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<div class="ChapSects"><a href="chap3_mj.html#X7C4165447B8AB223">3 <span class="Heading">Irreducible representations of prime-power level</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X82DA126D7F755B1C">3.1 <span class="Heading">Representations of type D</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X845D92CB7841CB0B">3.1-1 SL2ModuleD</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7FDB517981A2C091">3.1-2 SL2IrrepD</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7CD74CFC84B38042">3.2 <span class="Heading">Representations of type N</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7A50CC5A7933E207">3.2-1 SL2ModuleN</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X81D60FE878F02838">3.2-2 SL2IrrepN</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7D9759387C499FA0">3.3 <span class="Heading">Representations of type R</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7B10D99E7AEAC411">3.3-1 SL2ModuleR</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X80961A2C7C5F632E">3.3-2 SL2IrrepR</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7C94E3007A1BEE85">3.3-3 SL2IrrepRUnary</a></span>
</div></div>
</div>
<h3>3 <span class="Heading">Irreducible representations of prime-power level</span></h3>
<p>Methods for generating individual irreducible representations of <span class="SimpleMath">\(\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})\)</span> for a given level <span class="SimpleMath">\(p^\lambda\)</span>.</p>
<p>After generating a representation <span class="SimpleMath">\(\rho\)</span> by means of the bases in <a href="chapBib_mj.html#biBNW76">[NW76]</a>, we perform a change of basis that results in a symmetric representation equivalent to <span class="SimpleMath">\(\rho\)</span>.</p>
<p>In each case (except the unary type <span class="SimpleMath">\(R\)</span>, for which see <code class="func">SL2IrrepRUnary</code> (<a href="chap3_mj.html#X7C94E3007A1BEE85"><span class="RefLink">3.3-3</span></a>)), the underlying module <span class="SimpleMath">\(M\)</span> is of rank <span class="SimpleMath">\(2\)</span>, so its elements have the form <span class="SimpleMath">\((x,y)\)</span> and are thus represented by lists <code class="code">[x,y]</code>.</p>
<p>Characters of the abelian group <span class="SimpleMath">\(\mathfrak{A} = \langle\alpha\rangle \times \langle\beta\rangle\)</span> have the form <span class="SimpleMath">\(\chi_{i,j}\)</span>, given by</p>
<p class="center">\[\chi_{i,j}(\alpha^{v}\beta^{w}) \mapsto \mathbf{e}\left(\frac{vi}{|\alpha|}\right) \mathbf{e}\left(\frac{wj}{|\beta|}\right)~,\]</p>
<p>where <span class="SimpleMath">\(i\)</span> and <span class="SimpleMath">\(j\)</span> are integers. We therefore represent each character by a list <code class="code">[i,j]</code>. Note that in some cases <span class="SimpleMath">\(\alpha\)</span> or <span class="SimpleMath">\(\beta\)</span> is trivial, and the corresponding index <span class="SimpleMath">\(i\)</span> or <span class="SimpleMath">\(j\)</span> is therefore irrelevant.</p>
<p>We write <code class="code">p=</code><span class="SimpleMath">\(p\)</span>, <code class="code">lambda=</code><span class="SimpleMath">\(\lambda\)</span>, <code class="code">sigma=</code><span class="SimpleMath">\(\sigma\)</span>, and <code class="code">chi=</code><span class="SimpleMath">\(\chi\)</span>.</p>
<p><a id="X82DA126D7F755B1C" name="X82DA126D7F755B1C"></a></p>
<h4>3.1 <span class="Heading">Representations of type D</span></h4>
<p>See Section <a href="chap2_mj.html#X7AEC048F793F1D79"><span class="RefLink">2.2-1</span></a>.</p>
<p><a id="X845D92CB7841CB0B" name="X845D92CB7841CB0B"></a></p>
