The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions
I have been very gratified by the response to the first edition, which has resulted in it being sold out. This put some pressure on me to come out with a second edition and now, finally, here it is. The original text has stayed much the same, the major change being in the treatment of the hook formula which is now based on the beautiful Novelli-Pak-Stoyanovskii bijection (NPS 97]. I have also added a chapter on applications of the material from the first edition. This includes Stanley's theory of differential posets (Stn 88, Stn 90] and Fomin's related concept of growths (Fom 86, Fom 94, Fom 95], which extends some of the combinatorics of Sn-representations. Next come a couple of sections showing how groups acting on posets give rise to interesting representations that can be used to prove unimodality results (Stn 82]. Finally, we discuss Stanley's symmetric function analogue of the chromatic polynomial of a graph (Stn 95, Stn ta]. I would like to thank all the people, too numerous to mention, who pointed out typos in the first edition. My computer has been severely reprimanded for making them. Thanks also go to Christian Krattenthaler, Tom Roby, and Richard Stanley, all of whom read portions of the new material and gave me their comments. Finally, I would like to give my heartfelt thanks to my editor at Springer, Ina Lindemann, who has been very supportive and helpful through various difficult times.
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The Symmetric Group: Representations, Combinatorial Algorithms, and ...
Bruce E. Sagan
Limited preview - 2013
action acts algebra algorithm appears applying associated basis bijection called cell Chapter character coefficient color column complete composition compute conjugacy classes consider construct contains Corollary corresponding covered cycle defined Definition denoted determinant diagram dual easy elements entries equal equation equivalence example Exercise fact Finally finite fixed formula G-module give given hook increasing induction inner product insertion integers Introduction irreducible isomorphic Knuth Lemma length linear matrix module multiplication Note obtain operator pair partial partition path permutation poset positive possible Proof Proposition prove reader relation replaced representation respectively result reverse rule satisfies semistandard sequence shape skew slide standard submodule subsequence suffices Suppose symmetric functions tableau tabloid Theorem Theory trivial vector space verify weakly write yields Young zero
Page 225 - Calderbank, P. Hanlon and RW Robinson, Partitions into even and odd block size and some unusual characters of the symmetric groups, Proc.
Page 227 - AP Hillman and RM Grassl, Reverse plane partitions and tableau hook numbers, J. Combin. Theory Ser. A 21 (1976), 216-221. [Knl] DE Knuth, Permutations, matrices, and generalized Young tableaux, Pacific J. Math. 34 (1970), 709-727. [Kul] JPS Kung, "Young Tableaux in Combinatorics, Invariant Theory, and Algebra," Academic Press, New York, 1982.