The ring of sections of a complete symmetric variety

Chirivì, Rocco; Maffei, Andrea

doi:10.1016/s0021-8693(02)00676-2

Cited by 14 publications

(21 citation statements)

References 16 publications

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“…In this case, V (̟) G θ is one-dimensional and θ(̟) = −̟, so ̟ belongs to χ(S) R . One can show that set of dominant weights of R G,θ is the set of spherical weights and that C + is the intersection of [ChMa03], Theorem 2.3 or [T06], Proposition 26.4). We say that a spherical weight is regular if it is strictly dominant as weight of the restricted root system.…”

Section: Restricted Root Systemmentioning

confidence: 99%

Fano Symmetric Varieties with Low Rank

Ruzzi¹

2012

Publ. Res. Inst. Math. Sci.

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The symmetric projective varieties of rank one are all smooth and Fano by a classic result of Akhiezer. We classify the locally factorial (respectively smooth) projective symmetric G-varieties of rank 2 which are Fano. When G is semisimple we classify also the locally factorial (respectively smooth) projective symmetric G-varieties of rank 2 which are only quasi-Fano. Moreover, we classify the Fano symmetric G-varieties of rank 3 obtainable from a wonderful variety by a sequence of blow-ups along G-stable varieties. Finally, we classify the Fano symmetric varieties of arbitrary rank which are obtainable from a wonderful variety by a sequence of blow-ups along closed orbits.

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Section: Restricted Root Systemmentioning

confidence: 99%

Fano Symmetric Varieties with Low Rank

Ruzzi¹

2012

Publ. Res. Inst. Math. Sci.

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“…De Concini and Procesi [7] defined the wonderful compactificationX of G/H wherē H is the normalizer of H. In [3] the total ring of sections Γ = ⊕ M∈Pic(X) Γ(X, M) and a canonical set of generators for these rings had been introduced. The computation of the relations among these generators is equivalent to the computation of the relations in the ring k[G/H] above.…”

Section: Richardson Variety R Inmentioning

confidence: 99%

Equations Defining Symmetric Varieties and Affine Grassmannians

Chirivì¹,

Littelmann²,

Maffei³

2008

International Mathematics Research Notices

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Abstract. Let σ be a simple involution of an algebraic semisimple group G and let H be the subgroup of G of points fixed by σ. If the restricted root system is of type A, C or BC and G is simply connected or if the restricted root system is of type B and G is adjoint, then we describe a standard monomial theory and the equations for the coordinate ring k[G/H] using the standard monomial theory and the Plücker relations of an appropriate (maybe infinite dimensional) Grassmann variety.The aim of this paper is the description of the coordinate ring of the symmetric varieties and of certain rings related to their wonderful compactification. The main tool to achieve this goal is a (possibly infinite dimensional) Grassmann variety associated to a pair consisting of a symmetric space and a spherical representation.More precisely, let G be a semisimple algebraic group over an algebraically closed field k of characteristic 0 and let σ be a simple involution of G (i.e. G ⋊{id, σ} acts irreducibly on the Lie algebra of G). Let H = G σ be the fixed point subgroup. Fix a spherical dominant weight ε in Ω + . We add a node n 0 to the Dynkin diagram of G and, for all simple roots α, we join n 0 with the node n α of the simple root α by ε(α ∨ ) lines, and we put an arrow in direction of n α if ε(α ∨ ) ≥ 2. In the cases relevant for us, the Kac-Moody group e G associated to the extended diagram will be of finite or affine type. Let L be the ample generator of Pic(Gr) for the generalized Grassmann varietyP . The homogeneous coordinate ring Γ Gr = j≥0 Γ(Gr, L j ) is the quotient of the symmetric algebra S(Γ(Gr, L)) by an ideal generated by quadratic relations, the generalized Plücker relations.Since our aim is to relate these Plücker relations to k[G/H], we say that the monoid Ω + is quadratic if (it is free and) its basis has the following property with respect to the dominant order of the restricted root system: any element of Ω + that is less than the

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“…Given a dominant weight λ of G, we denote by V (λ) the irreducible representation of highest weight λ. See [ChMa03], Theorem 2.3 or [T06], Proposition 26.4 for an explicit description of the dominant weights of R G,θ . They are called spherical weights and are also dominant weights of R G .…”

Section: Notationmentioning

confidence: 99%