<p>The group <span class="Math">\mathrm{SL}_2(\mathbb{Z})</span> is generated by <span class="Math">\mathfrak{s}</span> = <code class="code">[[0,1],[-1,0]]</code> and <span class="Math">\mathfrak{t}</span> = <code class="code">[[1,1],[0,1]]</code> (which satisfy the relations <span class="Math">\mathfrak{s}^4 = (\mathfrak{st})^3 = \mathrm{id}</span>). Thus, any complex representation <span class="Math">\rho</span> of <span class="Math">\mathrm{SL}_2(\mathbb{Z})</span> on <span class="Math">\mathbb{C}^n</span> (where <span class="Math">n \in \mathbb{Z}^+</span> is called the <em>degree</em> or <em>dimension</em> of <span class="Math">\rho</span>) is determined by the <span class="Math">n \times n</span> matrices <span class="Math">S = \rho(\mathfrak{s})</span> and <span class="Math">T = \rho(\mathfrak{t})</span>.</p>
<p>This package constructs irreducible representations of <span class="Math">\mathrm{SL}_2(\mathbb{Z})</span> which factor through <span class="Math">\mathrm{SL}_2(\mathbb{Z}/\ell\mathbb{Z})</span> for some <span class="Math">\ell \in \mathbb{Z}^+</span>; the smallest such <spanclass="Math">\ell</span> is called the <em>level</em> of the representation, and is equal to the order of <span class="Math">T</span>. One may equivalently say that the kernel of the representation is a congruence subgroup. Such representations are called <em>congruent</em> representations. A congruent representation <span class="Math">\rho</span> is called <em>symmetric</em> if <span class="Math">S = \rho(\mathfrak{s})</span> is a symmetric, unitary matrix and <span class="Math">T = \rho(\mathfrak{t})</span> is a diagonal matrix; it was proved by the authors that every congruent representation is equivalent to a symmetric one (see <a href="chap2.html#X785B74657C952121"><span class="RefLink">2.1-4</span></a>). Any representation of <span class="Math">\mathrm{SL}_2(\mathbb{Z})</span> arising from a modular tensor category is symmetric <a href="chapBib.html#biBDLN15">[DLN15]</a>.</p>
<p>We therefore present representations in the form of a record <code class="code">rec(S, T, degree, level, name)</code>, where the name follows the conventions of <a href="chapBib.html#biBNW76">[NW76]</a>.</p>
<p>Note that our definition of <span class="Math">\mathfrak{s}</span> follows that of <a href="chapBib.html#biBNobs1">[Nob76]</a>; other authors prefer the inverse, i.e. <span class="Math">\mathfrak{s}</span> = <code class="code">[[0,-1],[1,0]]</code> (under which convention the relations are <span class="Math">\mathfrak{s}^4 = \mathrm{id}</span>, <span class="Math">(\mathfrak{s}\mathfrak{t})^3 = \mathfrak{s}^2</span>). When working with that convention, one must invert the <span class="Math">S</span> matrices output by this package.</p>
<p>Throughout, we denote by <span class="Math">\mathbf{e}</span> the map <span class="Math">k \mapsto e^{2 \pi i k}</span> (an isomorphism from <span class="Math">\mathbb{Q}/\mathbb{Z}</span> to the group of finite roots of unity in <span class="Math">\mathbb{C}</span>). For a group <span class="Math">G</span>, we denote by <span class="Math">\widehat{G}</span> the character group <span class="Math">\operatorname{Hom}(G, \mathbb{C}^\times)</span>.</p>
<p>Any representation <span class="Math">\rho</span> of <span class="Math">\mathrm{SL}_2(\mathbb{Z})</span> can be decomposed into a direct sum of irreducible representations (irreps). Further, if <span class="Math">\rho</span> 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 <span class="Math">\mathrm{SL}_2(\mathbb{Z})</span> of finite level, it suffices to consider irreps of <span class="Math">\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})</span> for primes <span class="Math">p</span> and positive integers <span class="Math">\lambda</span>.</p>
