Methods for generating individual irreducible representations of \(\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})\) for a given level \(p^\lambda\).
After generating a representation \(\rho\) by means of the bases in [NW76], we perform a change of basis that results in a symmetric representation equivalent to \(\rho\).
In each case (except the unary type \(R\), for which see SL2IrrepRUnary (3.3-3)), the underlying module \(M\) is of rank \(2\), so its elements have the form \((x,y)\) and are thus represented by lists [x,y].
Characters of the abelian group \(\mathfrak{A} = \langle\alpha\rangle \times \langle\beta\rangle\) have the form \(\chi_{i,j}\), given by
\[\chi_{i,j}(\alpha^{v}\beta^{w}) \mapsto \mathbf{e}\left(\frac{vi}{|\alpha|}\right) \mathbf{e}\left(\frac{wj}{|\beta|}\right)~,\]
where \(i\) and \(j\) are integers. We therefore represent each character by a list [i,j]. Note that in some cases \(\alpha\) or \(\beta\) is trivial, and the corresponding index \(i\) or \(j\) is therefore irrelevant.
We write p=\(p\), lambda=\(\lambda\), sigma=\(\sigma\), and chi=\(\chi\).
See Section 2.2-1.
‣ SL2ModuleD( p, lambda ) | ( function ) |
Returns: a record rec(Agrp, Bp, Char, IsPrim) describing \((M,Q)\).
Constructs information about the underlying quadratic module \((M,Q)\) of type \(D\), for \(p\) a prime and \(\lambda \geq 1\).
Agrp is a list describing the elements of \(\mathfrak{A}\). Each element \(a \in \mathfrak{A}\) is represented in Agrp by a list [v, a, a_inv], where v is a list defined by \(a = \alpha^{\mathtt{v[1]}} \beta^{\mathtt{v[2]}}\). Note that \(\beta\) is trivial, and hence v[2] is irrelevant, when \(\mathfrak{A}\) is cyclic.
Bp is a list of representatives for the \(\mathfrak{A}\)-orbits on \(M^\times\), which correspond to a basis for the \(\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})\)-invariant subspace associated to any primitive character \(\chi \in \widehat{\mathfrak{A}}\) with \(\chi^2 \not\equiv 1\). This is the basis given by [NW76], which may result in a non-symmetric representation; if this occurs, we perform a change of basis in SL2IrrepD (3.1-2) to obtain a symmetric representation. For non-primitive characters, we must use different bases which are particular to each case.
Char(i,j) converts two integers \(i\), \(j\) to a function representing the character \(\chi_{i,j} \in \widehat{\mathfrak{A}}\).
IsPrim(chi) tests whether the output of Char(i,j) represents a primitive character.
‣ SL2IrrepD( p, lambda, chi_index ) | ( function ) |
Returns: a list of lists of the form \([S,T]\).
Constructs the modular data for the irreducible representation(s) of type \(D\) with level \(p^\lambda\), for \(p\) a prime and \(\lambda \geq 1\), corresponding to the character \(\chi\) indexed by chi_index = [i,j] (see the discussion of Char(i,j) in SL2ModuleD (3.1-1)).
Here \(S\) is symmetric and unitary and \(T\) is diagonal.
Depending on the parameters, \(W(M,Q)\) will contain either 1 or 2 such irreps.
See Section 2.2-2.
‣ SL2ModuleN( p, lambda ) | ( function ) |
Returns: a record rec(Agrp, Bp, Char, IsPrim, Nm, Prod) describing \((M,Q)\).
Constructs information about the underlying quadratic module \((M,Q)\) of type \(N\), for \(p\) a prime and \(\lambda \geq 1\).
Agrp is a list describing the elements of \(\mathfrak{A}\). Each element \(a \in \mathfrak{A}\) is represented in Agrp by a list [v, a], where v is a list defined by \(a = \alpha^{\mathtt{v[1]}} \beta^{\mathtt{v[2]}}\). Note that \(\alpha\) is trivial, and hence v[1] is irrelevant, when \(\mathfrak{A}\) is cyclic.
Bp is a list of representatives for the \(\mathfrak{A}\)-orbits on \(M^\times\), which correspond to a basis for the \(\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})\)-invariant subspace associated to any primitive character \(\chi \in \widehat{\mathfrak{A}}\) with \(\chi^2 \not\equiv 1\). This is the basis given by [NW76], which may result in a non-symmetric representation; if this occurs, we perform a change of basis in SL2IrrepD (3.1-2) to obtain a symmetric representation. For non-primitive characters, we must use different bases which are particular to each case.
Char(i,j) converts two integers \(i\), \(j\) to a function representing the character \(\chi_{i,j} \in \widehat{\mathfrak{A}}\).
IsPrim(chi) tests whether the output of Char(i,j) represents a primitive character.
