Semi-magic matrices for dihedral groups

TL;DR

This paper explores the structure of semi-magic matrices associated with dihedral groups using representation theory, providing new algebraic tools and generating functions for enumerating these matrices with fixed line sums.

Contribution

It introduces an intertwining operator linking the dihedral group algebra to semi-magic matrices and derives generating functions for their enumeration, extending the algebraic framework beyond circulant matrices.

Findings

Derived a generating function for semi-magic matrices with fixed line sum.

Established an algebra extending circulant matrices related to dihedral groups.

Connected representation theory with combinatorial enumeration of semi-magic squares.

Abstract

After reviewing the group structure and representation theory for the dihedral group $D_{2n},$ we consider an intertwining operator $\Phi_\rho$ from the group algebra $\mathbb{C}[D_{2n}]$ into a corresponding space of semi-magic matrices. From this intertwining operator, one obtains the generating function for enumerating the associated semi-magic squares with fixed line sum and an algebra extending the circulant matrices. While this work complements the approach to $D_{2n}$ through permutation polytopes, we use only methods from representation theory.