<h3>3 <span class="Heading">Irreducible representations of prime-power level</span></h3>
<p>Methods for generating individual irreducible representations of <span class="Math">\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})</span> for a given level <span class="Math">p^\lambda</span>.</p>
<p>After generating a representation <span class="Math">\rho</span> by means of the bases in <a href="chapBib.html#biBNW76">[NW76]</a>, we perform a change of basis that results in a symmetric representation equivalent to <span class="Math">\rho</span>.</p>
<p>In each case (except the unary type <span class="Math">R</span>, for which see <code class="func">SL2IrrepRUnary</code> (<a href="chap3.html#X7C94E3007A1BEE85"><span class="RefLink">3.3-3</span></a>)), the underlying module <span class="Math">M</span> is of rank <span class="Math">2</span>, so its elements have the form <span class="Math">(x,y)</span> and are thus represented by lists <code class="code">[x,y]</code>.</p>
<p>Characters of the abelian group <span class="Math">\mathfrak{A} = \langle\alpha\rangle \times \langle\beta\rangle</span> have the form <span class="Math">\chi_{i,j}</span>, given by</p>
<p>where <span class="Math">i</span> and <span class="Math">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="Math">\alpha</span> or <span class="Math">\beta</span> is trivial, and the corresponding index <span class="Math">i</span> or <span class="Math">j</span> is therefore irrelevant.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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="Math">(M,Q)</span>.</p>
<p>Constructs information about the underlying quadratic module <span class="Math">(M,Q)</span> of type <span class="Math">D</span>, for <span class="Math">p</span> a prime and <span class="Math">\lambda \geq 1</span>.</p>
<p><code class="code">Agrp</code> is a list describing the elements of <span class="Math">\mathfrak{A}</span>. Each element <span class="Math">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="Math">a = \alpha^{\mathtt{v[1]}} \beta^{\mathtt{v[2]}}</span>. Note that <span class="Math">\beta</span> is trivial, and hence <code class="code">v[2]</code> is irrelevant, when <span class="Math">\mathfrak{A}</span> is cyclic.</p>
<p><code class="code">Bp</code> is a list of representatives for the <span class="Math">\mathfrak{A}</span>-orbits on <span class="Math">M^\times</span>, which correspond to a basis for the <span class="Math">\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})</span>-invariant subspace associated to any primitive character <span class="Math">\chi \in \widehat{\mathfrak{A}}</span> with <span class="Math">\chi^2 \not\equiv 1</span>. This is the basis given by <a href="chapBib.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.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="Math">i</span>, <span class="Math">j</span> to a function representing the character <span class="Math">\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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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="Math">[S,T]</span>.</p>
<p>Constructs the modular data for the irreducible representation(s) of type <span class="Math">D</span> with level <span class="Math">p^\lambda</span>, for <span class="Math">p</span> a prime and <span class="Math">\lambda \geq 1</span>, corresponding to the character <span class="Math">\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.html#X845D92CB7841CB0B"><span class="RefLink">3.1-1</span></a>)).</p>
<p>Here <span class="Math">S</span> is symmetric and unitary and <span class="Math">T</span> is diagonal.</p>
<p>Depending on the parameters, <span class="Math">W(M,Q)</span> will contain either 1 or 2 such irreps.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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="Math">(M,Q)</span>.</p>
<p>Constructs information about the underlying quadratic module <span class="Math">(M,Q)</span> of type <span class="Math">N</span>, for <span class="Math">p</span> a prime and <span class="Math">\lambda \geq 1</span>.</p>
<p><code class="code">Agrp</code> is a list describing the elements of <span class="Math">\mathfrak{A}</span>. Each element <span class="Math">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="Math">a = \alpha^{\mathtt{v[1]}} \beta^{\mathtt{v[2]}}</span>. Note that <span class="Math">\alpha</span> is trivial, and hence <code class="code">v[1]</code> is irrelevant, when <span class="Math">\mathfrak{A}</span> is cyclic.</p>
<p><code class="code">Bp</code> is a list of representatives for the <span class="Math">\mathfrak{A}</span>-orbits on <span class="Math">M^\times</span>, which correspond to a basis for the <span class="Math">\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})</span>-invariant subspace associated to any primitive character <span class="Math">\chi \in \widehat{\mathfrak{A}}</span> with <span class="Math">\chi^2 \not\equiv 1</span>. This is the basis given by <a href="chapBib.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.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="Math">i</span>, <span class="Math">j</span> to a function representing the character <span class="Math">\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="Math">M</span>, respectively.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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="Math">[S,T]</span>.</p>
