On the irreducibility of the extensions of Burau and Gassner representations

Publication date: 18 October 2020

Abstract: We study the

n

th degree representations

h a t h o_{G}

of

C b_{n}

and

h a t h o_{B}

of

C_{n}

, defined by Valerij G. Bardakov, where

C b_{n}

is the group of basis conjugating automorphisms and

C_{n}

is the group of conjugating automorphisms. We prove that

h a t h o_{G}

is reducible and its

(n - 1)

th degree composition factor

h a t p h i_{G}

is irreducible if and only if

t_{i} e q 1

for all

1 l e q i l e q n

. Also we prove that

h a t h o_{B}

is reducible and its

(n - 1)

th degree composition factor

h a t p h i_{B}

is irreducible if and only if

t e q 1

. Moreover, for

n = 3

, we prove that

h a t p h i_{G} (t_{1}, t_{2}, t_{3}) o t i m e s h a t p h i_{G} (m_{1}, m_{2}, m_{3})

is irreducible if and only if

(t_{1}, t_{2}, t_{3})

and

(m_{1}, m_{2}, m_{3})

are distinct vectors, and the representation

h a t p h i_{B} (t) o t i m e s h a t p h i_{B} (m)

is irreducible if and only if

t e q m

.

This page was built for publication: On the irreducibility of the extensions of Burau and Gassner representations