Simple SL( n)-modules with normal closures of maximal torus orbits
Karine Kuyumzhiyan
DOI: 10.1007/s10801-009-0175-2
Abstract
Let T be the subgroup of diagonal matrices in the group SL( n). The aim of this paper is to find all finite-dimensional simple rational SL( n)-modules V with the following property: for each point v\in V the closure [ `( Tv)] \overline{Tv} of its T-orbit is a normal affine variety. Moreover, for any SL( n)-module without this property a T-orbit with non-normal closure is constructed. The proof is purely combinatorial: it deals with the set of weights of simple SL( n)-modules. The saturation property is checked for each subset in the set of weights.
Pages: 515–538
Keywords: keywords toric variety; normality; saturation
Full Text: PDF
References
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3. Carrell, J.B., Kurth, A.: Normality of torus orbit closures in G/P . J. Algebra 233, 122-134 (2000) J Algebr Comb (2009) 30: 515-538
4. Chindris, C.: Orbit semigroups and the representation type of quivers.
5. Dabrowski, R.: On normality of the closure of a generic torus orbit in G/P . Pac. J. Math. 192(2), 321-330 (1996)
6. Fulton, W.: Introduction to Toric Varieties. Princeton University Press, Princeton (1993)
7. Harary, F.: Graph Theory. Addison-Wesley, Reading (1969)
8. Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. GTM, vol.
9. Springer, Berlin (1978)
9. Kemph, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings I. LNM, vol.
339. Springer, Berlin (1973)
10. Klyachko, A.A.: Toric varieties and flag varieties. Tr. Mat. Inst. Steklova 208, 139-162 (1995) (in Russian), English transl.: Proc. Steklov Inst. Math. 208 (1995), 124-145
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