I am reading Ryba, A natural invariant algebra for the Baby Monster group, Journal of Group Theory, 2007, https://doi.org/10.1515/JGT.2007.006 The algebra is constructed using representation theory. The formula for multiplication in the algebra is
$\bar t*\bar{t'}=\bar{t'}(t+1) + (\bar{t'},\bar{t})\bar{t} + \lambda_t(\bar{t'})$
I used bar instead of underscore for axis of transposition $t$ in $B$. The drawback of this formula is $\lambda_t$ which is constructed by use of representation theory. I would like to see direct formula for multiplication within the algebra in order to see the symmetries of baby monster. I would like to obtain algebra in GAP in which I can test some hypothesis about the algebra.
I have following questions related to this algebra.
- Let $L$ denote $78$-dimensional simple Lie algebra generated by GAP command $SimpleLieAlgebra("E", 6, GF(2))$. What is the automorphism group of this Lie algebra - is it $^2E_6(2)$ or $E_6(2)$ ? How many generators this algebra have ? According to my test, 12 generators (basis elements 1..6 and 37..42) generate full algebra. I need to do more testing to see how to obtain subalgebra $L$ of algebra $V_{4370}$ for given transposition $t$.
- There is $782$-dimensional subalgebra invariant for $Fi_{23}$ subgroup. This subgroup does not have any axis fixed. This subalgebra is spanned by $31671$ axes of conjugacy class $2A$ of $Fi_{23}$. Using information from chapter 6 in this paper about "dihedral subalgebras" we could define such algebra, I believe.
- I am not sure how to define $5$-dimensional dihedral subalgebra for 4B case, because I don't follow how $l_t$ is defined in this paper.
- From direct calculations of eigen spaces in GAP I observed that given subgroup $H$ of $B$ has vectors fixed only when stabilizer of $H$ is not trivial. This is experimental data, I don't know how to prove it.
I will add more questions when I have time to think more about it.