<h5>3.1-1 SL2ModuleD</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SL2ModuleD</code>( <var class="Arg">p</var>, <var class="Arg">lambda</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a record <code class="code">rec(Agrp, Bp, Char, IsPrim)</code> describing <span class="SimpleMath">\((M,Q)\)</span>.</p>
<p>Constructs information about the underlying quadratic module <span class="SimpleMath">\((M,Q)\)</span> of type <span class="SimpleMath">\(D\)</span>, for <span class="SimpleMath">\(p\)</span> a prime and <span class="SimpleMath">\(\lambda \geq 1\)</span>.</p>
<p><code class="code">Agrp</code> is a list describing the elements of <span class="SimpleMath">\(\mathfrak{A}\)</span>. Each element <span class="SimpleMath">\(a \in \mathfrak{A}\)</span> is represented in <code class="code">Agrp</code> by a list <code class="code">[v, a, a_inv]</code>, where <code class="code">v</code> is a list defined by <span class="SimpleMath">\(a = \alpha^{\mathtt{v[1]}} \beta^{\mathtt{v[2]}}\)</span>. Note that <span class="SimpleMath">\(\beta\)</span> is trivial, and hence <code class="code">v[2]</code> is irrelevant, when <span class="SimpleMath">\(\mathfrak{A}\)</span> is cyclic.</p>
<p><code class="code">Bp</code> is a list of representatives for the <span class="SimpleMath">\(\mathfrak{A}\)</span>-orbits on <span class="SimpleMath">\(M^\times\)</span>, which correspond to a basis for the <span class="SimpleMath">\(\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})\)</span>-invariant subspace associated to any primitive character <span class="SimpleMath">\(\chi \in \widehat{\mathfrak{A}}\)</span> with <span class="SimpleMath">\(\chi^2 \not\equiv 1\)</span>. This is the basis given by <a href="chapBib_mj.html#biBNW76">[NW76]</a>, which may result in a non-symmetric representation; if this occurs, we perform a change of basis in <code class="func">SL2IrrepD</code> (<a href="chap3_mj.html#X7FDB517981A2C091"><span class="RefLink">3.1-2</span></a>) to obtain a symmetric representation. For non-primitive characters, we must use different bases which are particular to each case.</p>
<p><code class="code">Char(i,j)</code> converts two integers <span class="SimpleMath">\(i\)</span>, <span class="SimpleMath">\(j\)</span> to a function representing the character <span class="SimpleMath">\(\chi_{i,j} \in \widehat{\mathfrak{A}}\)</span>.</p>
<p><code class="code">IsPrim(chi)</code> tests whether the output of <code class="code">Char(i,j)</code> represents a primitive character.</p>
<p><a id="X7FDB517981A2C091" name="X7FDB517981A2C091"></a></p>
<h5>3.1-2 SL2IrrepD</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SL2IrrepD</code>( <var class="Arg">p</var>, <var class="Arg">lambda</var>, <var class="Arg">chi_index</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list of lists of the form <span class="SimpleMath">\([S,T]\)</span>.</p>
<p>Constructs the modular data for the irreducible representation(s) of type <span class="SimpleMath">\(D\)</span> with level <span class="SimpleMath">\(p^\lambda\)</span>, for <span class="SimpleMath">\(p\)</span> a prime and <span class="SimpleMath">\(\lambda \geq 1\)</span>, corresponding to the character <span class="SimpleMath">\(\chi\)</span> indexed by <code class="code">chi_index = [i,j]</code> (see the discussion of <code class="code">Char(i,j)</code> in <code class="func">SL2ModuleD</code> (<a href="chap3_mj.html#X845D92CB7841CB0B"><span class="RefLink">3.1-1</span></a>)).</p>
<p>Here <span class="SimpleMath">\(S\)</span> is symmetric and unitary and <span class="SimpleMath">\(T\)</span> is diagonal.</p>
<p>Depending on the parameters, <span class="SimpleMath">\(W(M,Q)\)</span> will contain either 1 or 2 such irreps.</p>
<p><a id="X7CD74CFC84B38042" name="X7CD74CFC84B38042"></a></p>
<h4>3.2 <span class="Heading">Representations of type N</span></h4>
<p>See Section <a href="chap2_mj.html#X84E15A1E8326AF4F"><span class="RefLink">2.2-2</span></a>.</p>
