<p>The group <span class="SimpleMath">\(\mathrm{SL}_2(\mathbb{Z})\)</span> is generated by <span class="SimpleMath">\(\mathfrak{s}\)</span> = <code class="code">[[0,1],[-1,0]]</code> and <span class="SimpleMath">\(\mathfrak{t}\)</span> = <code class="code">[[1,1],[0,1]]</code> (which satisfy the relations <span class="SimpleMath">\(\mathfrak{s}^4 = (\mathfrak{st})^3 = \mathrm{id}\)</span>). Thus, any complex representation <span class="SimpleMath">\(\rho\)</span> of <span class="SimpleMath">\(\mathrm{SL}_2(\mathbb{Z})\)</span> on <span class="SimpleMath">\(\mathbb{C}^n\)</span> (where <span class="SimpleMath">\(n \in \mathbb{Z}^+\)</span> is called the <em>degree</em> or <em>dimension</em> of <span class="SimpleMath">\(\rho\)</span>) is determined by the <span class="SimpleMath">\(n \times n\)</span> matrices <span class="SimpleMath">\(S = \rho(\mathfrak{s})\)</span> and <span class="SimpleMath">\(T = \rho(\mathfrak{t})\)</span>.</p>
<p>This package constructs irreducible representations of <span class="SimpleMath">\(\mathrm{SL}_2(\mathbb{Z})\)</span> which factor through <span class="SimpleMath">\(\mathrm{SL}_2(\mathbb{Z}/\ell\mathbb{Z})\)</span> for some <span class="SimpleMath">\(\ell \in \mathbb{Z}^+\)</span>; the smallest such <span class="SimpleMath">\(\ell\)</span> is called the <em>level</em> of the representation, and is equal to the order of <span class="SimpleMath">\(T\)</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="SimpleMath">\(\rho\)</span> is called <em>symmetric</em> if <span class="SimpleMath">\(S = \rho(\mathfrak{s})\)</span> is a symmetric, unitary matrix and <span class="SimpleMath">\(T = \rho(\mathfrak{t})\)</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_mj.html#X785B74657C952121"><span class="RefLink">2.1-4</span></a>). Any representation of <span class="SimpleMath">\(\mathrm{SL}_2(\mathbb{Z})\)</span> arising from a modular tensor category is symmetric <a href="chapBib_mj.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_mj.html#biBNW76">[NW76]</a>.</p>
<p>Note that our definition of <span class="SimpleMath">\(\mathfrak{s}\)</span> follows that of <a href="chapBib_mj.html#biBNobs1">[Nob76]</a>; other authors prefer the inverse, i.e. <span class="SimpleMath">\(\mathfrak{s}\)</span> = <code class="code">[[0,-1],[1,0]]</code> (under which convention the relations are <span class="SimpleMath">\(\mathfrak{s}^4 = \mathrm{id}\)</span>, <span class="SimpleMath">\((\mathfrak{s}\mathfrak{t})^3 = \mathfrak{s}^2\)</span>). When working with that convention, one must invert the <span class="SimpleMath">\(S\)</span> matrices output by this package.</p>
<p>Throughout, we denote by <span class="SimpleMath">\(\mathbf{e}\)</span> the map <span class="SimpleMath">\(k \mapsto e^{2 \pi i k}\)</span> (an isomorphism from <span class="SimpleMath">\(\mathbb{Q}/\mathbb{Z}\)</span> to the group of finite roots of unity in <span class="SimpleMath">\(\mathbb{C}\)</span>). For a group <span class="SimpleMath">\(G\)</span>, we denote by <span class="SimpleMath">\(\widehat{G}\)</span> the character group <span class="SimpleMath">\(\operatorname{Hom}(G, \mathbb{C}^\times)\)</span>.</p>
