# representation theory – Characterization of nilpotent elements even in \$ mathfrak {sl} _n \$

According to Collingwood-McGovern (1993, Corollary 3.8.8), an element is even if and all the labels of its weighted Dynkin diagram are 0 or 2. For $$[d_1,dots,d_k,0,dots,0]$$ these labels are the $$h_i-h_ {i + 1}$$, or $$h_1 geqslant h_2 geqslant dots geqslant h_n$$ is a reorganization of $$bigcup_ {i = 1} ^ k {d_i-1, d_i-3, dots, -d_i + 1 }$$ (ibid., Lemma 3.6.4).