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Algebraic Combinatorics
The academic and research portion of the 2013 MSRI-UP will be led by Prof. Rosa Orellana from Dartmouth College. Professor Orellana has supervised over 30 undergraduate student research projects, several of which resulted in senior thesis containing original research. Many of her students have continued their mathematical education in PhD programs.
Algebraic combinatorics is an area of mathematics that studies objects that have combinatorial and algebraic properties. An example of such object is the ring of symmetric functions. In algebraic combinatorics, we use algebraic methods to answer combinatorial questions, and conversely, apply combinatorial techniques to problems in algebra.
Let be commuting variables, a polynomial is symmetric if = _{ }for all permutations . The space of all symmetric polynomials forms a ring, . This simply says that if we multiply two symmetric functions we get another symmetric function. has several distinguished bases that are indexed by partitions. One of the most important bases is Schur's basis: . The objective of the summer is to learn about and work on open problems involving symmetric polynomials.
Sample research project I. In 1995 Stanley described a symmetric polynomial associated to a graph. This polynomial is called the symmetric chromatic polynomial. Here the variables represent distinct colors and the monomials in the polynomial correspond to proper colorings. To obtain the symmetric polynomial we sum all monomials corresponding to proper colorings. A hard open problem is a conjecture of Stanley that says that the symmetric polynomial is an invariant of trees. Until recently it was not known if the chromatic symmetric polynomial was also an invariant of unicyclic graphs. Geofrey Scott (Ph.D. student at the Univ. of Michigan) found a method to construct non-isomorphic graphs containing a triangle that have the same chromatic polynomial. Students in this research project will generalize Scott's results to unicylic graphs containing larger cycles and discover properties under which two unicyclic graphs will have the same chromatic polynomial.
Sample research project II. A symmetric polynomial is called Schur positive if when written as a linear combination of Schur polynomials all the coefficients are nonnegative. Students will study examples of symmetric polynomials of the form and determine conditions so that such a product is Schur positive.