GAP (SL2Reps) - Chapter 3: Irreducible representations of prime-power level

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3 Irreducible representations of prime-power level
3.1 Representations of type D
3.1-1 SL2ModuleD
3.1-2 SL2IrrepD
3.2 Representations of type N
3.2-1 SL2ModuleN
3.2-2 SL2IrrepN
3.3 Representations of type R
3.3-1 SL2ModuleR
3.3-2 SL2IrrepR
3.3-3 SL2IrrepRUnary

### 3 Irreducible representations of prime-power level

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=`\(\chi\).

#### 3.1 Representations of type D

See Section 2.2-1.

##### 3.1-1 SL2ModuleD

`‣ 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.

##### 3.1-2 SL2IrrepD

`‣ 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.

#### 3.2 Representations of type N

See Section 2.2-2.

##### 3.2-1 SL2ModuleN

`‣ 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.

##### 3.2-2 SL2IrrepN

`‣ 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.

#### 3.3 Representations of type R

See Section 2.2-3.

##### 3.3-1 SL2ModuleR

`‣ 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 \(M - pM\).

##### 3.3-2 SL2IrrepR

`‣ 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).

##### 3.3-3 SL2IrrepRUnary

`‣ 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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