The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric FunctionsSpringer Science & Business Media, 2013 M03 9 - 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
1 | |
Combinatorial Algorithms | 3 |
Young Subgroups Tableaux and Tabloids | 53 |
Symmetric Functions | 141 |
10 | 146 |
13 | 156 |
18 | 164 |
Other editions - View all
The Symmetric Group: Representations, Combinatorial Algorithms, and ... Bruce Sagan Limited preview - 2001 |
Common terms and phrases
action of G basis bijection cell character table coefficient color combinatorial compute conjugacy classes consider Corollary corresponding coset defining representation Definition denoted dual Knuth entries equation equivalence class example fact Ferrers diagram finite G-homomorphism G-module given graded poset group algebra group G hook formula hooklengths increasing subsequence induction inequivalent irreducible inner product insertion integers irreducible characters irreducible representations isomorphic jeu de taquin Knuth equivalence Knuth relations Lemma length Let G linear Littlewood-Richardson rule Maschke's theorem matrix representation monomial multiplication notation Note obtain pair partial order plane partitions polynomial polytabloids Proof Proposition reader representation of G respectively result 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 Theory trivial unimodal vector space verify X-tableaux Young tableau zero