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.
‣ 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.
‣ 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.
‣ 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.
‣ 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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