<p>Any representation <span class="SimpleMath">\(\rho\)</span> of <span class="SimpleMath">\(\mathrm{SL}_2(\mathbb{Z})\)</span> can be decomposed into a direct sum of irreducible representations (irreps). Further, if <span class="SimpleMath">\(\rho\)</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="SimpleMath">\(\mathrm{SL}_2(\mathbb{Z})\)</span> of finite level, it suffices to consider irreps of <span class="SimpleMath">\(\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})\)</span> for primes <span class="SimpleMath">\(p\)</span> and positive integers <span class="SimpleMath">\(\lambda\)</span>.</p>
<p>Such representations may be constructed using Weil representations as described in <a href="chapBib_mj.html#biBNobs1">[Nob76, Section 1]</a>. We give a brief summary of the process here. First, if <span class="SimpleMath">\(M\)</span> is any additive abelian group, a <em>quadratic form</em> on <span class="SimpleMath">\(M\)</span> is a map <span class="SimpleMath">\(Q : M \to \mathbb{Q}/\mathbb{Z}\)</span> such that</p>
<ul>
<li><p><span class="SimpleMath">\(Q(-x) = Q(x)\)</span> for all <span class="SimpleMath">\(x \in M\)</span>, and</p>
</li>
<li><p><span class="SimpleMath">\(B(x,y) = Q(x+y) - Q(x) - Q(y)\)</span> defines a <span class="SimpleMath">\(\mathbb{Z}\)</span>-bilinear map <span class="SimpleMath">\(M \times M \to \mathbb{Q}/\mathbb{Z}\)</span>.</p>
</li>
</ul>
<p>Now let <span class="SimpleMath">\(p\)</span> be a prime number and <span class="SimpleMath">\(\lambda \in \mathbb{Z}^+\)</span>. Choose a <span class="SimpleMath">\(\mathbb{Z}/p^\lambda\mathbb{Z}\)</span>-module <span class="SimpleMath">\(M\)</span> and a quadratic form <span class="SimpleMath">\(Q\)</span> on <span class="SimpleMath">\(M\)</span> such that the pair <span class="SimpleMath">\((M,Q)\)</span> is of one of the three types described in Section <a href="chap2_mj.html#X861BA4A8800E1A08"><span class="RefLink">2.2</span></a>. Each such <span class="SimpleMath">\(M\)</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="SimpleMath">\(M\)</span>. Then the <em>quadratic module</em> <span class="SimpleMath">\((M,Q)\)</span> gives rise to a representation of <span class="SimpleMath">\(\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})\)</span> on the vector space <span class="SimpleMath">\(V = \mathbb{C}^M\)</span> of complex-valued functions on <span class="SimpleMath">\(M\)</span>. This representation is denoted <span class="SimpleMath">\(W(M,Q)\)</span>. Note that the <em>central charge</em> of <span class="SimpleMath">\((M,Q)\)</span> is given by <span class="SimpleMath">\(S_Q(-1) = \frac{1}{\sqrt{|M|}} \sum_{x \in M} \mathbf{e}(Q(x))\)</span>.</p>
<h5>2.1-2 <span class="Heading">Character subspaces and primitive characters</span></h5>
<p>A family of subrepresentations <span class="SimpleMath">\(W(M,Q,\chi)\)</span> of <span class="SimpleMath">\(W(M,Q)\)</span> may be constructed as follows. Denote</p>
<p class="center">\[\operatorname{Aut}(M,Q) = \{ \varepsilon \in \operatorname{Aut}(M) \mid Q(\varepsilon x) = Q(x) \text{ for all } x \in M\}~.\]</p>
