Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces. A representation makes an abstract algebraic object concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication. Representation theory is a powerful tool because it reduces problems in abstract algebra to problems in linear algebra, which is well understood. Successively pushed forward throughout the past decades, the concept of modeling representation theoretical invariants in a combinatorial, computational and geometric manner is a main strand of modern representation theory with many applications in numerous areas of mathematics and in other basic sciences as well. This is why representation theory has become the crossroad of many subjects and ideas in mathematics, including geometry, linear algebra, combinatorics and mathematical physics. The inspiration for the current research comes from the connections of the representation theory of symmetric groups to the representation theory of Kac-Moody algebras. These connections include new algorithms developed to study important invariants, such as the crystal decomposition numbers for the quantized affine Lie algebras. These invariants are very important in the related fields of mathematical physics, with their main applications being related to the phenomena of black holes.

In our research we will focus on the construction of new algorithms for computation of the crystal decomposition matrices for certain classes of quantized affine Lie algebras. Since these matrices are not defined over a field, but rather over rings of polynomials, it is necessary to develop different approaches from the classical ones when dealing with this kind of mathematical objects. The starting point for our research will be the basic representation of the quantized affine Lie algebra of type A, which is known as the Fock space. Because these objects are very large in terms of computational power needed to deal with them, we will attempt to find more efficient algorithms for their computation than the ones used at the moment, such as the LLT algorithm. Our approach to the above families of algebras involves representation theoretical, combinatorial, homological, geometrical and computational methods. These methods represent a combination of classical methods, and new methods originating in Kac-Moody algebras and quantum groups.