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).
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.
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)).
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.
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).
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)].
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.
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.
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 |
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.
generated by GAPDoc2HTML