<p><a id="X7A50CC5A7933E207" name="X7A50CC5A7933E207"></a></p>
<h5>3.2-1 SL2ModuleN</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SL2ModuleN</code>( <var class="Arg">p</var>, <var class="Arg">lambda</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a record <code class="code">rec(Agrp, Bp, Char, IsPrim, Nm, Prod)</code> describing <span class="SimpleMath">\((M,Q)\)</span>.</p>
<p>Constructs information about the underlying quadratic module <span class="SimpleMath">\((M,Q)\)</span> of type <span class="SimpleMath">\(N\)</span>, for <span class="SimpleMath">\(p\)</span> a prime and <span class="SimpleMath">\(\lambda \geq 1\)</span>.</p>
<p><code class="code">Agrp</code> is a list describing the elements of <span class="SimpleMath">\(\mathfrak{A}\)</span>. Each element <span class="SimpleMath">\(a \in \mathfrak{A}\)</span> is represented in <code class="code">Agrp</code> by a list <code class="code">[v, a]</code>, where <code class="code">v</code> is a list defined by <span class="SimpleMath">\(a = \alpha^{\mathtt{v[1]}} \beta^{\mathtt{v[2]}}\)</span>. Note that <span class="SimpleMath">\(\alpha\)</span> is trivial, and hence <code class="code">v[1]</code> is irrelevant, when <span class="SimpleMath">\(\mathfrak{A}\)</span> is cyclic.</p>
<p><code class="code">Bp</code> is a list of representatives for the <span class="SimpleMath">\(\mathfrak{A}\)</span>-orbits on <span class="SimpleMath">\(M^\times\)</span>, which correspond to a basis for the <span class="SimpleMath">\(\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})\)</span>-invariant subspace associated to any primitive character <span class="SimpleMath">\(\chi \in \widehat{\mathfrak{A}}\)</span> with <span class="SimpleMath">\(\chi^2 \not\equiv 1\)</span>. This is the basis given by <a href="chapBib_mj.html#biBNW76">[NW76]</a>, which may result in a non-symmetric representation; if this occurs, we perform a change of basis in <code class="func">SL2IrrepD</code> (<a href="chap3_mj.html#X7FDB517981A2C091"><span class="RefLink">3.1-2</span></a>) to obtain a symmetric representation. For non-primitive characters, we must use different bases which are particular to each case.</p>
<p><code class="code">Char(i,j)</code> converts two integers <span class="SimpleMath">\(i\)</span>, <span class="SimpleMath">\(j\)</span> to a function representing the character <span class="SimpleMath">\(\chi_{i,j} \in \widehat{\mathfrak{A}}\)</span>.</p>
<p><code class="code">IsPrim(chi)</code> tests whether the output of <code class="code">Char(i,j)</code> represents a primitive character.</p>
<p><code class="code">Nm(a)</code> and <code class="code">Prod(a,b)</code> are the norm and product functions on <span class="SimpleMath">\(M\)</span>, respectively.</p>
<p><a id="X81D60FE878F02838" name="X81D60FE878F02838"></a></p>
<h5>3.2-2 SL2IrrepN</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SL2IrrepN</code>( <var class="Arg">p</var>, <var class="Arg">lambda</var>, <var class="Arg">chi_index</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list of lists of the form <span class="SimpleMath">\([S,T]\)</span>.</p>
<p>Constructs the modular data for the irreducible representation(s) of type <span class="SimpleMath">\(N\)</span> with level <span class="SimpleMath">\(p^\lambda\)</span>, for <span class="SimpleMath">\(p\)</span> a prime and <span class="SimpleMath">\(\lambda \geq 1\)</span>, corresponding to the character <span class="SimpleMath">\(\chi\)</span> indexed by <code class="code">chi_index = [i,j]</code> (see the discussion of <code class="code">Char(i,j)</code> in <code class="func">SL2ModuleN</code> (<a href="chap3_mj.html#X7A50CC5A7933E207"><span class="RefLink">3.2-1</span></a>)).</p>
<p>Here <span class="SimpleMath">\(S\)</span> is symmetric and unitary and <span class="SimpleMath">\(T\)</span> is diagonal.</p>
