GAP (SL2Reps) - Chapter 2: Description

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2 Description
2.1 Construction
2.1-1 Weil representations
2.1-2 Character subspaces and primitive characters
2.1-3 Irrep Types
2.1-4 S and T matrices
2.2 Weil representation types
2.2-1 Type D
2.2-2 Type N
2.2-3 Type R, generic cases
2.2-4 Type R, unary and extremal cases

### 2 Description

The group \(\mathrm{SL}_2(\mathbb{Z})\) is generated by \(\mathfrak{s}\) = `[[0,1],[-1,0]]` and \(\mathfrak{t}\) = `[[1,1],[0,1]]` (which satisfy the relations \(\mathfrak{s}^4 = (\mathfrak{st})^3 = \mathrm{id}\)). Thus, any complex representation \(\rho\) of \(\mathrm{SL}_2(\mathbb{Z})\) on \(\mathbb{C}^n\) (where \(n \in \mathbb{Z}^+\) is called the degree or dimension of \(\rho\)) is determined by the \(n \times n\) matrices \(S = \rho(\mathfrak{s})\) and \(T = \rho(\mathfrak{t})\).

This package constructs irreducible representations of \(\mathrm{SL}_2(\mathbb{Z})\) which factor through \(\mathrm{SL}_2(\mathbb{Z}/\ell\mathbb{Z})\) for some \(\ell \in \mathbb{Z}^+\); the smallest such \(\ell\) is called the level of the representation, and is equal to the order of \(T\). One may equivalently say that the kernel of the representation is a congruence subgroup. Such representations are called congruent representations. A congruent representation \(\rho\) is called symmetric if \(S = \rho(\mathfrak{s})\) is a symmetric, unitary matrix and \(T = \rho(\mathfrak{t})\) is a diagonal matrix; it was proved by the authors that every congruent representation is equivalent to a symmetric one (see 2.1-4). Any representation of \(\mathrm{SL}_2(\mathbb{Z})\) arising from a modular tensor category is symmetric [DLN15].

We therefore present representations in the form of a record `rec(S, T, degree, level, name)`, where the name follows the conventions of [NW76].

Note that our definition of \(\mathfrak{s}\) follows that of [Nob76]; other authors prefer the inverse, i.e. \(\mathfrak{s}\) = `[[0,-1],[1,0]]` (under which convention the relations are \(\mathfrak{s}^4 = \mathrm{id}\), \((\mathfrak{s}\mathfrak{t})^3 = \mathfrak{s}^2\)). When working with that convention, one must invert the \(S\) matrices output by this package.

Throughout, we denote by \(\mathbf{e}\) the map \(k \mapsto e^{2 \pi i k}\) (an isomorphism from \(\mathbb{Q}/\mathbb{Z}\) to the group of finite roots of unity in \(\mathbb{C}\)). For a group \(G\), we denote by \(\widehat{G}\) the character group \(\operatorname{Hom}(G, \mathbb{C}^\times)\).

#### 2.1 Construction

Any representation \(\rho\) of \(\mathrm{SL}_2(\mathbb{Z})\) can be decomposed into a direct sum of irreducible representations (irreps). Further, if \(\rho\) has finite level, each irrep can be factorized into a tensor product of irreps whose levels are powers of distinct primes (using the Chinese remainder theorem). Therefore, to characterize all finite-dimensional representations of \(\mathrm{SL}_2(\mathbb{Z})\) of finite level, it suffices to consider irreps of \(\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})\) for primes \(p\) and positive integers \(\lambda\).

##### 2.1-1 Weil representations

Such representations may be constructed using Weil representations as described in [Nob76, Section 1]. We give a brief summary of the process here. First, if \(M\) is any additive abelian group, a quadratic form on \(M\) is a map \(Q : M \to \mathbb{Q}/\mathbb{Z}\) such that

- \(Q(-x) = Q(x)\) for all \(x \in M\), and

- \(B(x,y) = Q(x+y) - Q(x) - Q(y)\) defines a \(\mathbb{Z}\)-bilinear map \(M \times M \to \mathbb{Q}/\mathbb{Z}\).

