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 ). Any representation of \mathrm{SL}_2(\mathbb{Z}) arising from a modular tensor category is symmetric .

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

Note that our definition of \mathfrak{s} follows that of ; 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).

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.

Weil representations

Such representations may be constructed using Weil representations as described in . 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 . 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)).

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 . 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 ), 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 and may be used to determine the primitive characters.

Irrep Types

All irreps of prime-power level and finite degree may then be constructed in one of three ways (): 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 ), 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 .

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 ). 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 ; 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 (, 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)]`.

Weil representation types

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 , and subrepresentations of D(p,\lambda) with level p^\lambda may be constructed via .

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 . 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.

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 , and subrepresentations of N(p,\lambda) with level p^\lambda may be constructed via .

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 .

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.

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 .

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

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 . 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 . 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 . 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

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 (which is called by 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 ) fails, as there are no elements of \mathfrak{A} fixing 2M pointwise. The extremal family is handled by and , just like the generic case.