<p>Constructs the modular data for the irreducible representation(s) of type <span class="Math">N</span> with level <span class="Math">p^\lambda</span>, for <span class="Math">p</span> a prime and <span class="Math">\lambda \geq 1</span>, corresponding to the character <span class="Math">\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.html#X7A50CC5A7933E207"><span class="RefLink">3.2-1</span></a>)).</p>
<p>Here <span class="Math">S</span> is symmetric and unitary and <span class="Math">T</span> is diagonal.</p>
<p>Depending on the parameters, <span class="Math">W(M,Q)</span> will contain either 1 or 2 such irreps.</p>
<p>Constructs information about the underlying quadratic module <span class="Math">(M,Q)</span> of type <span class="Math">R</span>, for <span class="Math">p</span> a prime. The additional parameters <span class="Math">\lambda</span>, <span class="Math">\sigma</span>, <span class="Math">r</span>, and <span class="Math">t</span> should be integers chosen as follows.</p>
<p>If <span class="Math">p</span> is an odd prime, let <span class="Math">\lambda \geq 2</span>, <span class="Math">\sigma \in \{1, \dots, \lambda - 1\}</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>. Note that <span class="Math">\sigma = \lambda</span> is a valid choice for type <span class="Math">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.html#X7C94E3007A1BEE85"><span class="RefLink">3.3-3</span></a>) instead.</p>
<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>.</p>
<p><code class="code">Agrp</code> is a list describing the elements of <span class="Math">\mathfrak{A}</span>. Each element <span class="Math">a</span> of <span class="Math">\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="Math">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="Math">\mathfrak{A}</span>-orbits on <span class="Math">M^\times</span>, which correspond to a basis for the <span class="Math">\mathrm{SL}_2(\mathbb{Z}/p^\lambda\mathbb{Z})</span>-invariant subspace associated to any primitive character <span class="Math">\chi \in \widehat{\mathfrak{A}}</span> with <span class="Math">\chi^2 \not\equiv 1</span>. This is the basis given by <a href="chapBib.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.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="Math">i</span>, <span class="Math">j</span> to a function representing the character <span class="Math">\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="Math">M</span>, respectively.</p>
<p><code class="code">c</code> is a scalar used in calculating the <span class="Math">S</span>-matrix; namely <span class="Math">c = \frac{1}{|M|} \sum_{x \in M} \mathbf{e}(Q(x))</span>. Note that this is equal to <span class="Math">S_Q(-1) / \sqrt{|M|}</span>, where <span class="Math">S_Q(-1)</span> is the central charge (see Section <a href="chap2.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="Math">M - pM</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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="Math">[S,T]</span>.</p>
<p>Constructs the modular data for the irreducible representation(s) of type <span class="Math">R</span> with parameters <span class="Math">p</span>, <span class="Math">\lambda</span>, <span class="Math">\sigma</span>, <span class="Math">r</span>, and <span class="Math">t</span>, corresponding to the character <span class="Math">\chi</span> indexed by <code class="code">chi_index = [i,j]</code> (see the discussions of <span class="Math">\sigma</span>, <span class="Math">r</span>, <span class="Math">t</span>, and <code class="code">Char(i,j)</code> in <code class="func">SL2ModuleR</code> (<a href="chap3.html#X7B10D99E7AEAC411"><span class="RefLink">3.3-1</span></a>)).</p>
<p>Here <span class="Math">S</span> is symmetric and unitary and <span class="Math">T</span> is diagonal.</p>
<p>Depending on the parameters, <span class="Math">W(M,Q)</span> will contain either 1 or 2 such irreps.</p>
<p>If <span class="Math">\sigma = \lambda</span> for <span class="Math">p \neq 2</span>, then the second factor of <span class="Math">M</span> is trivial (and hence <span class="Math">t</span> is irrelevant), so this falls through to <code class="func">SL2IrrepRUnary</code> (<a href="chap3.html#X7C94E3007A1BEE85"><span class="RefLink">3.3-3</span></a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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="Math">[S,T]</span>.</p>
<p>Constructs the modular data for the irreducible representation(s) of unary type <span class="Math">R</span> (that is, the special case where <span class="Math">\sigma = \lambda</span>) with <span class="Math">p</span> an odd prime, <span class="Math">\lambda</span> a positive integer, and <span class="Math">r \in \{1,u\}</span> with <span class="Math">u</span> a quadratic non-residue mod <span class="Math">p</span>.</p>
<p>Here <span class="Math">S</span> is symmetric and unitary and <span class="Math">T</span> is diagonal.</p>
<p>In this case, <span class="Math">W(M,Q)</span> always contains exactly 2 such irreps.</p>
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