<p>Such representations may be constructed using Weil representations as described in <a href="chapBib.html#biBNobs1">[Nob76, Section 1]</a>. We give a brief summary of the process here. First, if <span class="Math">M</span> is any additive abelian group, a <em>quadratic form</em> on <span class="Math">M</span> is a map <span class="Math">Q : M \to \mathbb{Q}/\mathbb{Z}</span> such that</p>
<ul>
<li><p><span class="Math">Q(-x) = Q(x)</span> for all <span class="Math">x \in M</span>, and</p>
</li>
<li><p><span class="Math">B(x,y) = Q(x+y) - Q(x) - Q(y)</span> defines a <span class="Math">\mathbb{Z}</span>-bilinear map <span class="Math">M \times M \to \mathbb{Q}/\mathbb{Z}</span>.</p>
</li>
</ul>
<p>Now let <span class="Math">p</span> be a prime number and <span class="Math">\lambda \in \mathbb{Z}^+</span>. Choose a <span class="Math">\mathbb{Z}/p^\lambda\mathbb{Z}</span>-module <span class="Math">M</span> and a quadratic form <span class="Math">Q</span> on <span class="Math">M</span> such that the pair <span class="Math">(M,Q)</span> is of one of the three types described in Section <a href="chap2.html#X861BA4A8800E1A08"><span class="RefLink">2.2</span></a>. Each such <span class="Math">M</span> 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 <span class="Math">M</span>. Then the <em>quadratic module</em> <span class="Math">(M,Q)</span> gives rise to a representation of <span class="Math">\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})</span> on the vector space <span class="Math">V = \mathbb{C}^M</span> of complex-valued functions on <span class="Math">M</span>. This representation is denoted <span class="Math">W(M,Q)</span>. Note that the <em>central charge</em> of <span class="Math">(M,Q)</span> is given by <span class="Math">S_Q(-1) = \frac{1}{\sqrt{|M|}} \sum_{x \in M} \mathbf{e}(Q(x))</span>.</p>
<h5>2.1-2 <span class="Heading">Character subspaces and primitive characters</span></h5>
<p>A family of subrepresentations <span class="Math">W(M,Q,\chi)</span> of <span class="Math">W(M,Q)</span> may be constructed as follows. Denote</p>
<p class="pcenter">\operatorname{Aut}(M,Q) = \{ \varepsilon \in \operatorname{Aut}(M) \mid Q(\varepsilon x) = Q(x) \text{ for all } x \in M\}~.</p>
<p>We then associate to <span class="Math">(M,Q)</span> an abelian subgroup <span class="Math">\mathfrak{A} \leq \operatorname{Aut}(M,Q)</span>; the structure of this group depends on <span class="Math">(M,Q)</span> and is described in Section <a href="chap2.html#X861BA4A8800E1A08"><span class="RefLink">2.2</span></a>. Note that <span class="Math">\mathfrak{A}</span> has at most two cyclic factors, whose generators we denote by <span class="Math">\alpha</span> and <span class="Math">\beta</span>. Now, let <span class="Math">\chi \in \widehat{\mathfrak{A}}</span> be a 1-dimensional representation (<em>character</em>) of <span class="Math">\mathfrak{A}</span>, and define</p>
<p class="pcenter">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}\}~,</p>
<p>which is a <span class="Math">\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})</span>-invariant subspace of <span class="Math">V</span>. We then denote by <span class="Math">W(M,Q,\chi)</span> the subrepresentation of <span class="Math">W(M,Q)</span> on <span class="Math">V_\chi</span>. Note that <span class="Math">W(M,Q,\chi) \cong W(M,Q,\overline{\chi})</span>.</p>
