ModularMatrix

An element of $\mathrm{PSL}(2,\mathbb{Z})$ — or an $LR$ word standing for one. Wolfram has no dedicated modular-group heads: elements multiply, invert and raise to a power exactly like any other integer matrix, via Dot, Inverse and MatrixPower — which is why those are widened in place to recognise a ModularMatrix or a word, rather than declared afresh.

ModularMatrix(a, b, c, d)the matrix

\begin{pmatrix}a&b\\c&d\end{pmatrix}

, determinant 1

Dot(m, n)the group product — compute-engine's own head

MatrixPower(m, k)

m

to the

k

-th power, for any integer

k

— compute-engine's own head

Inverse(m)the group inverse — compute-engine's own head

Typed signatures

Type	Package	Applies
`((string) -> expression<ModularMatrix>) & ((integer, integer, integer, integer) -> expression<ModularMatrix>)`	`modular`	always

Domain: The modular group

Details

Every head here also accepts a WORD in place of a matrix — ModularMatrix("LR") or just the string
A plain nested-list matrix pairs with a ModularMatrix too, since it canonicalises the same way; a Tuple or a Vector does not, and is left unevaluated — there's no settled convention for what that pairing should mean
$M$ and $-M$ are the same element of $\mathrm{PSL}(2,\mathbb{Z})$ , so Inverse and MatrixPower may hand back the negated matrix