On The Combinatorics Of Littlewood-Richardson Coefficients

Littlewood-Richardson coefficient is an important object in combinatorics, it also plays a crucial role in algebra and algebraic geometry. In combinatorics, Littlewood-Richardson coefficients arise as the coefficients in the expansion of a skew Schur function in terms of ordinary Schur functions. These coefficients have various combinatorial interpretations, the most well-known one is that the coefficient c_μv^λequals the number of semistandard Young tableaux of shapeλ/μand type v whose reverse reading word is a lattice permutation, which was first discovered by Littlewood and Richardson. In representation theory, they give the multiplicity of the irreducible polynomial representationsφ^λin the tensor product ofφ^μandφ^v. In the theory of algebraic geometry, they are just the coefficients in the product of two Schubert classes.The main results of this thesis are about some properties and the applications of Littlewood-Richardson coefficients in combinatorics. First, we present some properties of Littlewood-Richardson coefficient by employing the method of hive model and give the characterization of the skew Schur functions which are multiplicity-free in their expansions in terms of Schur functions, that is, each of the coefficients in the resulting Schur function expansions is 0 or 1. Second, we demonstrate the applications of Littlewood-Richardson coefficients by studying a q-log-convexity problem. We prove a conjecture on the q-log-convexity of the polynomial sequence (?) in which an important property of Littlewood-Richardson coefficient known as the dual Pieri's rule plays an important role.In Chapter 1, we first give a background on symmetric functions, especially on Schur functions and the Littlewood-Richardson rule, as well as a background on q-log-convexity and q-log-concavity. Then we introduce some notations and definitions which will be used in this thesis.In Chapter 2, we discuss the multiplicity-free problem of skew Schur function expansions by virtue of the hive model. The hive model is a combinatorial device that may be used to determine Littlewood-Richardson coefficients and study their properties. It represents an alternative to the use of the Littlewood-Richardson rule. Resorting to this combinatorial tool, we give the sufficient and necessary conditions for a skew Schur function s_λ/μ being multiplicity-free. One may see a number of advantages of this hive model method, including the fact that it allows a direct proof that all the cases enumerated in our main result are indeed multiplicity-free. Also, the hive model offers some insight into the origin of the breakdown of multiplicity-freeness for expansions of skew Schur functions.In Chapter 3, we employ some properties of Littlewood-Richardson coefficients to prove a conjecture of Liu and Wang on the q-log-convexity of the polynomial sequence (?). The polynomial is the generating function of the lattice of noncrossing partitions of type B on [n], and it also arises in the theory of growth series of the root lattice. For the generating function of the lattice of noncrossing partitions of type A on [n], which is known as the Narayana polynomial, Chen, Wang and Yang have proved its q-log-convexity. By using Pieri's rule and the Jacobi-Trudi identity for Schur functions, we obtain an expansion of a sum of products of elementary symmetric functions in terms of Schur functions with nonnegative coefficients. Then the principal specialization leads to the q-log-convexity. We also prove that a technical condition of Liu and Wang holds for the squares of the binomial coefficients. Hence we deduce that the linear transformation with respect to the triangular array (?) is log-convexity preserving.Finally, we provide a hive model proof of a pair of inequalities on Littlewood-Richardson coefficients in the Appendix.