Normality of Torus Orbit Closures in G/P

Abstract

The purpose of this note is to classify the torus orbit closures in an arbitrary algebraic homogeneous space G/P that are toric varieties.

Simple SL(n)-modules with normal closures of maximal torus orbits

“…Indeed, the cases (7, 2) and (8, 3) coincide with Examples 3.1 and 3.2 respectively. The cases (7,3) and (9,4) are the particular cases of n = 2k + 1 (Example 3.4). The case (9, 2) can be obtained from (7, 2) (Example 3.1) using the Step procedure.…”

“…It is well known that M(V ) contains the only maximal element λ with respect to , it is called the highest weight of the module. The weight λ is dominant; moreover, for any dominant weight λ ∈ X(T ) there exists a unique simple SL(n)-module V (λ) with the highest weight λ (see [16,Chapter 4,§3,Thm.11] or [8, § §20-21]). The role of the Weyl group W is played here by the permutation group S n , which acts on Z n by permutations of coordinates.…”

“…Group Highest weight G-module SL (2) 3π 1 S 3 k 2 SL (2) 4π 1 S 4 k 2 SL (3) 2π 1 S 2 k 3 SL(4) π 2 2 k 4 SL(5) π 2 2 k 5 SL (6) π 2 2 k 6 SL (6) π 3 3 k 6…”

Projective normality of torus quotients of flag varieties

“…327]. In [3], Carrell and Kurth proved that if G is of type A n , D 4 or B 2 and P is any maximal parabolic subgroup of G, then every T orbit closure in G/P is normal. In the context of a problem on projective normality for torus actions, Howard proved that for any parabolic subgroup P of SL n (C), the Zariski closure T.x of the T -orbit of any point x in SL n (C)/P is projectively normal for the choice of any ample line bundle L on SL n (C)/P .…”

“…In this paper we show that the quotient T \\G/P α 1 ∩ P α 2 is projectively normal with respect to the descent of a suitable line bundle, where P α i is the maximal parabolic subgroup in G associated to the simple root α i , i = 1, 2. We give a degree bound of the generators of the homogeneous coordinate ring of T \\(G 3,6 ) ss T (L 2̟ 3 ). For G is of type B 3 , we give a degree bound of the generators of the homogeneous coordinate ring of T \\(G/P α 2 ) ss T (L 2̟ 2 ) whereas we prove that T \\(G/P α 3 ) ss T (L 4̟ 3 ) is projectively normal with respect to the descent of the line bundles L 4̟ 3 .…”