Using the theory of representations of the symmetric group, we propose an algorithm to compute the invariant ring of a permutation group. Our approach have the goal to reduce the amount of linear algebra computations and exploit a thinner combinatorial description of the invariant ring. Computational Invariant Theory, representation theory, permutation group This is a drafty old version : full ...

Symmetry has sung its siren song to Physicists since the beginning of time, or at least since before there were Physicists. Today the ideas of symmetry are incorporated into a subject with the less imaginative and suggestive name of Group Theory. This Chapter introduces many of the ideas of group theory that are important in the natural sciences. Natural philosophers in the past have come up wi...

The first version of these notes was written for a first-year graduate algebra course. As in most such courses, the notes concentrated on abstract groups and, in particular, on finite groups. However, it is not as abstract groups that most mathematicians encounter groups, but rather as algebraic groups, topological groups, or Lie groups, and it is not just the groups themselves that are of inte...

The notion of modular arithmetic is related to that of the remainder in Euclidean division. The operation of finding the remainder is sometimes referred to as the modulo operation, and denoted with ”mod” used as an infix operator. For example, the remainder of the division of 16 by 12 is denoted by 16 mod 12; as this remainder is 4, we have 16 mod 12 = 4. The congruence, indicated by ”≡” follow...