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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, representations of the symmetric and general linear groups, and the geometry of flag varieties.
With his usual lucidity, Fulton brings together the surprisingly wide area of mathematics concerned with Young tableaux. These are combinatorial patterns which index basis vectors of group representations (either of the symmetric group or the general linear group). These vectors can be seen as Plucker coordinate functions on non-linear representations, namely homogeneous spaces (Grassmannians and flag varieties). Thus, Young tableaux form an invaluable tool to examine these representations and varieties in concrete detail. Fulton also gives a good exposition of the combinatorial operations on tableaux which reflect the crystal basis structure from quantum GL(n), though Fulton does not explicitly discuss quantum groups. Other good expositions of these topics, from a more algebraic and combinatorial point of view, are Sagan's newly revised "The Symmetric Group", and Stanley's "Enumerative Combinatorics", Vol 2.
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