Representation theory of the symmetric group

Polynomials are an age-old subject. Defined in terms of a base ring and finitely many indeterminates, polynomials are obtained by using only a finite number of additions and multiplications of the elements of the base ring and the indeterminates. Symmetry is a fundamental concept of mathematics. At the heart of symmetry one wants to understand what remains invariant under a group action. Symmetric polynomials are the polynomials which are invariant under the permutation action of the symmetric group. Catalan numbers is a sequence of positive integers which enumerate seemingly unrelated combinatorial objects. This thesis is devoted to two subjects. First is the verification of an algorithm to calculate the expansion coefficients of a product of a Jack polynomial $J_\mu$ and a power-sum symmetric polynomial $p_n$ in the Jack polynomial basis for various partitions $\mu$ and integers $n$. The second is a new bijection between two sets of objects enumerated by Catalan numbers, solving an earlier conjecture in the paper "From Parking Functions to Gelfand Pairs" by K. Aker and M. B. Can in 2012. Symmetric polynomials appear naturally in mathematics. For instance, the coefficients of a polynomial are symmetric polynomials of the roots of the given polynomials. These symmetric polynomials are called elementary symmetric polynomials, $e_n$. Summing up all monomials of a fixed total degree produces complete symmetric polynomials, $h_n$. Summing up the $n$-th power of indeterminates produces the power-sum symmetric polynomials, $p_n$. Symmetrizing monomials produces monomial symmetric polynomials, $m_\lambda$, where $\lambda$ is any partition. In fact, the monomial symmetric polynomials form a basis of the ring of symmetric polynomials. By extending the definitions of elementary, complete and power-sum symmetric polynomials appropriately, each of theses sets of polynomials constitute a basis for the symmetric polynomials. In addition to these four bases, Schur polynomials, originally defined as the quotient of two determinants, form a fifth basis. Schur polynomials happen to be the most coveted basis among these five classical basis of symmetric polynomials. In representation theory, they correspond to the irreducible characters of the symmetric group. It is possible to study representation theory of symmetric groups by studying Schur polynomials. When dealing with symmetric polynomials, it is best to work with infinitely many indeterminates to avoid accidental identities and to clarify the underlying structure. In terms of modern algebra, a symmetric function is an element of the symmetric function ring and the symmetric function ring is defined as the projective limit of rings of symmetric polynomials as the number of indeterminates tend to infinity. Alternatively, a symmetric function can be thought of as a symmetric polynomial with finitely many indeterminates where the number of indeterminates can change. From this point of view, the crucial bit is that any identity involving symmetric functions must be valid as the number of indeterminates changes. The product in the polynomials is so easy -- the product of two monomials is again a monomial, add up the corresponding exponents -- that we hardly think about it twice. How about the product in the ring of symmetric functions? What are the structure constants for the ring of symmetric functions? This turns out to be either rather straightforward to be interesting as in the case of polynomials, or rather hard. For elementary, complete, power-sum symmetric functions, the answer is straightforward akin to the product of two monomials. The structure constants for the product of the two monomial symmetric functions is only obtained in 2001 in "A MAPLE program for calculations with Schur functions" by M.J. Carvalho, S. D'Agostino. The structure constants for the product of two Schur functions is calculated by the Littlewood-Richardson rule in the paper "Group Characters and Algebra" by D. E. Littlewood and A. R. Richardson. First stated by D. E. Littlewood and A. R. Richardson in 1934, it took 4 decades for complete proofs to show up. How about multiplying elements in different basis? How about multiplying Schur functions with other basis elements expanding in the Schur basis again? The product of a Schur function with a complete symmetric function is calculated by the Pieri rule. The product of a Schur function with the power-sum symmetric functions is calculated by the Murnaghan-Nakayama rule and it calculates the values of the irreducible characters of the symmetric group. Various other symmetric functions arose in mathematics in the twentieth century, such as Hall polynomials, zonal polynomials, Jack polynomials, Macdonald polynomials etc. They arose as the invariant polynomials for action of some group. For instance, zonal polynomials were first introduced in relation to orthogonal groups. These functions can be seen as extensions of Schur functions. Schur functions, when expanded in terms of monomial functions, have triangular matrices. The set of Schur functions is orthonormal. When a normalization is imposed, Schur functions are uniquely defined by these properties. The above functions, albeit defined typically over a different base ring, can be characterized similarly. This is the approach followed in Macdonald's book. In this thesis, our first consideration is to study the product of Jack polynomials with power-sum symmetric polynomials in terms of Jack polynomials. We search how to calculate the coefficients of this product and compute these coefficients step by step. Firstly, we prove that the product of an arbitrary Jack polynomial with a power sum symmetric function $p_1$ in the basis of Jack polynomials. Secondly, we prove that this product for the power sum symmetric function $p_2$. Then we find the formulas of the product of an arbitrary Jack polynomial with a power sum symmetric function for some cases. Thus the formula of the product $J_\mu\,p_n$ for any partition $\mu$ is generalized partially. Catalan numbers pop up in all parts of mathematics. They enumerate a variety of different mathematical objects which seem unrelated at first impression. In the second problem, we prove a conjecture about the equality of two generating functions described in the paper "From Parking Functions to Gelfand Pairs" by K. Aker and M. B. Can in 2012 attached to two sets whose cardinalities are given by Catalan numbers: We establish a new combinatorial bijection between the two sets which proves the equality of the corresponding generating functions.