# Notes for users of other computer algebra systems

## General differences

• Julia evaluates 2^100 to 0 because 2 is regarded as a 64 bit integer. Write ZZRingElem(2)^100 to get a long.

## Notes for GAP users

This section describes differences between GAP and Oscar. (Hints about using GAP in Oscar can be found in the section about GAP Integration.)

• The syntax of the languages is slightly different.

• In GAP, equality of two objects is checked with =, and one assigns a value to a variable with :=. In Julia, equality is checked with ==, and = denotes assignment. Similarly, inequality of objects is checked with <> in GAP and with != in Julia.

• In GAP, the operator not is used to negate boolean expressions, whereas ! is used in Julia.

• In GAP, object identity is checked with the function IsIdenticalObj, whereas the infix operator === (with negation !==) is used in Julia.

• In GAP, if statements have the form

if condition1 then
statements1
elif condition2 then
statements2
else
statements3
fi;

whereas the Julia syntax is

if condition1
statements1
elseif condition2
statements2
else
statements3
end

Similarly, GAP's for loops have the form

for var in list do
statements
od;

whereas the Julia syntax is

for var in list
statements
end

(The situation with while loops is analogous.)

• The interactive sessions behave differently.

When an error occurs or when the user hits ctrl-C in a GAP session, usually a break loop is entered, from which one can either try to continue the computations, by entering return, or return to the GAP prompt, by entering quit; in the latter case, some objects may be corrupted afterwards.

In a Julia session, one gets automatically back to the Julia prompt when an error occurs or when the user hits ctrl-C, and again some objects may be corrupted afterwards.

• Variable names in GAP and Julia are recommended to be written in camel case and snake case, respectively, see Naming conventions. For example, the GAP function SylowSubgroup corresponds to Oscar's sylow_subgroup.

Thus the GAP rule that the names of user variables should start with a lowercase letter, in order to avoid clashes with system variables, does not make sense in Julia.

Moreover, global Oscar variables are not write protected, contrary to most global GAP variables. Thus there is always the danger that assignments overwrite Julia functions. For example, it is tempting to use gens, hom, and map as names for variables, but Julia or Oscar define them already.

(Also copying some lines of code from an Oscar function into a Julia session can be dangerous in this sense, because some names of local variables of the function may coincide with the names of global variables.)

• GAP provides natural embeddings of many algebraic structures. For example, two finite fields of the same characteristic are embedded into each other whenever this makes sense, and the elements of the smaller field are regarded also as elements of the larger field. Analogously, subfields of cyclotomic fields are naturally embedded into each other, and in fact their elements are internally represented w.r.t. the smallest possible cyclotomic field.

In Oscar, this is not the case. Each element of an algebraic structure has a parent, and operations involving several elements (such as arithmetic operations) are usually restricted to the situation that their parents coincide. One has to explicitly coerce a given element into a different parent if necessary.

The consequences can be quite subtle. Each permutation group in Oscar has a fixed degree, and the function is_transitive checks whether its argument is transitive on the points from 1 to the degree. In GAP, however, the function IsTransitive, called with a permutation group, checks whether this group is transitive on the points which are moved by it. Thus the group generated by the permutation (1, 2, 4) is regarded as transitive in GAP but as intransitive in Oscar.

## Notes for Polymake users

• OSCAR (and Julia) is 1-based, meaning that it counts from 1, rather than from 0 like polymake. For most properties we have taken care of the translation but be aware that it might pop up at some point and generate confusion.

For convenience, Polymake.jl provides Polymake.to_one_based_indexing and Polymake.to_zero_based_indexing.

• Polyhedra and polyhedral complexes in OSCAR are represented inhomogeneously, i.e. without the leading 1 for vertices or 0 for rays. Hence constructors take points, rays, and lineality generators separately.

• user_methods cannot be accessed via Julia's dot syntax, i.e. something like

c = Polymake.polytope.cube(3)
c.AMBIENT_DIM

will not work. Instead user_methods are attached as Julia functions in their respective application. They are always written in lowercase. In the example the following works:

c = Polymake.polytope.cube(3)
Polymake.polytope.ambient_dim(c)