Lifting a 5-dimensional representation of $M_{11}$ to a complex unitary representation of a certain amalgam

TL;DR

This paper constructs a complex unitary representation of an amalgam from a 5-dimensional representation of M11 and explores its reductions modulo various primes, advancing understanding of representations of complex groups.

Contribution

It introduces a method to lift a 5-dimensional representation of M11 to a complex unitary representation of a specific amalgam, and analyzes its reductions mod p.

Findings

Successfully lifted the representation to a complex unitary form.

Analyzed the reduction behavior modulo different odd primes.

Provided new insights into the structure of representations of the amalgam.

Abstract

We lift the $5$-dimensional characteristic $3$ representation of $M_{11}$ to a complex representation of the amalgam ${\rm GL}(2,3)*_{D_8}S_{4}$, and consider its reduction (mod $p$) for other odd primes.