<p>We then associate to <span class="SimpleMath">\((M,Q)\)</span> an abelian subgroup <span class="SimpleMath">\(\mathfrak{A} \leq \operatorname{Aut}(M,Q)\)</span>; the structure of this group depends on <span class="SimpleMath">\((M,Q)\)</span> and is described in Section <a href="chap2_mj.html#X861BA4A8800E1A08"><span class="RefLink">2.2</span></a>. Note that <span class="SimpleMath">\(\mathfrak{A}\)</span> has at most two cyclic factors, whose generators we denote by <span class="SimpleMath">\(\alpha\)</span> and <span class="SimpleMath">\(\beta\)</span>. Now, let <span class="SimpleMath">\(\chi \in \widehat{\mathfrak{A}}\)</span> be a 1-dimensional representation (<em>character</em>) of <span class="SimpleMath">\(\mathfrak{A}\)</span>, and define</p>
<p class="center">\[V_\chi = \{f \in V \mid f(\varepsilon x) = \chi(\varepsilon) f(x) \text{ for all } x \in M \text{ and } \varepsilon \in \mathfrak{A}\}~,\]</p>
<p>which is a <span class="SimpleMath">\(\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})\)</span>-invariant subspace of <span class="SimpleMath">\(V\)</span>. We then denote by <span class="SimpleMath">\(W(M,Q,\chi)\)</span> the subrepresentation of <span class="SimpleMath">\(W(M,Q)\)</span> on <span class="SimpleMath">\(V_\chi\)</span>. Note that <span class="SimpleMath">\(W(M,Q,\chi) \cong W(M,Q,\overline{\chi})\)</span>.</p>
<p>For the abelian groups <span class="SimpleMath">\(\mathfrak{A} \leq \operatorname{Aut}(M,Q)\)</span>, we will frequently refer to a character <span class="SimpleMath">\(\chi \in \widehat{\mathfrak{A}}\)</span> as being <em>primitive</em>. With the exception of a single family of modules of type <span class="SimpleMath">\(R\)</span> (the <em>extremal</em> case, for which see Section <a href="chap2_mj.html#X7B1E74AC85CBD40B"><span class="RefLink">2.2-4</span></a>), primitivity amounts to the following: there exists some <span class="SimpleMath">\(\varepsilon \in \mathfrak{A}\)</span> such that <span class="SimpleMath">\(\chi(\varepsilon) \neq 1\)</span> and <span class="SimpleMath">\(\varepsilon\)</span> fixes the submodule <span class="SimpleMath">\(pM \subset M\)</span> pointwise. There exists a subgroup <span class="SimpleMath">\(\mathfrak{A}_0 \leq \mathfrak{A}\)</span> such that a non-trivial <span class="SimpleMath">\(\chi \in \widehat{\mathfrak{A}}\)</span> is primitive if and only if <span class="SimpleMath">\(\chi\)</span> is injective on <span class="SimpleMath">\(\mathfrak{A}_0\)</span> (or, equivalently, if <span class="SimpleMath">\(\mathfrak{A}_0 \cap \operatorname{ker} \chi\)</span> is trivial).</p>
<p>Explicit descriptions of the group <span class="SimpleMath">\(\mathfrak{A}_0\)</span> for each type are given in Section <a href="chap2_mj.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_mj.html#biBNW76">[NW76, Hauptsatz 2]</a>):</p>
<ul>
<li><p>The overwhelming majority are of the form <span class="SimpleMath">\(W(M,Q,\chi)\)</span> for <span class="SimpleMath">\(\chi\)</span> primitive and <span class="SimpleMath">\(\chi^2 \neq 1\)</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_mj.html#X7B1E74AC85CBD40B"><span class="RefLink">2.2-4</span></a>), are instead constructed by using non-primitive characters or primitive characters <span class="SimpleMath">\(\chi\)</span> with <span class="SimpleMath">\(\chi^2 = 1\)</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_mj.