Semistandard \(k\)-tableaux: Covering relations (Q1275303)

The semistandard \(k\)-tableaux parametrize a decomposition of the variety \(G_k^u\) of \(k\)-dimensional subspaces of a fixed vector space which are fixed by a given nilpotent endomorphism \(u\) [see \textit{N. Shimomura}, J. Math. Soc. Japan 37, No. 3, 537-556 (1985, Zbl 0566.14023)]. In this paper, the authors compute the dimension of \(G_k^u\) by means of a greedy algorithm, by characterising the covering relations of the natural partial order on the set of semistandard \(k\)-tableaux---this leads to a proof that the dimension of the partial order is equal to the dimension of the variety \(G_k^u\). The proof of these results is combinatorial, and involves careful consideration of so-called `elementary operations' on semistandard \(k\)-tableaux and use of their properties as developed in \textit{J. Martínez-Bernal} [Bol. Soc. Mat. Mex., III. Ser. 4, No. 2, 203-210 (1998; Zbl 0926.05036)]. The authors also show how to use this result to prove that if the Jordan partition of \(u\) is of the form \(\lambda=(p,p, \ldots ,p)\), then the corresponding poset is lexicographically shellable, generalising the well-known case when \(u\) is zero (i.e. \(p=0\)).