Inclusion relation among permutation representations of S6

Abstract

Permutation representations arise from the action of groups on sets. Every conjugacy class C of a finite group G has a transitive action on G:g.x=gxg-1 for xɛC and gɛG. In this paper we consider the group S6, group of all permutations on 6 symbols and completely describe the inter relationship among permutation representations arising from all its conjugacy classes. We have achieved a complete specification of these permutation representations: how they are mutually disposed to each other. For this purpose, we define subordination relationship between two conjugacy classes.Given two conjugacy classes C1 and C2, we say that C1 is subordinate to C2 if the permutation representation corresponding to C1 is a subrepresentation included in that of C2. As in whole of Mathematics relationship that are expressed in the form of functions will have special significance when these functions turn out to be injective. Our subordinate relationship naturally captures these significant cases. This relation is reflexive and transitive. Though not antisymmetric it has many desirable properties. The Hasse diagram has a very nice visual symmetry. In this paper we determine which conjugacy classes are subordinate to which for the case of the group G=S6. Our method involves averaging process as used in Reynold’s operator. Another important feature employed in our methodology is using duality which converts the problem of constructing an injective homomorphism to constructing a surjective homomorphism in the opposite direction.