A partial order on Littlewood–Richardson tableaux

For a nilpotent linear transformation f:V→V of a vector space V of Jordan type λ=(λ1,…,λl), i.e. the sizes of the Jordan blocks equal to the conjugate of λ, let X(λ,d) be the set of f-stable subspaces of dimension d. X(λ,d) is partitioned indexed by Littlewood–Richardson tableaux (LR-tableau) T of shapes λ/μ's with |λ/μ|=d, each of which corresponds to the set S(V,T) consisting of W∈X(λ,d) such that the dimension of the vector space fr−1V∩ft−1W/〈frV∩ft−1W,fr−1V∩ftW〉 is equal to the number of cells (squares) in the rth row of T filled with the letter t for all r,t⩾1. We define a partial order on the set LR(λ,d) of LR-tableaux of shapes λ/μ's with |λ/μ|=d by T⩾T′ if S(V,T′) is contained in the closure of S(V,T) in the Grassmaniann G(V,d). We give a certain sufficient condition (Corollary C), and a necessary condition (Corollary D) for T⩾T′. For this we introduce the notions of generic vectors, the generic subspace WT, the rational function field FT, and the LR-tableau associated to a tableau T, and then show; the map π:S(V,T)→S(V,Tlet>1), π(W)=fW, where Tlet>1 is the subtableau of an LR-tableau T consisting of the cells filled with the letters greater than 1, is a fiber bundle with fiber isomorphic to a union of Schubert cells and S(V,T) is nonsingular (Theorem A); the generic subspace WT of a tableau T is a generic point of whose function field isomorphic to the rational function field associated to the LR-tableau (Theorem B).