For a nilpotent class in a semi-simple Lie algebra that is well determined by its weight Dynkin diagram, according to Bala-Carter, how can I determine the associated nilpotent matrix? For example: which nilpotent element (matrix) $e\in so_{7}(C)=b_3$ associated with (WDD 1----0----1)?
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I don't understand why you refer here to Bala-Carter, since their method provides a different approach to the traditional Dynkin classification and is formulated for semisimple groups (over algebraically closed fields in most characteristics) rather than for semisimple Lie algebras (in characteristic 0). Aside from that, for classical types you should study Chapter 5 of the 1993 book Nilpotent Orbits in Semisimple Lie Algebras by Collingwood and McGovern. There the Dynkin method is made explicit in terms of the way nilpotent matrices look for the possible weighted Dynkin diagrams, using the Jacobson/Morozov embedding of a nilpotent into a 3-dimensional simple Lie algebra. But it does take some work to correlate nilpotent matrices with weighted diagrams, since the diagonal element of a TDS intervenes in an essential way.
ADDED: Concerning your specific example for type $B_3$, there are just 7 nilpotent orbits here of dimensions 18, 16, 14, 12, 10, 8, 0, arranged linearly in the usual partial ordering. Here the partitions of 7 involved are just those with an even number of even parts. From the data you give, I expect the nilpotent element you are interested in has Jordan blocks of sizes 3, 2, 2. This partition $[3,2^2]$ corresponds to the middle orbit, of dimension 12. Here the alternative parametrization you give in terms of pairs of partitions would involve the singletons $[2]$, $[3]$. (This has to be checked more carefully but is likely to fit the Dynkin diagram labelling you specify.)
answered Mar 26, 2017 at 0:05
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$\begingroup$ Thank you Jim, I'm who asked question. Using the Bala-Carter theorem we can associate to this unipotent class a Levi subgroup of $G$ and a distinguished parabolic subgroup of the Levi subgroup in (P. Bala and R.W. Carter. Classes of unipotent elements 1 and 2) the WDD 1--0--1 is the $A_2\oplus B_3$ class ( here we have pair of partition calss (2,3 )). we use the notation of Nathan Jacobson in Lie Algebras book, we have (cont.) $\endgroup$ Commented Mar 26, 2017 at 15:58
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$\begingroup$ $\mathfrak{b}_3=so_7(\mathbb{C})=\star H_1 \oplus \star H_2 \oplus \star H_3 \oplus \star G_{12}^{+} \oplus\star G_{13}^{+} \oplus\star G_{23}^{+} \oplus \star G_{21}^{+} \oplus\star G_{31}^{+} \oplus\star G_{32}^{+} \oplus \star G_{12}^{-} \oplus\star G_{13}^{-} \oplus\star G_{23}^{-} \oplus \star G_{21}^{-} \oplus \star G_{31}^{-} \oplus\star G_{32}^{-}\oplus \star D_{1}^{+}\oplus \star D_{2}^{+}\oplus \star D_{3}^{+}\oplus \star D_{1}^{-}\oplus \star D_{2}^{-}\oplus \star D_{3}^{-} $ Which nilpotent element $e$ here represent 1--0--1 for example? i need to understand all this. $\endgroup$ Commented Mar 26, 2017 at 15:58