Now let \(p\) be a prime number and \(\lambda \in \mathbb{Z}^+\). Choose a \(\mathbb{Z}/p^\lambda\mathbb{Z}\)-module \(M\) and a quadratic form \(Q\) on \(M\) such that the pair \((M,Q)\) is of one of the three types described in Section 2.2. Each such \(M\) is a ring, and has at most 2 cyclic factors as an additive group. Those with 2 cyclic factors may be identified with a quotient of the quadratic integers, giving a norm on \(M\). Then the quadratic module \((M,Q)\) gives rise to a representation of \(\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})\) on the vector space \(V = \mathbb{C}^M\) of complex-valued functions on \(M\). This representation is denoted \(W(M,Q)\). Note that the central charge of \((M,Q)\) is given by \(S_Q(-1) = \frac{1}{\sqrt{|M|}} \sum_{x \in M} \mathbf{e}(Q(x))\).

##### 2.1-2 Character subspaces and primitive characters

A family of subrepresentations \(W(M,Q,\chi)\) of \(W(M,Q)\) may be constructed as follows. Denote

\[\operatorname{Aut}(M,Q) = \{ \varepsilon \in \operatorname{Aut}(M) \mid Q(\varepsilon x) = Q(x) \text{ for all } x \in M\}~.\]

We then associate to \((M,Q)\) an abelian subgroup \(\mathfrak{A} \leq \operatorname{Aut}(M,Q)\); the structure of this group depends on \((M,Q)\) and is described in Section 2.2. Note that \(\mathfrak{A}\) has at most two cyclic factors, whose generators we denote by \(\alpha\) and \(\beta\). Now, let \(\chi \in \widehat{\mathfrak{A}}\) be a 1-dimensional representation (character) of \(\mathfrak{A}\), and define

\[V_\chi = \{f \in V \mid f(\varepsilon x) = \chi(\varepsilon) f(x) \text{ for all } x \in M \text{ and } \varepsilon \in \mathfrak{A}\}~,\]

which is a \(\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})\)-invariant subspace of \(V\). We then denote by \(W(M,Q,\chi)\) the subrepresentation of \(W(M,Q)\) on \(V_\chi\). Note that \(W(M,Q,\chi) \cong W(M,Q,\overline{\chi})\).

For the abelian groups \(\mathfrak{A} \leq \operatorname{Aut}(M,Q)\), we will frequently refer to a character \(\chi \in \widehat{\mathfrak{A}}\) as being primitive. With the exception of a single family of modules of type \(R\) (the extremal case, for which see Section 2.2-4), primitivity amounts to the following: there exists some \(\varepsilon \in \mathfrak{A}\) such that \(\chi(\varepsilon) \neq 1\) and \(\varepsilon\) fixes the submodule \(pM \subset M\) pointwise. There exists a subgroup \(\mathfrak{A}_0 \leq \mathfrak{A}\) such that a non-trivial \(\chi \in \widehat{\mathfrak{A}}\) is primitive if and only if \(\chi\) is injective on \(\mathfrak{A}_0\) (or, equivalently, if \(\mathfrak{A}_0 \cap \operatorname{ker} \chi\) is trivial).

Explicit descriptions of the group \(\mathfrak{A}_0\) for each type are given in Section 2.2 and may be used to determine the primitive characters.

##### 2.1-3 Irrep Types

All irreps of prime-power level and finite degree may then be constructed in one of three ways ([NW76, Hauptsatz 2]):

- The overwhelming majority are of the form \(W(M,Q,\chi)\) for \(\chi\) primitive and \(\chi^2 \neq 1\); we call these standard. This includes the primitive characters from the extremal case.

- A finite number, and a single infinite family arising from the extremal case (Section 2.2-4), are instead constructed by using non-primitive characters or primitive characters \(\chi\) with \(\chi^2 = 1\). We call these non-standard.

- Finally, 18 exceptional irreps are constructed as tensor products of two irreps from the other two cases. A full list of these may be constructed by `SL2IrrepsExceptional` (4.3-1).

