The Symmetric Group:Representations, Combinatorial Algorithms, and Symmetric Functions

Graduate Texts in Mathematics #203

by Carl Sagan

This text is an introduction to the representation theory of the symmetric group from three different points of view: via general representation theory, via combinatorial algorithms, and via symmetric functions. It is the only book to deal with all three aspects of this subject at once. The style of presentation is relaxed yet rigorous and the prerequisites have been kept to a minimum¿undergraduate courses in linear algebra and group theory will suffice.

Genres: Mathematics, Algebra

197 pages, Hardcover

First published February 28, 1991

About the author

In 1934, scientist Carl Sagan was born in Brooklyn, N.Y. After earning bachelor and master's degrees at Cornell, Sagan earned a double doctorate at the University of Chicago in 1960. He became professor of astronomy and space science and director of the Laboratory for Planetary Studies at Cornell University, and co-founder of the Planetary Society. A great popularizer of science, Sagan produced the PBS series, "Cosmos," which was Emmy and Peabody award-winning, and was watched by 500 million people in 60 countries. A book of the same title came out in 1980, and was on The New York Times bestseller list for 7 weeks. Sagan was author, co-author or editor of 20 books, including The Dragons of Eden (1977), which won a Pulitzer, Pale Blue Dot (1995) and The Demon-Haunted World: Science As a Candle in the Dark (1996), his hardest-hitting on religion. With his wife, Ann Druyan, he was co-producer of the popular motion picture, "Contact," which featured a feminist, atheist protagonist played by Jodie Foster (1997). The film came out after Sagan's death, following a 2-year struggle with a bone marrow disease. Sagan played a leading role in NASA's Mariner, Viking, Voyager, and Galileo expeditions to other planets. Ann Druyan, in the epilogue to Sagan's last book, Billions and Billions: Thoughts on Life and Death at the Brink of the Millennium (published posthumously in 1997), gives a moving account of Carl's last days: "Contrary to the fantasies of the fundamentalists, there was no deathbed conversion, no last minute refuge taken in a comforting vision of a heaven or an afterlife. For Carl, what mattered most was what was true, not merely what would make us feel better. Even at this moment when anyone would be forgiven for turning away from the reality of our situation, Carl was unflinching. As we looked deeply into each other's eyes, it was with a shared conviction that our wondrous life together was ending forever."

For his work, Dr. Sagan received the NASA medals for Exceptional Scientific Achievement and (twice) for Distinguished Public Service, as well as the NASA Apollo Achievement Award. Asteroid 2709 Sagan is named after him. He was also awarded the John F. Kennedy Astronautics Award of the American Astronautical Society, the Explorers Club 75th Anniversary Award, the Konstantin Tsiolkovsky Medal of the Soviet Cosmonauts Federation, and the Masursky Award of the American Astronomical Society, ("for his extraordinary contributions to the development of planetary science…As a scientist trained in both astronomy and biology, Dr. Sagan has made seminal contributions to the study of planetary atmospheres, planetary surfaces, the history of the Earth, and exobiology. Many of the most productive planetary scientists working today are his present and former students and associates").

He was also a recipient of the Public Welfare Medal, the highest award of the National Academy of Sciences.

Dr. Sagan was elected Chairman of the Division of Planetary Sciences of the American Astronomical Society, President of the Planetology Section of the American Geophysical Union, and Chairman of the Astronomy Section of the American Association for the Advancement of Science. For twelve years he was the editor-in-chief of Icarus, the leading professional journal devoted to planetary research. He was cofounder and President of the Planetary Society, a 100,000-member organization that is the largest space-interest group in the world; and Distinguished Visiting Scientist, Jet Propulsion Laboratory, California Institute of Technology.

In their posthumous award to Dr. Sagan of their highest honor, the National Science Foundation declared that his "research transformed planetary science… his gifts to mankind were infinite." D. 1996.

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Reviews

[Chris · Rating 3 out of 5]

I refuse to tolerate any mathematician who says "All of science"---get over yourself---but these specialty topics (jeu de taquin, schur lemma, group characters) are ably covered and deserve wider recognition.


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By the way, ETS (the people who process childrens' test scores to prevent them from getting ahead in life) don't know this material, and should. Read upon linking, aligning, scores, scaling, Rasch models, and "equating", and you'll find that there's nothing to it but nonparametric guesswork bullshit. The theory of the symmetric group is quite deep but expositions of it are, unfortunately, hidden away.

The chances that psychometricians will read this review and act on my advice are nil, so prepare for another century of your kids' test scores based in pseudoscience.
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That said, Sagan covers a ton of interesting material. Who knows if it will apply to your area, but


The symmetric group as a topic demonstrates what mathematics is like generally. Technically, the symmetric group is "just a" subset of linear algebra, considering 0/1 matrices which are almost entirely sparse except for a single 1 in each row/column, like:

0 1 0
1 0 0
0 0 1

. However, contemplating the general linear group as shuffles, and asking how these could be, e.g., factored, leads somewhere entirely different.





Interesting subject matter, writing leaves much to be desired.