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="SimpleMath">\(W(M,Q)(\mathfrak{s})(f)\)</span> and <span class="SimpleMath">\(W(M,Q)(\mathfrak{t})(f)\)</span> may be calculated for any <span class="SimpleMath">\(f \in V\)</span> (see <a href="chapBib_mj.html#biBNobs1">[Nob76, Satz 2]</a>). Thus, to construct <span class="SimpleMath">\(S\)</span> and <span class="SimpleMath">\(T\)</span> matrices for the irreducible subrepresentations of <span class="SimpleMath">\(W(M,Q)\)</span>, it suffices to find bases for the <span class="SimpleMath">\(W(M,Q)\)</span>-invariant subspaces of <span class="SimpleMath">\(V\)</span>. Choices for such bases are given by <a href="chapBib_mj.html#biBNW76">[NW76]</a>; however, these often result in non-symmetric <span class="SimpleMath">\(S\)</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="SimpleMath">\(V\)</span> such that <span class="SimpleMath">\(S\)</span> is symmetric and unitary and <span class="SimpleMath">\(T\)</span> is diagonal (<a href="chapBib_mj.html#biBNWWi">[NWW21]</a>, in preparation). In particular, <span class="SimpleMath">\(S\)</span> is always either a real matrix or <span class="SimpleMath">\(i\)</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="SimpleMath">\(\mathrm{SL}_2(\mathbb{Z})\)</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="SimpleMath">\(p\)</span> be prime. If <span class="SimpleMath">\(p=2\)</span> or <span class="SimpleMath">\(p=3\)</span>, let <span class="SimpleMath">\(\lambda \geq 2\)</span>; otherwise, let <span class="SimpleMath">\(\lambda \geq 1\)</span>. Then the Weil representation arising from the quadratic module with</p>
<p>is said to be of type <span class="SimpleMath">\(D\)</span> and denoted <span class="SimpleMath">\(D(p,\lambda)\)</span>. Information on type <span class="SimpleMath">\(D\)</span> quadratic modules may be obtained via <code class="func">SL2ModuleD</code> (<a href="chap3_mj.html#X845D92CB7841CB0B"><span class="RefLink">3.1-1</span></a>), and subrepresentations of <span class="SimpleMath">\(D(p,\lambda)\)</span> with level <span class="SimpleMath">\(p^\lambda\)</span> may be constructed via <code class="func">SL2IrrepD</code> (<a href="chap3_mj.html#X7FDB517981A2C091"><span class="RefLink">3.1-2</span></a>).</p>
<p>acts on <span class="SimpleMath">\(M\)</span> by <span class="SimpleMath">\(a(x,y) = (a^{-1}x, ay)\)</span> and is thus identified with a subgroup of <span class="SimpleMath">\(\operatorname{Aut}(M,Q)\)</span>; see <a href="chapBib_mj.html#biBNW76">[NW76, Section 2.1]</a>. The group <span class="SimpleMath">\(\mathfrak{A}\)</span> has order <span class="SimpleMath">\(p^{\lambda-1}(p-1)\)</span> and <span class="SimpleMath">\(\mathfrak{A} = \langle\alpha\rangle \times \langle\beta\rangle\)</span>. The relevant information for type <span class="SimpleMath">\(D\)</span> quadratic modules is as follows:</p>
<p>When <span class="SimpleMath">\(\mathfrak{A}_0\)</span> is trivial, every non-trivial character <span class="SimpleMath">\(\chi \in \widehat{\mathfrak{A}}\)</span> is primitive.</p>
<p>Let <span class="SimpleMath">\(p\)</span> be prime and <span class="SimpleMath">\(\lambda \geq 1\)</span>. If <span class="SimpleMath">\(p \neq 2\)</span>, let <span class="SimpleMath">\(u\)</span> be a positive integer so that <span class="SimpleMath">\(u \equiv 3\)</span> mod 4 with <span class="SimpleMath">\(-u\)</span> a quadratic non-residue mod <span class="SimpleMath">\(p\)</span>; if <span class="SimpleMath">\(p = 2\)</span>, let <span class="SimpleMath">\(u=3\)</span>. Then the Weil representation arising from the quadratic module with</p>
