Julia and OSCAR

This manual assumes familiarity with categories, but no previous experience with Julia. We introduce the language features we need as we go. For more about the language itself, see Getting Started in the Julia manual.

Julia's type system and multiple dispatch let us use the same mathematical operation for quite different representations of categories.

Starting a session

After following the installation instructions, start Julia. At the interactive Julia prompt (the REPL), load TensorCategories.jl and OSCAR with

using TensorCategories, Oscar

Installing a package and loading it are different operations; you need not install it again each session.

Integers, types, and parents

Julia can be used as a calculator:

julia> 1 + 1
2

julia> BigInt(2)^64
18446744073709551616

Ordinary integer literals have type Int, usually a 64-bit machine integer. Arithmetic can overflow: on a 64-bit system, 2^64 is 0. Use BigInt or OSCAR's integers ZZ for integers of unbounded size. The Julia expression 1//2 constructs the exact rational number one half. OSCAR also has its own rational field QQ; the next page uses QQ(1)/3 when the parent field matters.

Every Julia value has a type, which determines the applicable methods. An algebraic element also has a parent: for example, a polynomial belongs to a particular polynomial ring. Elements of distinct rings can have the same Julia type. The same distinction matters for categories.

Computer algebra

OSCAR supplies the rings, fields, matrices, groups, and algebra algorithms used by TensorCategories.jl:

using TensorCategories, Oscar
R, x = polynomial_ring(ZZ, "x")
f = x^2 + 2*x + 1
f^2

x^4 + 4*x^3 + 6*x^2 + 4*x + 1

Here R, x = ... assigns two returned values to two variables. The name x denotes an element of R, not an unspecified complex number. ZZ denotes the integers and QQ the rational field.

Reading Julia examples

Indices start at 1. A vector is written [a, b, c]; a matrix is written [a b; c d]. A semicolon separates rows. Use matrix(K, ...) to construct an OSCAR matrix over a specified field:

A = matrix(QQ, [1 2; 3 4])
@assert A[1, 2] == 2
size(A)

(2, 2)

@assert condition checks that condition is true and raises an error if it is not. A successful assertion produces no output. The manual uses assertions to record the mathematical result expected from an example.

An exclamation mark, as in sort!, conventionally indicates mutation. A dot applies an operation elementwise: sqrt.([1,4,9]) applies sqrt to every entry. A trailing semicolon suppresses the display of a result. In a function call, arguments following a semicolon are keyword arguments; for example, sort([3,1,2]; rev=true) requests descending order.

Enter ? at the REPL to switch to help mode, then type a name to see its documentation. Type \otimes followed by Tab to enter ⊗; tensor_product(X,Y) is its spelled-out form. Similarly, \oplus and \circ produce ⊕ and ∘.

Continue with Base fields.