The Character( ct, k ) command creates the kth character of the character table ct, which may be constructed by using the CharacterTable command from a finite group.

Characters are implemented as Maple objects and support several object methods, outlined below.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{Alt}\left(4\right)$

$G≔{\mathbf{A}}_4$

$\mathrm{ct}≔\mathrm{CharacterTable}\left(G\right)$

$\mathrm{ct}≔\left[\begin{array}{ccccc}C& \mathrm{1a}& \mathrm{2a}& \mathrm{3a}& \mathrm{3b}\\ \mathrm{|C|}& 1& 3& 4& 4\\ \mathrm{X1}& 1& 1& 1& 1\\ \mathrm{X2}& 1& 1& -\frac{1}{2}-\frac{\mathrm{I}\sqrt{3}}{2}& -\frac{1}{2}+\frac{\mathrm{I}\sqrt{3}}{2}\\ \mathrm{X3}& 1& 1& -\frac{1}{2}+\frac{\mathrm{I}\sqrt{3}}{2}& -\frac{1}{2}-\frac{\mathrm{I}\sqrt{3}}{2}\\ \mathrm{X4}& 3& \mathrm{-1}& 0& 0\end{array}\right]$

$\mathrm{Character}\left(\mathrm{ct}\,4\right)$

$⟨\text{character:}\text{1a}\to 3\,\text{2a}\to \mathrm{-1}\,\text{3a}\to 0\,\text{3b}\to 0\text{for}{\mathbf{A}}_4⟩$

$\mathrm{c1},\mathrm{c2},\mathrm{c3},\mathrm{c4}≔\mathrm{op}\left(\mathrm{map2}\left(\mathrm{Character}\,\mathrm{ct}\,\left[\mathrm{seq}\right]\left(1..4\right)\right)\right)$

$\mathrm{c1},\mathrm{c2},\mathrm{c3},\mathrm{c4}≔⟨\text{character:}\text{1a}\to 1\,\text{2a}\to 1\,\text{3a}\to 1\,\text{3b}\to 1\text{for}{\mathbf{A}}_4⟩,⟨\text{character:}\text{1a}\to 1\,\text{2a}\to 1\,\text{3a}\to -\frac{1}{2}-\frac{\mathrm{I}\sqrt{3}}{2}\,\text{3b}\to -\frac{1}{2}+\frac{\mathrm{I}\sqrt{3}}{2}\text{for}{\mathbf{A}}_4⟩,⟨\text{character:}\text{1a}\to 1\,\text{2a}\to 1\,\text{3a}\to -\frac{1}{2}+\frac{\mathrm{I}\sqrt{3}}{2}\,\text{3b}\to -\frac{1}{2}-\frac{\mathrm{I}\sqrt{3}}{2}\text{for}{\mathbf{A}}_4⟩,⟨\text{character:}\text{1a}\to 3\,\text{2a}\to \mathrm{-1}\,\text{3a}\to 0\,\text{3b}\to 0\text{for}{\mathbf{A}}_4⟩$

$\mathrm{c4}·\mathrm{c4}$

$1$

$\mathrm{c3}·\mathrm{c4}$

$0$

$\mathrm{Matrix}\left(4\,4\,\left(i\,j\right)\mapsto c\Vert i·c\Vert j\right)$

$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$

$K≔\mathrm{Kernel}\left(\mathrm{c3}\right)$

$K≔⟨\left(1\,4\right)\left(2\,3\right)\,\left(1\,2\right)\left(3\,4\right)⟩$

$\mathrm{IsNormal}\left(K\,G\right)$

$\mathrm{true}$

$\mathrm{GetValues}\left(\mathrm{c3}\right)$

$\left[1\,1\,-\frac{1}{2}+\frac{\mathrm{I}\sqrt{3}}{2}\,-\frac{1}{2}-\frac{\mathrm{I}\sqrt{3}}{2}\right]$

$\mathrm{ctQ}≔\mathrm{CharacterTable}\left(\mathrm{QuaternionGroup}\left(\right)\right)\:$

$\mathrm{c1Q},\mathrm{c2Q},\mathrm{c3Q},\mathrm{c4Q},\mathrm{c5Q}≔\mathrm{op}\left(\mathrm{map2}\left(\mathrm{Character}\,\mathrm{ctQ}\,\left[\mathrm{seq}\right]\left(1..5\right)\right)\right)\:$

$\mathrm{ctD4}≔\mathrm{CharacterTable}\left(\mathrm{DihedralGroup}\left(4\right)\right)\:$

$\mathrm{c1D4},\mathrm{c2D4},\mathrm{c3D4},\mathrm{c4D4},\mathrm{c5D4}≔\mathrm{op}\left(\mathrm{map2}\left(\mathrm{Character}\,\mathrm{ctD4}\,\left[\mathrm{seq}\right]\left(1..5\right)\right)\right)\:$

$\mathrm{map}\left(\mathrm{Indicator}\,\left[\mathrm{c1Q}\,\mathrm{c2Q}\,\mathrm{c3Q}\,\mathrm{c4Q}\,\mathrm{c5Q}\right]\right)$

$\left[1\,1\,1\,1\,\mathrm{-1}\right]$

$\mathrm{map}\left(\mathrm{Indicator}\,\left[\mathrm{c1D4}\,\mathrm{c2D4}\,\mathrm{c3D4}\,\mathrm{c4D4}\,\mathrm{c5D4}\right]\right)$

$\left[1\,1\,1\,1\,1\right]$

The GroupTheory[Character] command was introduced in Maple 2017.

For more information on Maple 2017 changes, see Updates in Maple 2017.