements)

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“…Recall that a symmetric determinantal variety is . It is irreducible, normal, and [ 13 , 6.2.5]. Set .…”

Section: On the Intersections Of With Andmentioning

confidence: 99%

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Projections of the minimal nilpotent orbit in a simple Lie algebra and secant varieties

Panyushev¹

2023

Math. Proc. Camb. Phil. Soc.

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Let G be a simple algebraic group with ${\mathfrak g}={\textrm{Lie }} G$ and ${\mathcal O}_{\textsf{min}}\subset{\mathfrak g}$ the minimal nilpotent orbit. For a ${\mathbb Z}_2$ -grading ${\mathfrak g}={\mathfrak g}_0\oplus{\mathfrak g}_1$ , let $G_0$ be a connected subgroup of G with ${\textrm{Lie }} G_0={\mathfrak g}_0$ . We study the $G_0$ -equivariant projections $\varphi\,:\,\overline{{\mathcal O}_{\textsf{min}}}\to {\mathfrak g}_0$ and $\psi:\overline{{\mathcal O}_{\textsf{min}}}\to{\mathfrak g}_1$ . It is shown that the properties of $\overline{\varphi({\mathcal O}_{\textsf{min}})}$ and $\overline{\psi({\mathcal O}_{\textsf{min}})}$ essentially depend on whether the intersection ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1$ is empty or not. If ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1\ne\varnothing$ , then both $\overline{\varphi({\mathcal O}_{\textsf{min}})}$ and $\overline{\psi({\mathcal O}_{\textsf{min}})}$ contain a 1-parameter family of closed $G_0$ -orbits, while if ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1=\varnothing$ , then both are $G_0$ -prehomogeneous. We prove that $\overline{G{\cdot}\varphi({\mathcal O}_{\textsf{min}})}=\overline{G{\cdot}\psi({\mathcal O}_{\textsf{min}})}$ . Moreover, if ${\mathcal O}_{\textsf{min}}\cap{\mathfrak g}_1\ne\varnothing$ , then this common variety is the affine cone over the secant variety of ${\mathbb P}({\mathcal O}_{\textsf{min}})\subset{\mathbb P}({\mathfrak g})$ . As a digression, we obtain some invariant-theoretic results on the affine cone over the secant variety of the minimal orbit in an arbitrary simple G-module. In conclusion, we discuss more general projections that are related to either arbitrary reductive subalgebras of ${\mathfrak g}$ in place of ${\mathfrak g}_0$ or spherical nilpotent G-orbits in place of ${\mathcal O}_{\textsf{min}}$ .

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“…Recall that a symmetric determinantal variety is . It is irreducible, normal, and [ 13 , 6.2.5]. Set .…”

Section: On the Intersections Of With Andmentioning

confidence: 99%

“…Set . Using the interpretation of as the categorical quotient , where [ 13 , theorem 12·1·7·2], one readily realises that there is a normal irreducible hypersurface such that . Hence the latter is also irreducible and normal, with dimension one less.…”

Section: On the Intersections Of With Andmentioning

confidence: 99%

Projections of the minimal nilpotent orbit in a simple Lie algebra and secant varieties

Panyushev¹

2023

Math. Proc. Camb. Phil. Soc.

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“…We simply define μ = μ w . The defining equations of (opposite) Schubert varieties, as subvarieties of P( m C n ) via the Plücker embedding, are just given by (see, e.g., [28,Subsection 4.3.4])…”

Section: Toric Degeneration Of Schubert Varietiesmentioning

confidence: 99%

W-translated Schubert divisors and transversal intersections

Hwang

Lee

et al. 2022

Sci. China Math.

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We study the toric degeneration of Weyl group translated Schubert divisors of a partial flag variety F n 1 ,...,n k ;n via Gelfand-Cetlin polytopes. We propose a conjecture that Schubert varieties of appropriate dimensions intersect transversally up to translation by Weyl group elements, and verify it in various cases, including the complex Grassmannian Gr(2, n) and the complete flag variety F 1,2,3;4 .

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“…Evaluating coefficients gives the invariantf 12357 . Using the (1, 2, 3) (3,4,5,7,9) with c 24 = 0. Since r 5 = r p 2 1 , we conclude that r 5 = r 1 .…”

Section: The Generic Casementioning

confidence: 99%