A partial order on Littlewood-Richardson tableaux
Let \(f:V\rightarrow V\) be a nilpotent linear transformation on a vector space \(V\) over \(\mathbb C\) of Jordan type \(\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_\ell)\), i.e. the conjugate of \(\lambda\) gives the sizes of the Jordan blocks. For an integer \(d\) such that \(0<d<n=\text{dim}\,V\), let \(X(\lambda,d)\) be the set of \(f\)-stable subspaces \(W\) of dimension \(d\); \(X(\lambda,d)\) is partitioned by certain sets \(S(V,T)\), which are indexed by the set \(\text{LR}(\lambda,d)\) of Littlewood-Richardson tableaux \(T\) of shape \(\lambda/\mu\) with \(|\lambda/\mu|=d\). The main of object of study is a partial order on \(\text{LR}(\lambda,d)\) defined in terms of the closure \(X(V,T)\) of \(S(V,T)\) in the Grassmannian \(G(V,d)\): \(T\geq T'\) if \(X(V,T)\supset X(V,T')\). Given a (semistandard) skew tableau \(T\), we let \(T_{\text{col}\leq s}\) (resp. \(T_{\text{row}\leq s}\)) denote the subtableau consisting of those cells of \(T\) whose column (resp. row) numbers are \(\leq s\), and let \(T^{\text{let}>1}\) denote the subtableau of cells of \(T\) filled with letters \(>1\). The author introduces the notions of generic vectors, the generic subspace \(W_T\) and the rational function field \(F_T\) associated to \(T\). An important construction is to apply a succession of coplactic operations (replacing a letter \(t\) in certain cells by \(t-1\)) to \(T\) to obtain an LR-tableau \(\hat{T}\). The main results are: Theorem A, which asserts that the map \(\pi:S(V,T)\rightarrow S(V,T^{\text{let}>1}), \pi(W)=fW\), is a fibre bundle with fibre isomorphic to a union of Schubert cells, and \(S(V,T)\) is nonsingular; and Theorem B, which asserts that the generic subspace \(W_T\) of a tableau \(T\) is a generic point of \(S(V,\hat{T})\), whose function field is isomorphic to the rational function field \(F_{\hat{T}}\). From these the author deduces the following necessary condition on the partial order (Corollary D): Suppose \(T\geq T'\). If \(|T_{\text{col}\leq s}|=|T'_{\text{col}\leq s}|\), then \(\hat{T}_{\text{col}\leq s}\geq \hat{T}'_{\text{col}\leq s}\). (There are similar conditions on analogous subtableaux of \(T\) and \(T'\).) He also obtains a sufficient condition (Corollary C): Suppose \(T\) is an LR-tableau with cell \(e\) filled with letter \(t\), and \(T'\) is obtained by replacing cell \(e\) by a cell \(f\) filled with letter \(t-1\). If \(T_{<\text{col}\,e}\) is LR, then \(T>T'\). (Here \(T_{<\text{col}\,e}\) denotes the subtableau of \(T\) consisting of cells whose column numbers are less than the column number of cell \(e\).) Several examples are included to illustrate the key concepts and results. The article concludes with a question to which an affirmative answer would allow the order relations among LR-tableaux of a fixed shape to be determined.
scientific article; zbMATH DE number 5303128