<p>For the abelian groups <span class="Math">\mathfrak{A} \leq \operatorname{Aut}(M,Q)</span>, we will frequently refer to a character <span class="Math">\chi \in \widehat{\mathfrak{A}}</span> as being <em>primitive</em>. With the exception of a single family of modules of type <span class="Math">R</span> (the <em>extremal</em> case, for which see Section <a href="chap2.html#X7B1E74AC85CBD40B"><span class="RefLink">2.2-4</span></a>), primitivity amounts to the following: there exists some <span class="Math">\varepsilon \in \mathfrak{A}</span> such that <span class="Math">\chi(\varepsilon) \neq 1</span> and <span class="Math">\varepsilon</span> fixes the submodule <span class="Math">pM \subset M</span> pointwise. There exists a subgroup <span class="Math">\mathfrak{A}_0 \leq \mathfrak{A}</span> such that a non-trivial <span class="Math">\chi \in \widehat{\mathfrak{A}}</span> is primitive if and only if <span class="Math">\chi</span> is injective on <span class="Math">\mathfrak{A}_0</span> (or, equivalently, if <span class="Math">\mathfrak{A}_0 \cap \operatorname{ker} \chi</span> is trivial).</p>
<p>Explicit descriptions of the group <span class="Math">\mathfrak{A}_0</span> for each type are given in Section <a href="chap2.html#X861BA4A8800E1A08"><span class="RefLink">2.2</span></a> and may be used to determine the primitive characters.</p>
<p>All irreps of prime-power level and finite degree may then be constructed in one of three ways (<a href="chapBib.html#biBNW76">[NW76, Hauptsatz 2]</a>):</p>
<ul>
<li><p>The overwhelming majority are of the form <span class="Math">W(M,Q,\chi)</span> for <span class="Math">\chi</span> primitive and <span class="Math">\chi^2 \neq 1</span>; we call these <em>standard</em>. This includes the primitive characters from the extremal case.</p>
</li>
<li><p>A finite number, and a single infinite family arising from the extremal case (Section <a href="chap2.html#X7B1E74AC85CBD40B"><span class="RefLink">2.2-4</span></a>), are instead constructed by using non-primitive characters or primitive characters <span class="Math">\chi</span> with <span class="Math">\chi^2 = 1</span>. We call these <em>non-standard</em>.</p>
</li>
<li><p>Finally, 18 <em>exceptional</em> irreps are constructed as tensor products of two irreps from the other two cases. A full list of these may be constructed by <code class="func">SL2IrrepsExceptional</code> (<a href="chap4.html#X87A4E88F784B5B9A"><span class="RefLink">4.3-1</span></a>).</p>
<h5>2.1-4 <span class="Heading">S and T matrices</span></h5>
<p>The images <span class="Math">W(M,Q)(\mathfrak{s})(f)</span> and <span class="Math">W(M,Q)(\mathfrak{t})(f)</span> may be calculated for any <span class="Math">f \in V</span> (see <a href="chapBib.html#biBNobs1">[Nob76, Satz 2]</a>). Thus, to construct <span class="Math">S</span> and <span class="Math">T</span> matrices for the irreducible subrepresentations of <span class="Math">W(M,Q)</span>, it suffices to find bases for the <span class="Math">W(M,Q)</span>-invariant subspaces of <span class="Math">V</span>. Choices for such bases are given by <a href="chapBib.html#biBNW76">[NW76]</a>; however, these often result in non-symmetric <span class="Math">S</span> 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 <span class="Math">V</span> such that <span class="Math">S</span> is symmetric and unitary and <span class="Math">T</span> is diagonal (<a href="chapBib.html#biBNWWi">[NWW21]</a>, in preparation). In particular, <span class="Math">S</span> is always either a real matrix or <span class="Math">i</span> times a real matrix. It follows that these properties hold for the exceptional irreps as well. This package therefore produces matrices with these properties.</p>
<p>All the finite-dimensional irreducible representations of <span class="Math">\mathrm{SL}_2(\mathbb{Z})</span> 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 <code class="code">[S1,T1]</code> and <code class="code">[S2,T2]</code>, then the pair for their tensor product may be calculated via the GAP command <code class="code">KroneckerProduct</code>, namely as <code class="code">[KroneckerProduct(S1,S2),KroneckerProduct(T1,T2)]</code>.</p>
