Two Interactions Between Combinatorics and Representation Theory :

This thesis consists of two independent parts. The first part concerns Stanley's symmetric function generalization of the chromatic polynomial, the series of immanant conjectures made by Stembridge, and Zaslavsky's theory of signed graphs. The conjectures made by Stembridge that the so-called "monomial" immanants are nonnegative on totally positive matrices and monomial positive on Jacobi-Trudi matrices are shown to hold for several infinite families of these monomial immanants. We make use of results of Stembridge and Goulden-Jackson which reduce both of these conjectures to a statement about some elements in the group algebra of the symmetric group. We also show that a more general conjecture by Stembridge concerning acyclic digraphs with certain path-intersection properties can be reduced to the conjectures considered above. A particular consequence is that the result of Greene that (ordinary) immanants of Jacobi-Trudi matrices are monomial positive can be extended to a wider class of combinatorially-defined matrices satisfying some simple conditions. Some new relationships between these conjectures and graph coloring are developed. Analogues of Stanley's chromatic symmetric function are given for signed graphs, and their basic properties are studied. There are some interesting connections with hyperplane arrangements. These also imply some new results about ordinary graphs. The second part concerns the so-called "Hodge-type" decompositions of Hochschild (co)homology. Let A be a commutative algebra over a field of characteristic zero, and M be a symmetric A-bimodule. Gerstenhaber and Schack have shown that there are decompositions H,(A, M) = ekHk,n-k(A, M), Hn(A, M) = (DkHk,n-k(A, M) of the Hochschild (co)homology. The first summands, H 1, 1(A, M) and H 1"n -1 (A, M), are known to be the Harrison (co)homology defined in terms of shuffles. We discuss interpretations of the decompositions in terms of k-shuffles and how these relate to versions of the Poincard-Birkoff-Witt theorem. We then turn to a detailed study of how the decomposition behaves with respect to the Gerstenhaber operations (cup and Lie products) in cohomology. We show by example that neither product is generally graded, but that .Fq = E,>q H*,'(A, A) are ideals for both products with Tp U .T C .Fp+q and [7p, .Fq] C Fp+q. The statements for the cup product were conjectured in this form by Gerstenhaber and Schack. The results in the second part were obtained in collaboration with Nantel Bergeron. Thesis Supervisor: Richard P. Stanley Title: Professor of Applied Mathematics