<p>Depending on the parameters, <span class="SimpleMath">\(W(M,Q)\)</span> will contain either 1 or 2 such irreps.</p>
<p><a id="X7D9759387C499FA0" name="X7D9759387C499FA0"></a></p>
<h4>3.3 <span class="Heading">Representations of type R</span></h4>
<p>See Section <a href="chap2_mj.html#X7B788E2F87AF51CA"><span class="RefLink">2.2-3</span></a>.</p>
<p><a id="X7B10D99E7AEAC411" name="X7B10D99E7AEAC411"></a></p>
<h5>3.3-1 SL2ModuleR</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SL2ModuleR</code>( <var class="Arg">p</var>, <var class="Arg">lambda</var>, <var class="Arg">sigma</var>, <var class="Arg">r</var>, <var class="Arg">t</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a record <code class="code">rec(Agrp, Bp, Char, IsPrim, Nm, Ord, Prod, c, tM)</code> describing <span class="SimpleMath">\((M,Q)\)</span>.</p>
<p>Constructs information about the underlying quadratic module <span class="SimpleMath">\((M,Q)\)</span> of type <span class="SimpleMath">\(R\)</span>, for <span class="SimpleMath">\(p\)</span> a prime. The additional parameters <span class="SimpleMath">\(\lambda\)</span>, <span class="SimpleMath">\(\sigma\)</span>, <span class="SimpleMath">\(r\)</span>, and <span class="SimpleMath">\(t\)</span> should be integers chosen as follows.</p>
<p>If <span class="SimpleMath">\(p\)</span> is an odd prime, let <span class="SimpleMath">\(\lambda \geq 2\)</span>, <span class="SimpleMath">\(\sigma \in \{1, \dots, \lambda - 1\}\)</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>. Note that <span class="SimpleMath">\(\sigma = \lambda\)</span> is a valid choice for type <span class="SimpleMath">\(R\)</span>, however, this gives the unary case, and so is not handled by this function, as it is decomposed in a different way; for this case, use <code class="func">SL2IrrepRUnary</code> (<a href="chap3_mj.html#X7C94E3007A1BEE85"><span class="RefLink">3.3-3</span></a>) instead.</p>
<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>.</p>
<p><code class="code">Agrp</code> is a list describing the elements of <span class="SimpleMath">\(\mathfrak{A}\)</span>. Each element <span class="SimpleMath">\(a\)</span> of <span class="SimpleMath">\(\mathfrak{A}\)</span> is represented in <code class="code">Agrp</code> by a list <code class="code">[v, a]</code>, where <code class="code">v</code> is a list defined by <span class="SimpleMath">\(a = \alpha^{\mathtt{v[1]}} \beta^{\mathtt{v[2]}}\)</span>.</p>
<p><code class="code">Bp</code> is a list of representatives for the <span class="SimpleMath">\(\mathfrak{A}\)</span>-orbits on <span class="SimpleMath">\(M^\times\)</span>, which correspond to a basis for the <span class="SimpleMath">\(\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})\)</span>-invariant subspace associated to any primitive character <span class="SimpleMath">\(\chi \in \widehat{\mathfrak{A}}\)</span> with <span class="SimpleMath">\(\chi^2 \not\equiv 1\)</span>. This is the basis given by <a href="chapBib_mj.html#biBNW76">[NW76]</a>, which may result in a non-symmetric representation; if this occurs, we perform a change of basis in <code class="func">SL2IrrepD</code> (<a href="chap3_mj.html#X7FDB517981A2C091"><span class="RefLink">3.1-2</span></a>) to obtain a symmetric representation. For non-primitive characters, we must use different bases which are particular to each case.</p>
<p><code class="code">Char(i,j)</code> converts two integers <span class="SimpleMath">\(i\)</span>, <span class="SimpleMath">\(j\)</span> to a function representing the character <span class="SimpleMath">\(\chi_{i,j} \in \widehat{\mathfrak{A}}\)</span>.</p>
<p><code class="code">IsPrim(chi)</code> tests whether the output of <code class="code">Char(i,j)</code> represents a primitive character.</p>
