Flag varieties and interpretations of Young tableau algorithms. (English) Zbl 0979.14025
From the introduction: The Schensted algorithm, which defines a bijective correspondence between permutations and pairs of (standard) Young tableaux, and the Schützenberger (or evacuation) algorithm, which defines a shape-preserving involution of the set of Young tableaux, can both be described using doubly indexed families of partitions that satisfy certain local rules, as described by H. A. A. Van Leeuwen [Electrons J. Comb. 3, No. 2, 391-422 (1996; Zbl 0852.05080)]. In this paper we show how both correspondences occur in relation to questions concerning varieties of flags stabilized by a fixed nilpotent transformation \(\eta\). The mentioned doubly indexed families of partitions arise very naturally in this context, and they provide detailed information concerning the internal steps of the algorithms, rather than just about the correspondences defined by them. As a consequence, the study of the Schützenberger algorithm also leads to an interpretation of jeu de taquin and of the Littlewood-Richardson coefficients. The connections between geometry and combinatorics presented here include and extend results due to R. Steinberg [J. Algebra 113, No. 2, 523-528 (1988; Zbl 0653.20039) and Invent. Math. 36, 209-224 (1976; Zbl 0352.20035)] and W. H. Hesselink [Compos. Math. 55, 89-133 (1985; Zbl 0579.15011)].
The basic fact underlying these interpretations is that the irreducible components of the variety \({\mathcal F}_\eta\) of \(\eta\)-stable complete flags is parametrized in a natural way by the set of standard Young tableaux of shape equal to the Jordan type \(J(\eta)\) of \(\eta\). In fact there are two dual parametrizations, and we show that the transition between them is given by the Schützenberger involution. Taking a projection on varieties of incomplete flags by forgetting the parts of flags below a certain dimension, we obtain from this an interpretation of jeu de taquin (operating on skew standard tableaux) and a bijection between Littlewood-Richardson tableaux and irreducible components of a variety of \(\eta\)-stable subspaces of fixed type and cotype.
The other interpretations involve the Robinson-Schensted algorithm and relative positions of flags. We give a derivation of Steinberg’s result that the relative position of two generically chosen flags in given irreducible components of \({\mathcal F}_\eta\) is the permutation related by the Robinson-Schensted correspondence to the pair of the tableaux parametrizing those components.
The basic fact underlying these interpretations is that the irreducible components of the variety \({\mathcal F}_\eta\) of \(\eta\)-stable complete flags is parametrized in a natural way by the set of standard Young tableaux of shape equal to the Jordan type \(J(\eta)\) of \(\eta\). In fact there are two dual parametrizations, and we show that the transition between them is given by the Schützenberger involution. Taking a projection on varieties of incomplete flags by forgetting the parts of flags below a certain dimension, we obtain from this an interpretation of jeu de taquin (operating on skew standard tableaux) and a bijection between Littlewood-Richardson tableaux and irreducible components of a variety of \(\eta\)-stable subspaces of fixed type and cotype.
The other interpretations involve the Robinson-Schensted algorithm and relative positions of flags. We give a derivation of Steinberg’s result that the relative position of two generically chosen flags in given irreducible components of \({\mathcal F}_\eta\) is the permutation related by the Robinson-Schensted correspondence to the pair of the tableaux parametrizing those components.
MSC:
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 05E10 | Combinatorial aspects of representation theory |
Keywords:
Young tableaux; Littlewood-Richardson tableaux; Robinson-Schensted algorithm; positions of flagsReferences:
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