Decomposition matrix: Difference between revisions

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In [[mathematics]], and in particular [[modular representation theory]], a '''decomposition matrix''' is a [[matrix (mathematics)|matrix]] that results from writing the irreducible [[ordinary character]]s in terms of the irreducible [[modular character]]s, where the entries of the two sets of characters are taken to be over all [[conjugacy classesclass]]es of elements of [[order (group theory)|order]] [[coprime]] to the [[characteristic (algebra)|characteristic]] of the [[field (mathematics)|field]]. All such entries in the matrix are non-negative integers[[integer]]s. The decomposition matrix, multiplied by its [[transpose]], forms the [[Cartan matrix]], listing the composition factors of the [[projective modulesmodule]]s.

{{Orphan|date=December 2012}}

In [[mathematics]], and in particular [[modular representation theory]], a '''decomposition matrix''' is a matrix that results from writing the irreducible [[ordinary character]]s in terms of the irreducible [[modular character]]s, where the entries of the two sets of characters are taken to be over all conjugacy classes of elements of order [[coprime]] to the characteristic of the field. All such entries in the matrix are non-negative integers. The decomposition matrix, multiplied by its transpose, forms the [[Cartan matrix]], listing the composition factors of the [[projective modules]].

==References==

* {{cite book | last=Webb | first=Peter | title=A Course in Finite Group Representation Theory | publisher=Cambridge University Press | publication-place=Cambridge | year=2016 | isbn=978-1-316-67721-6 | doi=10.1017/cbo9781316677216 | ref=harv}}

==See also==

*[[Matrix decomposition]]

[[Category:Representation theory of groups]]

[[Category:Matrices (mathematics)]]

{{Abstract-algebramatrix-stub}}