<p>is said to be of type <span class="SimpleMath">\(N\)</span> and denoted <span class="SimpleMath">\(N(p,\lambda)\)</span>. Information on type <span class="SimpleMath">\(N\)</span> quadratic modules may be obtained via <code class="func">SL2ModuleN</code> (<a href="chap3_mj.html#X7A50CC5A7933E207"><span class="RefLink">3.2-1</span></a>), and subrepresentations of <span class="SimpleMath">\(N(p,\lambda)\)</span> with level <span class="SimpleMath">\(p^\lambda\)</span> may be constructed via <code class="func">SL2IrrepN</code> (<a href="chap3_mj.html#X81D60FE878F02838"><span class="RefLink">3.2-2</span></a>).</p>
<p>The additive group <span class="SimpleMath">\(M\)</span> is a ring with multiplication given by</p>
<p>and identity element <span class="SimpleMath">\((1,0)\)</span>. We define a norm <span class="SimpleMath">\(\operatorname{Nm}(x,y) = x^2 + xy + \frac{1+u}{4}y^2\)</span> on <span class="SimpleMath">\(M\)</span>; then the multiplicative subgroup</p>
<p>of <span class="SimpleMath">\(M^\times\)</span> acts on <span class="SimpleMath">\(M\)</span> by multiplication and is identified with a subgroup of <span class="SimpleMath">\(\operatorname{Aut}(M,Q)\)</span>; see <a href="chapBib_mj.html#biBNW76">[NW76, Section 2.2]</a>.</p>
<p>The group <span class="SimpleMath">\(\mathfrak{A}\)</span> has order <span class="SimpleMath">\(p^{\lambda-1}(p+1)\)</span> and <span class="SimpleMath">\(\mathfrak{A} = \langle \alpha \rangle \times \langle \beta \rangle\)</span>. The relevant information for type <span class="SimpleMath">\(N\)</span> quadratic modules is as follows:</p>
<p>When <span class="SimpleMath">\(\mathfrak{A}_0\)</span> is trivial, every non-trivial character <span class="SimpleMath">\(\chi \in \widehat{\mathfrak{A}}\)</span> is primitive.</p>
<p>The structure of the quadratic module <span class="SimpleMath">\((M,Q)\)</span> of type <span class="SimpleMath">\(R\)</span> depends upon three additional parameters: <span class="SimpleMath">\(\sigma\)</span>, <span class="SimpleMath">\(r\)</span>, and <span class="SimpleMath">\(t\)</span>. Details are as follows:</p>
<ul>
<li><p>If <span class="SimpleMath">\(p\)</span> is odd, let <span class="SimpleMath">\(\lambda \geq 2\)</span>, <span class="SimpleMath">\(\sigma \in \{1, \dots, \lambda\}\)</span>, and <span class="SimpleMath">\(r,t \in \{1,u\}\)</span> with <span class="SimpleMath">\(u\)</span> a quadratic non-residue mod <span class="SimpleMath">\(p\)</span>. Then define</p>
<p>When <span class="SimpleMath">\(\sigma = \lambda\)</span>, the second factor of <span class="SimpleMath">\(M\)</span> is trivial, and <span class="SimpleMath">\((M,Q)\)</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="SimpleMath">\(p=2\)</span>, let <span class="SimpleMath">\(\lambda \geq 2\)</span>, <span class="SimpleMath">\(\sigma \in \{0, \dots, \lambda-2\}\)</span> and <span class="SimpleMath">\(r,t \in \{1,3,5,7\}\)</span>. Then define</p>
<p>When <span class="SimpleMath">\(\sigma = \lambda - 2\)</span>, the second factor of <span class="SimpleMath">\(M\)</span> is isomorphic to <span class="SimpleMath">\(\mathbb{Z}/2\mathbb{Z}\)</span>, and <span class="SimpleMath">\((M,Q)\)</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="SimpleMath">\(R\)</span> and denoted <span class="SimpleMath">\(R(p,\lambda,\sigma,r,t)\)</span>. The additive group <span class="SimpleMath">\(M\)</span> admits a ring structure with multiplication</p>