##### 2.1-4 S and T matrices

The images \(W(M,Q)(\mathfrak{s})(f)\) and \(W(M,Q)(\mathfrak{t})(f)\) may be calculated for any \(f \in V\) (see [Nob76, Satz 2]). Thus, to construct \(S\) and \(T\) matrices for the irreducible subrepresentations of \(W(M,Q)\), it suffices to find bases for the \(W(M,Q)\)-invariant subspaces of \(V\). Choices for such bases are given by [NW76]; however, these often result in non-symmetric \(S\) matrices. It has been proven by the authors of this package that, for all standard and non-standard irreps, there exists a basis for the corresponding subspace of \(V\) such that \(S\) is symmetric and unitary and \(T\) is diagonal ([NWW21], in preparation). In particular, \(S\) is always either a real matrix or \(i\) times a real matrix. It follows that these properties hold for the exceptional irreps as well. This package therefore produces matrices with these properties.

All the finite-dimensional irreducible representations of \(\mathrm{SL}_2(\mathbb{Z})\) of finite level can now be constructed by taking tensor products of these prime-power irreps. Note that, if two representations are determined by pairs `[S1,T1]` and `[S2,T2]`, then the pair for their tensor product may be calculated via the GAP command `KroneckerProduct`, namely as `[KroneckerProduct(S1,S2),KroneckerProduct(T1,T2)]`.

#### 2.2 Weil representation types

##### 2.2-1 Type D

Let \(p\) be prime. If \(p=2\) or \(p=3\), let \(\lambda \geq 2\); otherwise, let \(\lambda \geq 1\). Then the Weil representation arising from the quadratic module with

\[M = \mathbb{Z}/p^\lambda\mathbb{Z} \oplus \mathbb{Z}/p^\lambda\mathbb{Z} \qquad \text{and} \qquad Q(x,y) = \frac{xy}{p^\lambda}\]

is said to be of type \(D\) and denoted \(D(p,\lambda)\). Information on type \(D\) quadratic modules may be obtained via `SL2ModuleD` (3.1-1), and subrepresentations of \(D(p,\lambda)\) with level \(p^\lambda\) may be constructed via `SL2IrrepD` (3.1-2).

The group

\[\mathfrak{A} \cong (\mathbb{Z}/p^\lambda\mathbb{Z})^\times\]

acts on \(M\) by \(a(x,y) = (a^{-1}x, ay)\) and is thus identified with a subgroup of \(\operatorname{Aut}(M,Q)\); see [NW76, Section 2.1]. The group \(\mathfrak{A}\) has order \(p^{\lambda-1}(p-1)\) and \(\mathfrak{A} = \langle\alpha\rangle \times \langle\beta\rangle\). The relevant information for type \(D\) quadratic modules is as follows:

\(p\) | \(\lambda\) | \(\alpha\) | \(\beta\) | \(\mathfrak{A}_0\)
\(>2\) | \(1\) | \(1\) | \(|\beta| = p-1\) | \(\langle 1 \rangle\)
\(>2\) | \(>1\) | \(|\alpha| = p^{\lambda-1}\) (e.g. \(\alpha = 1 + p\)) | \(|\beta| = p-1\) | \(\langle \alpha \rangle\)
\(2\) | \(2\) | \(1\) | \(-1\) | \(\langle 1 \rangle\)
\(2\) | \(>2\) | \(|\alpha| = 2^{\lambda-2}\) (e.g. \(\alpha = 5\)) | \(-1\) | \(\langle \alpha \rangle\)

When \(\mathfrak{A}_0\) is trivial, every non-trivial character \(\chi \in \widehat{\mathfrak{A}}\) is primitive.

##### 2.2-2 Type N

Let \(p\) be prime and \(\lambda \geq 1\). If \(p \neq 2\), let \(u\) be a positive integer so that \(u \equiv 3\) mod 4 with \(-u\) a quadratic non-residue mod \(p\); if \(p = 2\), let \(u=3\). Then the Weil representation arising from the quadratic module with

\[M = \mathbb{Z}/p^\lambda\mathbb{Z} \oplus \mathbb{Z}/p^\lambda\mathbb{Z} \qquad \text{and} \qquad Q(x,y) = \frac{x^2 +xy+\frac{1+u}{4}y^2}{p^\lambda}\]

is said to be of type \(N\) and denoted \(N(p,\lambda)\). Information on type \(N\) quadratic modules may be obtained via `SL2ModuleN` (3.2-1), and subrepresentations of \(N(p,\lambda)\) with level \(p^\lambda\) may be constructed via `SL2IrrepN` (3.2-2).

