## On irreducible representations of Sn

##### Abstract

As mentioned in [8], A group representation can be thought of as an action
oi a group G on some vector space. Such actions arise naturally in many
branches oi mathematics and physics and as such it is important to study
and understand the theory of representations. Cayley's theorem asserts that
everv finite group can be embedded in a group of symmetries Sn for some n
- this makes the studies related to the symmetric group of much significance
since by the isomorphism guaranteed by the Cayley’s theorem we may un
derstand the properties of any other finite group.
A* an illustration, the solution space of a differential equation in a 3-dimensional
?pace having a rotational symmetry has its solution space invariant under ro
tations. Thus the space of solutions will constitute a representation of the
rotation group 50(3). If we know what all of the representations of 50(3)
are, then this can help immensely in narrowing down what the space of so
lutions can be.
In fact, one of the chief applications of representation theory is to exploit
symmetry. If a system has symmetry, then the set of symmetries will form a
group, and understanding the representations of the symmetry group allows
us to use that symmetry to simplify the problem
This report focuses on the representation theory of symmetric groups and
in particular on the construction of -ill irreducible modules of the symmet-
lic group 5;i 'Otherwise known as the Specht Modules). There are numerous
applications of representation of groups and in particular tiie representa
tion of the symmetric group. For instance, they arise in physics, probability
v
and statistics, topological graph theory, the theory of partially ordered sets
amongst many other areas.
This report contains 5 chapters - in chapter 1 we have provided elementary
definitions as well as basic results about the symmetric group Sn. Chapter 2
is on the ordinary representation of finite group with some small emphasis on
the symmetric group Sn. It should be noted that this chapter is not exhaus
tive as far as representation theory is concerned, we have only mentioned
some useful aspects that is in direct reference to the main objective of this
report. In chapter 3 we have dwelt on the construction of the ordinary irre
ducible modules of the symmetric group Sn popularly known as the Specht
modules, and a simple example for the case n = 3 has been given.
In representation theory of finite groups, it is useful to know which ordi
nary irreducible representations remain irreducible when reduced modulo a
prime p. In chapter 4. we have traced the history of classification of ordinary
irreducible modules that remain irreducible modulo p.

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