A partial order on Littlewood-Richardson tableaux
type: Item
instance of: scholarly article
title: A partial order on Littlewood-Richardson tableaux
zbMATH Open document ID: 1157.05052
author: Takashi Maeda
published in: Journal of Algebra
publication date: 2008-07-21T00:00:00Z
review text: 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}\le…
reviewed by: M. Rafiq Omar
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 14M15
zbMATH DE Number: 5303128
zbMATH Keywords: Jordan blocks
zbMATH Keywords: Grassmannians
zbMATH Keywords: Littlewood-Richardson tableaux
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.01.044
OpenAlex ID: W1992097048
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DOI: 10.1016/J.JALGEBRA.2008.01.044
description: scientific article; zbMATH DE number 5303128