Solutions for network games and symmetric group representations

Pfaffians and Representations of the Symmetric Group

Pfaffians of matrices with entries z[i, j]/(xi + xj), or determinants of matrices with entries z[i, j]/(xi−xj), where the antisymmetrical indeterminates z[i, j] satisfy the Plücker relations, can be identified with a trace in an irreducible representation of a product of two symmetric groups. Using Young’s orthogonal bases, one can write explicit expressions of such Pfaffians and determinants, ...

The symmetric group - representations, combinatorial algorithms, and symmetric functions

Application of the Representations of Symmetric Groups to Characterize Solutions of Games in Partition Function Form

A different perspective from the more “traditional” approaches to studying solutions of games in partition function form has been presented. We provide a decomposition of the space of such games under the action of the symmetric group, for the cases with three and four players. In particular, we identify all the irreducible subspaces that are relevant to the study of linear symmetric solutions....

Irreducible Representations of the Symmetric Group

We construct the Specht modules and prove that they completely characterize the irreducible representations of the symmetric group. We will prove certain properties of these representations using combinatorial tools (such as calculating the dimension using Hook’s length formula). Only an introductory knowledge of group theory and linear algebra will be assumed and representation theory concepts...

The Representations of the Symmetric Group

In this paper we classify the irreducible representations of the symmetric group Sn and give a proof of the hook formula for the dimension of each irreducible. 1. The Irreducible Representations of Sn We construct the irreducible representations of the symmetric group. The number of irreducible representations of Sn is the number of conjugacy classes of Sn, which is the number of partitions of ...