Nm(a) and Prod(a,b) are the norm and product functions on \(M\), respectively.
‣ SL2IrrepN( p, lambda, chi_index ) | ( function ) |
Returns: a list of lists of the form \([S,T]\).
Constructs the modular data for the irreducible representation(s) of type \(N\) with level \(p^\lambda\), for \(p\) a prime and \(\lambda \geq 1\), corresponding to the character \(\chi\) indexed by chi_index = [i,j] (see the discussion of Char(i,j) in SL2ModuleN (3.2-1)).
Here \(S\) is symmetric and unitary and \(T\) is diagonal.
Depending on the parameters, \(W(M,Q)\) will contain either 1 or 2 such irreps.
See Section 2.2-3.
‣ SL2ModuleR( p, lambda, sigma, r, t ) | ( function ) |
Returns: a record rec(Agrp, Bp, Char, IsPrim, Nm, Ord, Prod, c, tM) describing \((M,Q)\).
Constructs information about the underlying quadratic module \((M,Q)\) of type \(R\), for \(p\) a prime. The additional parameters \(\lambda\), \(\sigma\), \(r\), and \(t\) should be integers chosen as follows.
If \(p\) is an odd prime, let \(\lambda \geq 2\), \(\sigma \in \{1, \dots, \lambda - 1\}\), and \(r,t \in \{1,u\}\) with \(u\) a quadratic non-residue mod \(p\). Note that \(\sigma = \lambda\) is a valid choice for type \(R\), however, this gives the unary case, and so is not handled by this function, as it is decomposed in a different way; for this case, use SL2IrrepRUnary (3.3-3) instead.
If \(p=2\), let \(\lambda \geq 2\), \(\sigma \in \{0, \dots, \lambda-2\}\) and \(r,t \in \{1,3,5,7\}\).
Agrp is a list describing the elements of \(\mathfrak{A}\). Each element \(a\) of \(\mathfrak{A}\) is represented in Agrp by a list [v, a], where v is a list defined by \(a = \alpha^{\mathtt{v[1]}} \beta^{\mathtt{v[2]}}\).
Bp is a list of representatives for the \(\mathfrak{A}\)-orbits on \(M^\times\), which correspond to a basis for the \(\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})\)-invariant subspace associated to any primitive character \(\chi \in \widehat{\mathfrak{A}}\) with \(\chi^2 \not\equiv 1\). This is the basis given by [NW76], which may result in a non-symmetric representation; if this occurs, we perform a change of basis in SL2IrrepD (3.1-2) to obtain a symmetric representation. For non-primitive characters, we must use different bases which are particular to each case.
Char(i,j) converts two integers \(i\), \(j\) to a function representing the character \(\chi_{i,j} \in \widehat{\mathfrak{A}}\).
IsPrim(chi) tests whether the output of Char(i,j) represents a primitive character.
Nm(a), Ord(a), and Prod(a,b) are the norm, order, and product functions on \(M\), respectively.
c is a scalar used in calculating the \(S\)-matrix; namely \(c = \frac{1}{|M|} \sum_{x \in M} \mathbf{e}(Q(x))\). Note that this is equal to \(S_Q(-1) / \sqrt{|M|}\), where \(S_Q(-1)\) is the central charge (see Section 2.1-1).
tM is a list describing the elements of the group \(M - pM\).
‣ SL2IrrepR( p, lambda, sigma, r, t, chi_index ) | ( function ) |
Returns: a list of lists of the form \([S,T]\).
Constructs the modular data for the irreducible representation(s) of type \(R\) with parameters \(p\), \(\lambda\), \(\sigma\), \(r\), and \(t\), corresponding to the character \(\chi\) indexed by chi_index = [i,j] (see the discussions of \(\sigma\), \(r\), \(t\), and Char(i,j) in SL2ModuleR (3.3-1)).
Here \(S\) is symmetric and unitary and \(T\) is diagonal.
Depending on the parameters, \(W(M,Q)\) will contain either 1 or 2 such irreps.
If \(\sigma = \lambda\) for \(p \neq 2\), then the second factor of \(M\) is trivial (and hence \(t\) is irrelevant), so this falls through to SL2IrrepRUnary (3.3-3).
‣ SL2IrrepRUnary( p, lambda, r ) | ( function ) |
Returns: a list of lists of the form \([S,T]\).
Constructs the modular data for the irreducible representation(s) of unary type \(R\) (that is, the special case where \(\sigma = \lambda\)) with \(p\) an odd prime, \(\lambda\) a positive integer, and \(r \in \{1,u\}\) with \(u\) a quadratic non-residue mod \(p\).
Here \(S\) is symmetric and unitary and \(T\) is diagonal.
In this case, \(W(M,Q)\) always contains exactly 2 such irreps.
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