Young Tableaux: With Applications to Representation Theory and Geometry
Young tableaux are fillings of the boxes of diagrams that correspond to partitions with positive integers, that are strictly increasing down columns and weakly increasing along rows. The aim of this book is to develop the combinatorics of Young tableaux and to show them in action in the algebra of symmetric functions, the representations of the symmetric and general linear groups, and the geometry of flag varieties. Many of these applications have not been available in book form.
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Bumping and sliding
Words the plactic monoid
Increasing sequences proofs of the claims
The RobinsonSchenstedKnuth correspondence
The LittlewoodRichardson rule
Representations of the symmetric group
Representations of the general linear group
Schubert varieties and polynomials
Appendix A Combinatorial variations
Appendix B On the topology of algebraic varieties
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action algebraic array balls basic basis boxes bumping bundle called canonical Chapter closed column complex conjugate consider consists construction contains coordinate corner Corollary corresponding defined definition denote describe determines dimension dual elementary elements embedding entries equal equation exactly example Exercise fact fixed flag follows formula functions given gives homomorphism ideal identity increasing integers interchanging irreducible isomorphism Knuth equivalent Lemma length letters linear Littlewood-Richardson rule manifold matrix Note obtained occurs pair particular partition permutation polynomials position preceding projective Proof Proposition prove relations representation result reverse ring Schubert Schur seen sequence shape Show skew tableau slide space spanned standard tableau strictly subset subvariety suffices Suppose symmetric takes theorem unique variety vector weight word write zero