The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric FunctionsSpringer Science & Business Media, 2001 M04 20 - 240 pages 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. |
Contents
III | 1 |
IV | 4 |
V | 6 |
VI | 10 |
VII | 13 |
VIII | 18 |
IX | 23 |
X | 30 |
XXXI | 106 |
XXXII | 112 |
XXXIII | 116 |
XXXIV | 121 |
XXXV | 124 |
XXXVI | 132 |
XXXVII | 133 |
XXXVIII | 141 |
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The Symmetric Group: Representations, Combinatorial Algorithms, and ... Bruce E. Sagan Limited preview - 2013 |
Common terms and phrases
A-tableaux action of G basis bijection cell Chapter character table coefficient color column combinatorial compute conjugacy classes consider Corollary corresponding coset cycle Definition denoted entries equation example fact Ferrers diagram finite G-homomorphism G-module given graded poset graph group algebra group G hook formula hooklengths increasing subsequence induction inequivalent irreducible inner product insertion integers inverse irreducible characters irreducible representations isomorphic jeu de taquin Knuth equivalence lattice Lemma length Let G linear Littlewood-Richardson Littlewood-Richardson rule Maschke's theorem matrix representation monomial multiplication notation Note obtain pair partial order path permutation polynomial polytabloids Proof Proposition prove reader representation of G respectively result reverse rim hook Robinson-Schensted algorithm row word Schur functions semistandard tableaux sequence skew slide Specht modules standard tableau subgroup submodule subsets Suppose symmetric functions symmetric group tabloid unimodal vector space verify Young tableau zero
Popular passages
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.