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Symmetric Functions

### Class Information

COORDINATES: Classes meet Tuesdays and Thursdays 2:00–3:15 in MSB 411. The registrar calls this Section 001.

PREREQUISITES: Math 5210 or equivalent experience.

MAIN TEXTS: We will mostly be following chapter 7 of Richard P. Stanley: Enumerative Combinatorics 2. A more algebraic (and somewhat denser) treatment is given in I. G. Macdonald's Symmetric Functions and Hall Polynomials Oxford, 1979; 2nd Ed. 1995 whose BAMS review by Stanley gives a useful overview of the subject.

OTHER REFERENCES: Since the theory of symmetric functions lies at the intersection of combinatorics, representation theory, and algebraic geometry, there are many other references that provide useful points of view or connections. A sample are listed below:

WEBPAGE: The homepage for this course is http://www.math.uconn.edu/~troby/Math5020S15/. Some other web resources are listed below.

SOFTWARE: While doing some computations by hand is generally essential for learning, having software that can do bigger computations or check one's work is very useful. The free open-source computer algebra system Sage has an extensively-developed collection of objects and algorithms implemented for algebraic combinatorics with various documentation including thematic tutorials. More generally, Christian Krattenthaler has kindly compiled a comprehensive list of Combinatorial software and databases.

Specifically for symmetric functions, the two Sage tutorials called SF and SFA are good places to start. For Maple users, John Stembridge's excellent SF package is highly recommended.

GRADING: Your grade will be based on several HW assignments, and possibly a presentation given in class.

The text is loaded with exercises, most of them with solutions, though the latter often are brief outlines, sometimes with pointers to the research literature. Stanley's knowledge of the subject is encylopedic, and he was able include a large amount of interesting material this way. If I assign an exercise whose solution is already included, I'll expect much more detail in what you hand in. I recommend you browse through many of the unassigned exercises to get a sense for what's there.

You may find some homework problems to be challenging, and collaborating to find solutions is certainly fine. It's OK to get significant help from other resources, but in the end, please write your own solution in your own words. Copying someone else's work without credit is plagiarism and will be dealt with according to university policy. It also is a poor learning strategy.

ACADEMIC INTEGRITY: Please make sure you are familiar with and abide by The Student Code governing Academic Integrity in Undergraduate Education and Research.

ACKNOWLEDGEMENTS: Many people have influenced my approach to this material, particularly Richard Stanley and Ira Gessel.

CONTENT: The prehistory of symmetric functions goes back to 16th century Italian algebraists, particularly Cardano and Vieta, who were interested in how the roots of a polynomial related to its coefficients. The work of Girard in 1629 presaged the famous identities of Newton several decades later (though neither gave a proof). Many other famous mathematicians worked on symmetric polynomials, including Macaulay, Euler, Cauchy, and Jacobi. Meanwhile the parallel independent development of an equivalent theory was going on in enumerative algebraic geometry in what we now call the Schubert Calculus. That these two subjects were cryptomorphic was not noticed until 1947. All of this early work was in the context of symmetric polynomials in finitely many variables.

The modern approach to symmetric functions in countably many variables and a significant unification of the subject was achieved by Philip Hall in an overlooked 1959 paper. It was published in an obscure journal, so had little impact until R. Stanley's 1971 paper on plane partitions included an exposition of Hall's results. The objects of study are no longer polynomials nor even actual functions, but formal power series in infinitely many variables. This approach has the advantage that many useful results do not depend on the number of variables and can be stated more cleanly this way. It is easy to specialize back to finitely many variables when convenient.

ACCESSIBILITY & DISABILITY ISSUES: Please contact me and UConn's Center for Students with Disabilities as soon as possible if you have any accessibility issues, have a (documented) disability and wish to discuss academic accommodations, or if you would need assistance in the event of an emergency.

HOMEWORK ASSIGNMENTS: Homework will be assigned from this list of Supplementary Exercises (without solutions) provided by Stanley. Some of these problems are challenging, so don't hesitate to ask for a hint if you're stuck, or submit a partial solution.

• [Due 2/13] HW1: # 1, 2, 3, 4, 5, 6, 7, 9
• [Due 3/13] HW2: # 12, 14, 20, 23, 26, 27, 28, 30, 35
• [Due 4/10] HW3:
• [Due 5/01] HW4:

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