# On orbital variety closures in sln III. Geometric properties

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## Abstract

This is the third paper in the series. Here we define a few combinatorial orders on Young tableaux. The first order is obtained from the induced Duflo order by the extension with the help of Vogan Tα, β procedure. We call it the Duflo-Vogan order. The second order is obtained from the generalization of Spaltenstein's construction by consideration of an orbital variety as a double chain of nilpotent orbits. We call it the chain order. Again, we use Vogan's Tα, β procedure, however, this time to restrict the chain order. We call it the Vogan-chain order. The order on Young tableaux defined by the inclusion of orbital variety closures is called the geometric order and the order on Young tableaux defined by inverse inclusion of primitive ideals is called the algebraic order. We get the following relations between the orders: the Duflo-Vogan order is an extension of the induced Duflo order; the algebraic order is an extension of the Duflo-Vogan order; the geometric order is an extension of the algebraic order; the Vogan-chain order is an extension of the geometric order; and, finally, the chain order is an extension of the Vogan-chain order. The computations show that the Duflo-Vogan and the Vogan-chain orders coincide on sln for n ≤ 9 and in n = 10 there is one case (up to Tα, β procedure and transposition) where the chain-Vogan order is a proper extension of the Duflo-Vogan order. In this only case the algebraic order coincides with the Vogan-chain order. These computations permit us to conjecture that in sln the algebraic order coincides with the geometric order. As well we conjecture that the combinatorics of both the inclusions on primitive ideals and on orbital variety closures is defined by the Vogan-chain order on Young tableaux.

Original language English 68-97 30 Journal of Algebra 305 1 https://doi.org/10.1016/j.jalgebra.2006.01.010 Published - 1 Nov 2006

## ASJC Scopus subject areas

• Algebra and Number Theory

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