GAP (SL2Reps) - Chapter 5: Methods for testing

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5 Methods for testing
5.1 Testing
5.1-1 SL2WithConjClasses
5.1-2 SL2ChiST
5.1-3 SL2TestPositions
5.1-4 SL2TestSymmetry

### 5 Methods for testing

By the Chinese Remainder Theorem, it suffices to test irreps of prime power level, so those are the irreps handled by the functions in this section.

#### 5.1 Testing

##### 5.1-1 SL2WithConjClasses

`‣ SL2WithConjClasses`( p, lambda ) | ( function )

Returns: the group \(\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})\) with conjugacy classes set to the format we use.

##### 5.1-2 SL2ChiST

`‣ SL2ChiST`( S, T, p, lambda ) | ( function )

Returns: a list representing a character of \(\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})\).

Converts the modular data \((S,T)\), which must have level dividing \(p^\lambda\), into a character of \(\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})\), presented in a form matching the conjugacy classes used in `SL2WithConjClasses`.

##### 5.1-3 SL2TestPositions

`‣ SL2TestPositions`( p, lambda ) | ( function )

Returns: a boolean.

Constructs and tests all non-trivial irreps of level dividing \(p^\lambda\) by checking their positions in `Irr(G)` (see Section 71.8-2 of the GAP Manual). Note that this function will print information on the irreps involved if `InfoSL2Reps` is set to level 1 or higher; see Section 1.2.

##### 5.1-4 SL2TestSymmetry

`‣ SL2TestSymmetry`( p, lambda ) | ( function )

Returns: a boolean.

Constructs and tests all irreps of level \(p^\lambda\), confirming that the \(S\)-matrix is symmetric and unitary and the \(T\) matrix is diagonal. Note that this function will print information on the irreps involved if `InfoSL2Reps` is set to level 1 or higher; see Section 1.2.

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