2-Generated Axial Algebras of Monster Type

This is a Singular code for performing the computations needed for the paper

C. Franchi, M. Mainardis, and S. Shpectorov, 2-generated axial algebras of Monster type, arXiv:2101.10379

Content

• genericsakuma.s constructs the universal 2-generated primitive axial algebra of Monster type $(\alpha, \beta)$ when $\alpha\not \in$ { $2\beta, 4\beta$ }. This is used to obtain the formulas in Section 6 of the paper and produce the four polynomials poly1, poly2, poly3, poly4 (corresponding to the polynomials $p_1,p_2,p_3, p_4$ in Theorem 1.2 and Section 8).
• universalalgebra-al=4bt.s computes the formulas in Section 7 of the paper.
• checksSection9.s performs the computations needed in Section 9.
• majorana.s performs the computations needed in Section 10.

Code-2-paper dictionary

The following provides correspondence between the symbols used in the code and those used in the paper:

• al corresponds to $\alpha$ and bt corresponds to $\beta$
• lm, lmf, lm2, lm2f, lm3, lm3f, lm4, lm4f correspond to $\lambda_1$, $\lambda_1^f$, $\lambda_2$, $\lambda_2^f$, $\lambda_3$, $\lambda_3^f$, $\lambda_4$, $\lambda_4^f$, respectively;
• am4, am3, am2, am1, a0, a1, a2, a3, a4 correspond to $a_{-4}$, $a_{-3}$, $a_{-2}$, $a_{-1}$, $a_0$, $a_1$, $a_2$, $a_3$, $a_4$, respectively;
• s1, s2, s3, s4 correspond to $s_{0,1}$, $s_{0,2}$, $s_{0,3}$, $s_{0,4}$, respectively;
• s2f, s3f, s3t, s42 correspond to $s_{1,2}$, $s_{1,3}$, $s_{2,3}$, $s_{2,4}$, respectively;
• poly1, poly2, poly3, poly4 correspond to the polynomials $p_1$, $p_2$, $p_3$, $p_4$, respectively.

Examples

genericsakuma.s

To produce the polynomial $p_i$ (resp $q_i$) run the file genericsakuma.s in Singular and type

P(i); (resp Q(i);)

Note that Singular returns gen(i) for the ith entry of ( am2, am1, a0, a1, a2, s1, s2, s2f ).

universalalgebra-al=4bt.s

To produce the displayed formulas in Lemma 7.i ($i \in$ { $1,..., 11$ }) run the file universalalgebra-al=4bt.s in Singular and type

Formula(i);    

Note that Singular returns gen(i) for the ith entry of ( am4, am3, am2, am1, a0, a1, a2, a3, a4, s1, s2, s2f, s3, s3f, s3t, s4, s42 ).

For other products use the function Times: for example, to compute $a_0s_{1,3}$ in Lemma 7.6(iii) type

Times(a0,s3f);

To produce the value of the second member of Equation (i) ($i \in$ { $62, 66, 67, 68$ }) run the file universalalgebra-al=4bt.s in Singular and type

Eq(i);    

The other formulas in Section 7 can be easily computed or found in the code.

checksSection9.s

To check the result of Lemma 9.1, run the file genericsakuma.s and type

factorize(resultant(q1,q2,lm2)); 

to obtain the irreducible factors of the resultant of $q_1$ and $q_2$ with respect to $\lambda_2$. Considering the linear factors, we get the five values $1$, $0$, $\beta/2$, $\alpha/2$, $(3\alpha^2+3\alpha\beta-\alpha-2\beta)/(8\alpha-4)$ for $\lambda$ in the field. For each of the previous values we compute the corresponding value of $\lambda_2$ by typing

map val1=F, 1,1, lm2,lm2;
factorize(val1(q1)); 
factorize(val1(q2)); 


map val2=F, 0,0, lm2,lm2; 
factorize(val2(q1));  
factorize(val2(q2));  

majorana.s

Computations in the file majorana.s are similar to those in the previous files, performed in the special case where $(\alpha, \beta)=(1/4, 1/32)$ and the field has characteristic $0$.