<p>and identity element <span class="SimpleMath">\((1,0)\)</span>. We define a norm <span class="SimpleMath">\(\operatorname{Nm}(x,y) = x^2 + xy + p^\sigma t y^2\)</span> on <span class="SimpleMath">\(M\)</span>.</p>
<p>In this section, we detail generic type <span class="SimpleMath">\(R\)</span> quadratic modules. Information on the unary and extremal cases is covered in Section <a href="chap2_mj.html#X7B1E74AC85CBD40B"><span class="RefLink">2.2-4</span></a>.</p>
<p>Let <span class="SimpleMath">\((M,Q)\)</span> be a generic type <span class="SimpleMath">\(R\)</span> quadratic module. Information on <span class="SimpleMath">\((M,Q)\)</span> can be obtained via <code class="func">SL2ModuleR</code> (<a href="chap3_mj.html#X7B10D99E7AEAC411"><spanclass="RefLink">3.3-1</span></a>), and subrepresentations of <span class="SimpleMath">\(R(p,\lambda,\sigma,r,t)\)</span> with level <span class="SimpleMath">\(p^\lambda\)</span> may be constructed via <code class="func">SL2IrrepR</code> (<a href="chap3_mj.html#X80961A2C7C5F632E"><span class="RefLink">3.3-2</span></a>).</p>
<p>of <span class="SimpleMath">\(M^\times\)</span> acts on <span class="SimpleMath">\(M\)</span> by multiplication and is identified with a subgroup of <span class="SimpleMath">\(\operatorname{Aut}(M,Q)\)</span>; see <a href="chapBib_mj.html#biBNW76">[NW76, Section 2.3 - 2.4]</a>. The relevant information is as follows:</p>
<ul>
<li><p>If <span class="SimpleMath">\(p\)</span> is odd, <span class="SimpleMath">\(\mathfrak{A} = \langle\alpha\rangle \times \langle\beta\rangle\)</span> with order <span class="SimpleMath">\(2p^{\lambda-\sigma}\)</span>. In this case, for fixed <span class="SimpleMath">\(p\)</span>, <span class="SimpleMath">\(\lambda\)</span>, <span class="SimpleMath">\(\sigma\)</span>, each pair <span class="SimpleMath">\((r,t)\)</span> gives rise to a distinct quadratic module <a href="chapBib_mj.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="SimpleMath">\(p=2\)</span>, then the generic case occurs when <span class="SimpleMath">\(\lambda \geq 3\)</span> and <span class="SimpleMath">\(\sigma \in \{0,\dots,\lambda-3\}\)</span>. Again, <span class="SimpleMath">\(\mathfrak{A} = \langle\alpha\rangle \times \langle\beta\rangle\)</span>; the order is <span class="SimpleMath">\(2^{\lambda-\sigma-k}\)</span> with <span class="SimpleMath">\(k \in \{0,1,2\}\)</span> (as specified below). In this case, for fixed <span class="SimpleMath">\(p\)</span>, <span class="SimpleMath">\(\lambda\)</span>, <span class="SimpleMath">\(\sigma\)</span>, two pairs <span class="SimpleMath">\((r_1,t_1)\)</span> and <span class="SimpleMath">\((r_2,t_2)\)</span> may give rise to equivalent quadratic modules <a href="chapBib_mj.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="SimpleMath">\(R\)</span>.</p>