<p>Let <span class="Math">p</span> be prime. If <span class="Math">p=2</span> or <span class="Math">p=3</span>, let <span class="Math">\lambda \geq 2</span>; otherwise, let <span class="Math">\lambda \geq 1</span>. Then the Weil representation arising from the quadratic module with</p>
<p>is said to be of type <span class="Math">D</span> and denoted <span class="Math">D(p,\lambda)</span>. Information on type <span class="Math">D</span> quadratic modules may be obtained via <code class="func">SL2ModuleD</code> (<a href="chap3.html#X845D92CB7841CB0B"><span class="RefLink">3.1-1</span></a>), and subrepresentations of <span class="Math">D(p,\lambda)</span> with level <span class="Math">p^\lambda</span> may be constructed via <code class="func">SL2IrrepD</code> (<a href="chap3.html#X7FDB517981A2C091"><span class="RefLink">3.1-2</span></a>).</p>
<p>acts on <span class="Math">M</span> by <span class="Math">a(x,y) = (a^{-1}x, ay)</span> and is thus identified with a subgroup of <span class="Math">\operatorname{Aut}(M,Q)</span>; see <a href="chapBib.html#biBNW76">[NW76, Section 2.1]</a>. The group <span class="Math">\mathfrak{A}</span> has order <span class="Math">p^{\lambda-1}(p-1)</span> and <span class="Math">\mathfrak{A} = \langle\alpha\rangle \times \langle\beta\rangle</span>. The relevant information for type <span class="Math">D</span> quadratic modules is as follows:</p>
<p>When <span class="Math">\mathfrak{A}_0</span> is trivial, every non-trivial character <span class="Math">\chi \in \widehat{\mathfrak{A}}</span> is primitive.</p>
<p>Let <span class="Math">p</span> be prime and <span class="Math">\lambda \geq 1</span>. If <span class="Math">p \neq 2</span>, let <span class="Math">u</span> be a positive integer so that <span class="Math">u \equiv 3</span> mod 4 with <span class="Math">-u</span> a quadratic non-residue mod <span class="Math">p</span>; if <span class="Math">p = 2</span>, let <span class="Math">u=3</span>. Then the Weil representation arising from the quadratic module with</p>
<p>is said to be of type <span class="Math">N</span> and denoted <span class="Math">N(p,\lambda)</span>. Information on type <span class="Math">N</span> quadratic modules may be obtained via <code class="func">SL2ModuleN</code> (<a href="chap3.html#X7A50CC5A7933E207"><span class="RefLink">3.2-1</span></a>), and subrepresentations of <span class="Math">N(p,\lambda)</span> with level <span class="Math">p^\lambda</span> may be constructed via <code class="func">SL2IrrepN</code> (<a href="chap3.html#X81D60FE878F02838"><span class="RefLink">3.2-2</span></a>).</p>
<p>The additive group <span class="Math">M</span> is a ring with multiplication given by</p>
<p>and identity element <span class="Math">(1,0)</span>. We define a norm <span class="Math">\operatorname{Nm}(x,y) = x^2 + xy + \frac{1+u}{4}y^2</span> on <span class="Math">M</span>; then the multiplicative subgroup</p>
<p>of <span class="Math">M^\times</span> acts on <span class="Math">M</span> by multiplication and is identified with a subgroup of <span class="Math">\operatorname{Aut}(M,Q)</span>; see <a href="chapBib.html#biBNW76">[NW76, Section 2.2]</a>.</p>
<p>The group <span class="Math">\mathfrak{A}</span> has order <span class="Math">p^{\lambda-1}(p+1)</span> and <span class="Math">\mathfrak{A} = \langle \alpha \rangle \times \langle \beta \rangle</span>. The relevant information for type <span class="Math">N</span> quadratic modules is as follows:</p>
<p>When <span class="Math">\mathfrak{A}_0</span> is trivial, every non-trivial character <span class="Math">\chi \in \widehat{\mathfrak{A}}</span> is primitive.</p>
<p>The structure of the quadratic module <span class="Math">(M,Q)</span> of type <span class="Math">R</span> depends upon three additional parameters: <span class="Math">\sigma</span>, <span class="Math">r</span>, and <span class="Math">t</span>. Details are as follows:</p>
