Decomposition matrix

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, and in particular <a href="/facts/Modular_representation_theory/2FIf7j4j">modular representation theory</a>, a decomposition matrix is a <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> that results from writing the irreducible <a href="/facts/Ordinary_character/WMp5Q6YE">ordinary characters</a> in terms of the irreducible <a href="/facts/Modular_character/2FIf7j4j">modular characters</a>, where the entries of the two sets of characters are taken to be over all <a href="/facts/Conjugacy_class/AFuPIFNW">conjugacy classes</a> of elements of <a href="/facts/Order_(group_theory)/Cl3tKGFg">order</a> <a href="/facts/Coprime/bnCxCXxs">coprime</a> to the <a href="/facts/Characteristic_(algebra)/D2VzcQaG">characteristic</a> of the <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a>. All such entries in the matrix are non-negative <a href="/facts/Integer/PUwwrawx">integers</a>. The decomposition matrix, multiplied by its <a href="/facts/Transpose/8wmsagGS">transpose</a>, forms the <a href="/facts/Cartan_matrix/9KM1DPn4">Cartan matrix</a>, listing the composition factors of the <a href="/facts/Projective_module/lHymqhNC">projective modules</a>.
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<ul><li>Webb, Peter (2016). <i>A Course in Finite Group Representation Theory</i>. Cambridge: Cambridge University Press. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1017%2Fcbo9781316677216">10.1017/cbo9781316677216</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-316-67721-6.</li></ul>

Decomposition matrix open-in-new

Decomposition matrix