Bruhat Cells and Double Bruhat Cells for Gln : An interplay of the symmetric group and matrices

Some people think mathematics is nothing more than adding numbers. In fact, it is more about studying all kinds of abstract structures. Granted, often in the hope that it leads to some numbers we can add! A beautiful example of such a structure is the symmetric group and this is really the backbone of this book. The object is needed to eventually introduce Bruhat cells, a subject that is still very much alive in the world of research. However, on its own, the symmetric group is very interesting as well, as it tends to pop up everywhere in mathematics and physics and is often used as a 'tool' in the various fields. Although this may all sound very vague and sketchy, this book is really aiming at the opposite; using mathematically rigorous methods to guide you every step of the way. In here you will find correct and complete proves where possible alongside many examples to help you develop the needed intuition. Although this book is originally a bachelor's thesis, you only need basic knowledge of matrices and groups to fully understand it.