<ul>
<li><p>If <span class="Math">p</span> is odd, let <span class="Math">\lambda \geq 2</span>, <span class="Math">\sigma \in \{1, \dots, \lambda\}</span>, and <span class="Math">r,t \in \{1,u\}</span> with <span class="Math">u</span> a quadratic non-residue mod <span class="Math">p</span>. Then define</p>
<p>When <span class="Math">\sigma = \lambda</span>, the second factor of <span class="Math">M</span> is trivial, and <span class="Math">(M,Q)</span> is said to be in the <em>unary</em> family; otherwise, it is called <em>generic</em>.</p>
</li>
<li><p>If <span class="Math">p=2</span>, let <span class="Math">\lambda \geq 2</span>, <span class="Math">\sigma \in \{0, \dots, \lambda-2\}</span> and <span class="Math">r,t \in \{1,3,5,7\}</span>. Then define</p>
<p>When <span class="Math">\sigma = \lambda - 2</span>, the second factor of <span class="Math">M</span> is isomorphic to <span class="Math">\mathbb{Z}/2\mathbb{Z}</span>, and <span class="Math">(M,Q)</span> is said to be in the <em>extremal</em> family; otherwise, it is called <em>generic</em>.</p>
</li>
</ul>
<p>In all cases, the resulting representation is said to be of type <span class="Math">R</span> and denoted <span class="Math">R(p,\lambda,\sigma,r,t)</span>. The additive group <span class="Math">M</span> admits a ring structure with multiplication</p>
<p>and identity element <span class="Math">(1,0)</span>. We define a norm <span class="Math">\operatorname{Nm}(x,y) = x^2 + xy + p^\sigma t y^2</span> on <span class="Math">M</span>.</p>
<p>In this section, we detail generic type <span class="Math">R</span> quadratic modules. Information on the unary and extremal cases is covered in Section <a href="chap2.html#X7B1E74AC85CBD40B"><span class="RefLink">2.2-4</span></a>.</p>
<p>Let <span class="Math">(M,Q)</span> be a generic type <span class="Math">R</span> quadratic module. Information on <span class="Math">(M,Q)</span> can be obtained via <code class="func">SL2ModuleR</code> (<a href="chap3.html#X7B10D99E7AEAC411"><span class="RefLink">3.3-1</span></a>), and subrepresentations of <span class="Math">R(p,\lambda,\sigma,r,t)</span> with level <span class="Math">p^\lambda</span> may be constructed via <code class="func">SL2IrrepR</code> (<a href="chap3.html#X80961A2C7C5F632E"><span class="RefLink">3.3-2</span></a>).</p>
<p>of <span class="Math">M^\times</span> acts on <span class="Math">M</span> by multiplication and is identified with a subgroup of <span class="Math">\operatorname{Aut}(M,Q)</span>; see <a href="chapBib.html#biBNW76">[NW76, Section 2.3 - 2.4]</a>. The relevant information is as follows:</p>
<ul>
<li><p>If <span class="Math">p</span> is odd, <span class="Math">\mathfrak{A} = \langle\alpha\rangle \times \langle\beta\rangle</span> with order <span class="Math">2p^{\lambda-\sigma}</span>. In this case, for fixed <span class="Math">p</span>, <span class="Math">\lambda</span>, <span class="Math">\sigma</span>, each pair <span class="Math">(r,t)</span> gives rise to a distinct quadratic module <a href="chapBib.html#biBNobs1">[Nob76, Satz 4]</a>. The following table covers a complete list of representatives of equivalence classes of such modules.</p>
</li>
<li><p>If <span class="Math">p=2</span>, then the generic case occurs when <span class="Math">\lambda \geq 3</span> and <span class="Math">\sigma \in \{0,\dots,\lambda-3\}</span>. Again, <span class="Math">\mathfrak{A} = \langle\alpha\rangle \times \langle\beta\rangle</span>; the order is <span class="Math">2^{\lambda-\sigma-k}</span> with <span class="Math">k \in \{0,1,2\}</span> (as specified below). In this case, for fixed <span class="Math">p</span>, <span class="Math">\lambda</span>, <span class="Math">\sigma</span>, two pairs <span class="Math">(r_1,t_1)</span> and <span class="Math">(r_2,t_2)</span> may give rise to equivalent quadratic modules <a href="chapBib.html#biBNobs1">[Nob76, Satz 4]</a>. The following table covers a complete list of representatives of equivalence classes of such modules.</p>
