<p>Tells you if <var class="Arg">rho</var> is a linear representation of a finite group. The algorithms implemented in this package work on these homomorphisms only.</p>
<p>Uses a basic stabiliser chain for <span class="Math">G</span> to compute the sum described above. This trick requires <var class="Arg">summand</var> to be a function (in the GAP sense) that defines a monoid homomorphism (in the mathematical sense). The computation of the stabiliser chain assumes <var class="Arg">G</var> is a group. More precisely, if we have the basic stabiliser chain:</p>
<p>We traverse the chain from <span class="Math">G_1</span> to <span class="Math">G_n</span>, using the previous sum <span class="Math">G_{i-1}</span> to build the sum <span class="Math">G_i</span>. We do this by using the fact that (writing <span class="Math">f</span> for <var class="Arg">summand</var>)</p>
<p>where the <span class="Math">r_j</span> are right coset representatives of <span class="Math">G_{i-1}</span> in <span class="Math">G_i</span>. The condition on <var class="Arg">summand</var> is satisfied if, for example, it is a linear representation of a group <var class="Arg">G</var>.</p>
<h4>4.3 <span class="Heading">Space-efficient representation of tensors of matrices</span></h4>
<p>Suppose we have representations of <span class="Math">G</span>, <span class="Math">\rho</span> and <span class="Math">\tau</span>, with degree <span class="Math">n</span> and <span class="Math">m</span>. If we would like to construct the tensor product representation of <span class="Math">G</span>, <span class="Math">\rho \otimes \tau</span>, the usual way to do it would be to take the Kronecker product of the matrices. This means we now have to store very large <span class="Math">nm \times nm</span> matrices for each generator of <span class="Math">G</span>. This can be avoided by storing the tensor of matrices as pairs, essentially storing <span class="Math">A \otimes B</span> as a pair <span class="Math">(A,B)</span> and implementing group operations on top of these, along with some representation-theoretic functions. It is only possible to guarantee an economical representation for pure tensors, i.e. matrices of the form <span class="Math">A \otimes B</span>. These are closed under group operations, so it is natural to define a group structure.</p>
<p>Position <span class="Math">i</span> in this representation stores the matrix <span class="Math">A_i</span> in the tensor product <span class="Math">A_1 \otimes A_2</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorProductOfMatrices</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This uses the multiplicativity of characters when taking tensor products to avoid having to compute the trace of a big matrix.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorProductRepLists</code>( <var class="Arg">list1</var>, <var class="Arg">list2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: All possible tensor products given by <span class="Math">\rho \otimes \tau</span> where <span class="Math">\rho : G \to \mbox{GL}(V)</span> is taken from <var class="Arg">list1</var> and <span class="Math">\tau : H \to \mbox{GL}(W)</span> is taken from <var class="Arg">list2</var>. The result is then a list of representations of <span class="Math">G \times H</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DirectSumOfRepresentations</code>( <var class="Arg">list</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: Direct sum of the list of representations <var class="Arg">list</var></p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DegreeOfRepresentation</code>( <var class="Arg">rho</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: Degree of the representation <var class="Arg">rho</var>. That is, <span class="Math">\mbox{Tr}(\rho(e_G))</span>, where <span class="Math">e_G</span> is the identity of the group <spanclass="Math">G</span> that <var class="Arg">rho</var> has as domain.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsOrthonormalSet</code>( <var class="Arg">S</var>, <var class="Arg">prod</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: Whether <var class="Arg">S</var> is an orthonormal set with respect to the inner product <var class="Arg">prod</var>.</p>
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