<p>First, in the unary family, we have <span class="SimpleMath">\(p\)</span> odd and <span class="SimpleMath">\(\sigma = \lambda\)</span>. Then the second factor of <span class="SimpleMath">\(M\)</span> is trivial (and hence <span class="SimpleMath">\(t\)</span> is irrelevant). We then denote <span class="SimpleMath">\(R_{p^\lambda}(r) = R(p,\lambda,\lambda,r,t)\)</span>. In this case, we do not decompose <span class="SimpleMath">\(W(M,Q)\)</span> using characters: instead, if <span class="SimpleMath">\(\lambda \leq 2\)</span>, then <span class="SimpleMath">\(W(M,Q)\)</span> contains two distinct irreducible subrepresentations of level <span class="SimpleMath">\(p^\lambda\)</span>, denoted <span class="SimpleMath">\(R_{p^\lambda}(r)_{\pm}\)</span>; otherwise, it contains a single such subrepresentation, denoted <span class="SimpleMath">\(R_{p^\lambda}(r)_1\)</span>. The unary family is handled by <code class="func">SL2IrrepRUnary</code> (<a href="chap3_mj.html#X7C94E3007A1BEE85"><span class="RefLink">3.3-3</span></a>) (which is called by <code class="func">SL2IrrepR</code> (<a href="chap3_mj.html#X80961A2C7C5F632E"><span class="RefLink">3.3-2</span></a>) when appropriate).</p>
<p>Second, in the extremal family, we have <span class="SimpleMath">\(p=2\)</span>, <span class="SimpleMath">\(\lambda \geq 2\)</span>, and <span class="SimpleMath">\(\sigma = \lambda - 2\)</span>. Then the second factor of <span class="SimpleMath">\(M\)</span> is isomorphic to <span class="SimpleMath">\(\mathbb{Z}/2\mathbb{Z}\)</span>, and collapses in <span class="SimpleMath">\(2M\)</span>. Here, <span class="SimpleMath">\(\operatorname{Aut}(M,Q)\)</span> is itself abelian, so we let <span class="SimpleMath">\(\mathfrak{A} = \operatorname{Aut}(M,Q)\)</span>. This group has order 1, 2, or 4, with the following structure:</p>
<ul>
<li><p>For <span class="SimpleMath">\(\lambda = 2\)</span> and <span class="SimpleMath">\(t=1\)</span>, <span class="SimpleMath">\(\mathfrak{A} = \langle \tau \rangle\)</span> where <span class="SimpleMath">\(\tau : (x,y) \mapsto (y,x)\)</span>, and <span class="SimpleMath">\(\mathfrak{A}_0 = \mathfrak{A} = \langle\tau\rangle\)</span>.</p>
</li>
<li><p>For <span class="SimpleMath">\(\lambda = 2\)</span> and <span class="SimpleMath">\(t = 3\)</span>, <span class="SimpleMath">\(\mathfrak{A}\)</span> is trivial; there are no primitive characters.</p>
</li>
<li><p>For <span class="SimpleMath">\(\lambda = 3\)</span> or <span class="SimpleMath">\(4\)</span>, <span class="SimpleMath">\(\mathfrak{A} = \{\pm 1\}\)</span> acting on <span class="SimpleMath">\(M\)</span> by multiplication; there are no primitive characters.</p>
</li>
<li><p>Finally, for <span class="SimpleMath">\(\lambda \geq 5\)</span>, <span class="SimpleMath">\(\mathfrak{A} = \operatorname{Aut}(M,Q) = \langle \alpha \rangle \times \langle -1 \rangle\)</span> with <span class="SimpleMath">\(\alpha\)</span> of order 2, and <span class="SimpleMath">\(\mathfrak{A}_0 = \langle\alpha\rangle\)</span>. Note that, for this special case, the usual test for primitivity (described in Section <a href="chap2_mj.html#X7F6278CD87400D49"><spanclass="RefLink">2.1</span></a>) fails, as there are no elements of <span class="SimpleMath">\(\mathfrak{A}\)</span> fixing <span class="SimpleMath">\(2M\)</span> pointwise.</p>
</li>
</ul>
<p>The extremal family is handled by <code class="func">SL2ModuleR</code> (<a href="chap3_mj.html#X7B10D99E7AEAC411"><span class="RefLink">3.3-1</span></a>) and <code class="func">SL2IrrepR</code> (<a href="chap3_mj.html#X80961A2C7C5F632E"><span class="RefLink">3.3-2</span></a>), just like the generic case.</p>
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