The additive group \(M\) is a ring with multiplication given by

\[(x_1, y_1) \cdot (x_2, y_2) = (x_1x_2 - \frac{1+u}{4}y_1y_2, x_1y_2 + x_2y_1 + y_1y_2)\]

and identity element \((1,0)\). We define a norm \(\operatorname{Nm}(x,y) = x^2 + xy + \frac{1+u}{4}y^2\) on \(M\); then the multiplicative subgroup

\[\mathfrak{A} = \{\varepsilon \in M^\times \mid \operatorname{Nm}(\varepsilon) = 1 \}\]

of \(M^\times\) acts on \(M\) by multiplication and is identified with a subgroup of \(\operatorname{Aut}(M,Q)\); see [NW76, Section 2.2].

The group \(\mathfrak{A}\) has order \(p^{\lambda-1}(p+1)\) and \(\mathfrak{A} = \langle \alpha \rangle \times \langle \beta \rangle\). The relevant information for type \(N\) quadratic modules is as follows:

\(p\) | \(\lambda\) | \(\alpha\) | \(\beta\) | \(\mathfrak{A}_0\)
\(>2\) | \(1\) | \((1,0)\) | \(|\beta| = p+1\) | \(\langle (1,0) \rangle\)
\(>2\) | \(>1\) | \(|\alpha| = p^{\lambda-1}\) | \(|\beta| = p+1\) | \(\langle \alpha \rangle\)
\(2\) | \(1\) | \((1,0)\) | \(|\beta| = 3\) | \(\langle (1,0) \rangle\)
\(2\) | \(2\) | \((1,0)\) | \(|\beta| = 6\) | \(\langle (-1,0) \rangle\)
\(2\) | \(>2\) | \(|\alpha| = p^{\lambda-2}\) | \(|\beta| = 6\) | \(\langle \alpha \rangle\)

When \(\mathfrak{A}_0\) is trivial, every non-trivial character \(\chi \in \widehat{\mathfrak{A}}\) is primitive.

##### 2.2-3 Type R, generic cases

The structure of the quadratic module \((M,Q)\) of type \(R\) depends upon three additional parameters: \(\sigma\), \(r\), and \(t\). Details are as follows:

- If \(p\) is odd, let \(\lambda \geq 2\), \(\sigma \in \{1, \dots, \lambda\}\), and \(r,t \in \{1,u\}\) with \(u\) a quadratic non-residue mod \(p\). Then define

  \[M = \mathbb{Z}/p^\lambda\mathbb{Z} \oplus \mathbb{Z}/p^{\lambda-\sigma}\mathbb{Z} \qquad \text{and} \qquad Q(x,y) = \frac{r(x^2 + p^\sigma t y^2)}{p^\lambda}~.\]

  When \(\sigma = \lambda\), the second factor of \(M\) is trivial, and \((M,Q)\) is said to be in the unary family; otherwise, it is called generic.

- If \(p=2\), let \(\lambda \geq 2\), \(\sigma \in \{0, \dots, \lambda-2\}\) and \(r,t \in \{1,3,5,7\}\). Then define

  \[M = \mathbb{Z}/2^{\lambda-1}\mathbb{Z} \oplus \mathbb{Z}/2^{\lambda-\sigma-1}\mathbb{Z} \qquad \text{and} \qquad Q(x,y) = \frac{r(x^2 + 2^\sigma t y^2)}{2^\lambda}~.\]

  When \(\sigma = \lambda - 2\), the second factor of \(M\) is isomorphic to \(\mathbb{Z}/2\mathbb{Z}\), and \((M,Q)\) is said to be in the extremal family; otherwise, it is called generic.