<p><code class="code">Nm(a)</code>, <code class="code">Ord(a)</code>, and <code class="code">Prod(a,b)</code> are the norm, order, and product functions on <span class="SimpleMath">\(M\)</span>, respectively.</p>
<p><code class="code">c</code> is a scalar used in calculating the <span class="SimpleMath">\(S\)</span>-matrix; namely <span class="SimpleMath">\(c = \frac{1}{|M|} \sum_{x \in M} \mathbf{e}(Q(x))\)</span>. Note that this is equal to <span class="SimpleMath">\(S_Q(-1) / \sqrt{|M|}\)</span>, where <span class="SimpleMath">\(S_Q(-1)\)</span> is the central charge (see Section <a href="chap2_mj.html#X86466B2786DD47C4"><span class="RefLink">2.1-1</span></a>).</p>
<p><code class="code">tM</code> is a list describing the elements of the group <span class="SimpleMath">\(M - pM\)</span>.</p>
<p><a id="X80961A2C7C5F632E" name="X80961A2C7C5F632E"></a></p>
<h5>3.3-2 SL2IrrepR</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SL2IrrepR</code>( <var class="Arg">p</var>, <var class="Arg">lambda</var>, <var class="Arg">sigma</var>, <var class="Arg">r</var>, <var class="Arg">t</var>, <var class="Arg">chi_index</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list of lists of the form <span class="SimpleMath">\([S,T]\)</span>.</p>
<p>Constructs the modular data for the irreducible representation(s) of type <span class="SimpleMath">\(R\)</span> with parameters <span class="SimpleMath">\(p\)</span>, <span class="SimpleMath">\(\lambda\)</span>, <span class="SimpleMath">\(\sigma\)</span>, <span class="SimpleMath">\(r\)</span>, and <span class="SimpleMath">\(t\)</span>, corresponding to the character <span class="SimpleMath">\(\chi\)</span> indexed by <code class="code">chi_index = [i,j]</code> (see the discussions of <span class="SimpleMath">\(\sigma\)</span>, <span class="SimpleMath">\(r\)</span>, <span class="SimpleMath">\(t\)</span>, and <code class="code">Char(i,j)</code> in <code class="func">SL2ModuleR</code> (<a href="chap3_mj.html#X7B10D99E7AEAC411"><span class="RefLink">3.3-1</span></a>)).</p>
<p>Here <span class="SimpleMath">\(S\)</span> is symmetric and unitary and <span class="SimpleMath">\(T\)</span> is diagonal.</p>
<p>Depending on the parameters, <span class="SimpleMath">\(W(M,Q)\)</span> will contain either 1 or 2 such irreps.</p>
<p>If <span class="SimpleMath">\(\sigma = \lambda\)</span> for <span class="SimpleMath">\(p \neq 2\)</span>, then the second factor of <span class="SimpleMath">\(M\)</span> is trivial (and hence <span class="SimpleMath">\(t\)</span> is irrelevant), so this falls through to <code class="func">SL2IrrepRUnary</code> (<a href="chap3_mj.html#X7C94E3007A1BEE85"><span class="RefLink">3.3-3</span></a>).</p>
<p><a id="X7C94E3007A1BEE85" name="X7C94E3007A1BEE85"></a></p>
<h5>3.3-3 SL2IrrepRUnary</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SL2IrrepRUnary</code>( <var class="Arg">p</var>, <var class="Arg">lambda</var>, <var class="Arg">r</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list of lists of the form <span class="SimpleMath">\([S,T]\)</span>.</p>
<p>Constructs the modular data for the irreducible representation(s) of unary type <span class="SimpleMath">\(R\)</span> (that is, the special case where <span class="SimpleMath">\(\sigma = \lambda\)</span>) with <span class="SimpleMath">\(p\)</span> an odd prime, <span class="SimpleMath">\(\lambda\)</span> a positive integer, and <span class="SimpleMath">\(r \in \{1,u\}\)</span> with <span class="SimpleMath">\(u\)</span> a quadratic non-residue mod <span class="SimpleMath">\(p\)</span>.</p>
<p>Here <span class="SimpleMath">\(S\)</span> is symmetric and unitary and <span class="SimpleMath">\(T\)</span> is diagonal.</p>
<p>In this case, <span class="SimpleMath">\(W(M,Q)\)</span> always contains exactly 2 such irreps.</p>
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