<h5>2.2-4 <span class="Heading">Type R, unary and extremal cases</span></h5>
<p>This section covers the unary and extremal cases of type <span class="Math">R</span>.</p>
<p>First, in the unary family, we have <span class="Math">p</span> odd and <span class="Math">\sigma = \lambda</span>. Then the second factor of <span class="Math">M</span> is trivial (and hence <span class="Math">t</span> is irrelevant). We then denote <span class="Math">R_{p^\lambda}(r) = R(p,\lambda,\lambda,r,t)</span>. In this case, we do not decompose <span class="Math">W(M,Q)</span> using characters: instead, if <span class="Math">\lambda \leq 2</span>, then <span class="Math">W(M,Q)</span> contains two distinct irreducible subrepresentations of level <span class="Math">p^\lambda</span>, denoted <span class="Math">R_{p^\lambda}(r)_{\pm}</span>; otherwise, it contains a single such subrepresentation, denoted <span class="Math">R_{p^\lambda}(r)_1</span>. The unary family is handled by <code class="func">SL2IrrepRUnary</code> (<a href="chap3.html#X7C94E3007A1BEE85"><span class="RefLink">3.3-3</span></a>) (which is called by <code class="func">SL2IrrepR</code> (<a href="chap3.html#X80961A2C7C5F632E"><span class="RefLink">3.3-2</span></a>) when appropriate).</p>
<p>Second, in the extremal family, we have <span class="Math">p=2</span>, <span class="Math">\lambda \geq 2</span>, and <span class="Math">\sigma = \lambda - 2</span>. Then the second factor of <span class="Math">M</span> is isomorphic to <span class="Math">\mathbb{Z}/2\mathbb{Z}</span>, and collapses in <span class="Math">2M</span>. Here, <span class="Math">\operatorname{Aut}(M,Q)</span> is itself abelian, so we let <span class="Math">\mathfrak{A} = \operatorname{Aut}(M,Q)</span>. This group has order 1, 2, or 4, with the following structure:</p>
<ul>
<li><p>For <span class="Math">\lambda = 2</span> and <span class="Math">t=1</span>, <span class="Math">\mathfrak{A} = \langle \tau \rangle</span> where <span class="Math">\tau : (x,y) \mapsto (y,x)</span>, and <span class="Math">\mathfrak{A}_0 = \mathfrak{A} = \langle\tau\rangle</span>.</p>
</li>
<li><p>For <span class="Math">\lambda = 2</span> and <span class="Math">t = 3</span>, <span class="Math">\mathfrak{A}</span> is trivial; there are no primitive characters.</p>
</li>
<li><p>For <span class="Math">\lambda = 3</span> or <span class="Math">4</span>, <span class="Math">\mathfrak{A} = \{\pm 1\}</span> acting on <span class="Math">M</span> by multiplication; there are no primitive characters.</p>
</li>
<li><p>Finally, for <span class="Math">\lambda \geq 5</span>, <span class="Math">\mathfrak{A} = \operatorname{Aut}(M,Q) = \langle \alpha \rangle \times \langle -1 \rangle</span> with <span class="Math">\alpha</span> of order 2, and <span class="Math">\mathfrak{A}_0 = \langle\alpha\rangle</span>. Note that, for this special case, the usual test for primitivity (described in Section <a href="chap2.html#X7F6278CD87400D49"><span class="RefLink">2.1</span></a>) fails, as there are no elements of <span class="Math">\mathfrak{A}</span> fixing <span class="Math">2M</span> pointwise.</p>
</li>
</ul>
<p>The extremal family is handled by <code class="func">SL2ModuleR</code> (<a href="chap3.html#X7B10D99E7AEAC411"><span class="RefLink">3.3-1</span></a>) and <code class="func">SL2IrrepR</code> (<a href="chap3.html#X80961A2C7C5F632E"><span class="RefLink">3.3-2</span></a>), just like the generic case.</p>
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