In all cases, the resulting representation is said to be of type \(R\) and denoted \(R(p,\lambda,\sigma,r,t)\). The additive group \(M\) admits a ring structure with multiplication

\[(x_1, y_1) \cdot (x_2, y_2) = (x_1x_2 - p^\sigma ty_1y_2, x_1y_2 + x_2y_1)\]

and identity element \((1,0)\). We define a norm \(\operatorname{Nm}(x,y) = x^2 + xy + p^\sigma t y^2\) on \(M\).

In this section, we detail generic type \(R\) quadratic modules. Information on the unary and extremal cases is covered in Section 2.2-4.

Let \((M,Q)\) be a generic type \(R\) quadratic module. Information on \((M,Q)\) can be obtained via `SL2ModuleR` (3.3-1), and subrepresentations of \(R(p,\lambda,\sigma,r,t)\) with level \(p^\lambda\) may be constructed via `SL2IrrepR` (3.3-2).

The multiplicative subgroup

\[\mathfrak{A} = \{\varepsilon \in M^\times \mid \operatorname{Nm}(\varepsilon) = 1 \}\]

of \(M^\times\) acts on \(M\) by multiplication and is identified with a subgroup of \(\operatorname{Aut}(M,Q)\); see [NW76, Section 2.3 - 2.4]. The relevant information is as follows:

- If \(p\) is odd, \(\mathfrak{A} = \langle\alpha\rangle \times \langle\beta\rangle\) with order \(2p^{\lambda-\sigma}\). In this case, for fixed \(p\), \(\lambda\), \(\sigma\), each pair \((r,t)\) gives rise to a distinct quadratic module [Nob76, Satz 4]. The following table covers a complete list of representatives of equivalence classes of such modules.

  \(p\) | \(\lambda\) | \(\sigma\) | \((r,t)\) | \(\alpha\) | \(\beta\) | \(\mathfrak{A}_0\)
  \(3\) | \(2\) | \(1\) | \(r,t \in \{1,2\}\) | \(|\alpha| = 3\) | \((-1,0)\) | \(\langle \alpha \rangle\)
  \(3\) | \(\geq 3\) | \(1\) | \(t=1\), \(r \in \{1,2\}\) | \(|\alpha| = 3^{\lambda-\sigma-1}\) | \(|\beta| = 6\) | \(\langle \alpha \rangle\)
  \(3\) | \(\geq 3\) | \(1\) | \(t=2\), \(r \in \{1,2\}\) | \(|\alpha| = 3^{\lambda-\sigma}\) | \((-1,0)\) | \(\langle \alpha \rangle\)
  \(3\) | \(\geq 3\) | \(2,\dots,\lambda-1\) | \(r,t \in \{1,2\}\) | \(|\alpha| = 3^{\lambda-\sigma}\) | \((-1,0)\) | \(\langle \alpha \rangle\)
  \(\geq 5\) | \(\geq 2\) | \(1, \dots,\lambda - 1\) | \(r,t \in \{1,u\}\) | \(|\alpha| = p^{\lambda-\sigma}\) | \((-1,0)\) | \(\langle \alpha \rangle\)

- If \(p=2\), then the generic case occurs when \(\lambda \geq 3\) and \(\sigma \in \{0,\dots,\lambda-3\}\). Again, \(\mathfrak{A} = \langle\alpha\rangle \times \langle\beta\rangle\); the order is \(2^{\lambda-\sigma-k}\) with \(k \in \{0,1,2\}\) (as specified below). In this case, for fixed \(p\), \(\lambda\), \(\sigma\), two pairs \((r_1,t_1)\) and \((r_2,t_2)\) may give rise to equivalent quadratic modules [Nob76, Satz 4]. The following table covers a complete list of representatives of equivalence classes of such modules.

  \(\lambda\) | \(\sigma\) | \(r\) | \(t\) | \(\alpha = (x,y)\) | \(\beta\) | \(\mathfrak{A}_0\)
  \(3\) | \(0\) | \(1,3\) | \(1,5\) | \((1,0)\) | \((\frac{t-1}{2},1)\) | \(\langle (-1,0) \rangle\)
  \(3\) | \(0\) | \(1\) | \(3,7\) | \((1,0)\) | \((-1,0)\) | \(\langle (-1,0) \rangle\)
  \(4\) | \(0\) | \(1,3\) | \(5\) | \(x=2, y \equiv 3 \operatorname{mod} 4, |\alpha| = 2^{\lambda-2}\) | \((-1,0)\) | \(\langle -\alpha^2 \rangle\)
  \(\geq 4\) | \(0\) | \(1,3\) | \(1\) | \(x \equiv 1 \operatorname{mod} 4, y = 4, |\alpha| = 2^{\lambda-3}\) | \((0,1)\) | \(\langle \alpha \rangle\)
  \(\geq 4\) | \(0\) | \(1\) | \(3,7\) | \(x \equiv 1 \operatorname{mod} 4, y = 4, |\alpha| = 2^{\lambda-3}\) | \((-1,0)\) | \(\langle \alpha \rangle\)
  \(\geq 5\) | \(0\) | \(1,3\) | \(5\) | \(x=2, y \equiv 3 \operatorname{mod} 4, |\alpha| = 2^{\lambda-2}\) | \((-1,0)\) | \(\langle \alpha \rangle\)
  \(\geq 3\) | \(1,2\) | \(1,3,5,7\) | \(1,3,5,7\) | \(x\equiv 1 \operatorname{mod} 4, y=2, |\alpha| = 2^{\lambda-\sigma-2}\) | \((-1,0)\) | \(\langle \alpha \rangle\)
  \(\geq 3\) | \(\geq 3\) | \(1,3,5,7\) | \(1,3,5,7\) | \(x\equiv 1 \operatorname{mod} 4, y=1, |\alpha| = 2^{\lambda-\sigma-1}\) | \((-1,0)\) | \(\langle \alpha \rangle\)

##### 2.2-4 Type R, unary and extremal cases

This section covers the unary and extremal cases of type \(R\).

First, in the unary family, we have \(p\) odd and \(\sigma = \lambda\). Then the second factor of \(M\) is trivial (and hence \(t\) is irrelevant). We then denote \(R_{p^\lambda}(r) = R(p,\lambda,\lambda,r,t)\). In this case, we do not decompose \(W(M,Q)\) using characters: instead, if \(\lambda \leq 2\), then \(W(M,Q)\) contains two distinct irreducible subrepresentations of level \(p^\lambda\), denoted \(R_{p^\lambda}(r)_{\pm}\); otherwise, it contains a single such subrepresentation, denoted \(R_{p^\lambda}(r)_1\). The unary family is handled by `SL2IrrepRUnary` (3.3-3) (which is called by `SL2IrrepR` (3.3-2) when appropriate).

Second, in the extremal family, we have \(p=2\), \(\lambda \geq 2\), and \(\sigma = \lambda - 2\). Then the second factor of \(M\) is isomorphic to \(\mathbb{Z}/2\mathbb{Z}\), and collapses in \(2M\). Here, \(\operatorname{Aut}(M,Q)\) is itself abelian, so we let \(\mathfrak{A} = \operatorname{Aut}(M,Q)\). This group has order 1, 2, or 4, with the following structure:

- For \(\lambda = 2\) and \(t=1\), \(\mathfrak{A} = \langle \tau \rangle\) where \(\tau : (x,y) \mapsto (y,x)\), and \(\mathfrak{A}_0 = \mathfrak{A} = \langle\tau\rangle\).

- For \(\lambda = 2\) and \(t = 3\), \(\mathfrak{A}\) is trivial; there are no primitive characters.

- For \(\lambda = 3\) or \(4\), \(\mathfrak{A} = \{\pm 1\}\) acting on \(M\) by multiplication; there are no primitive characters.

- Finally, for \(\lambda \geq 5\), \(\mathfrak{A} = \operatorname{Aut}(M,Q) = \langle \alpha \rangle \times \langle -1 \rangle\) with \(\alpha\) of order 2, and \(\mathfrak{A}_0 = \langle\alpha\rangle\). Note that, for this special case, the usual test for primitivity (described in Section 2.1) fails, as there are no elements of \(\mathfrak{A}\) fixing \(2M\) pointwise.

The extremal family is handled by `SL2ModuleR` (3.3-1) and `SL2IrrepR` (3.3-2), just like the generic case.

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