Kaiqi Yang(杨凯奇)

I am a Research Assistant Professor in IMSA.

Example D5*D4

Group $D_5 \times D_4$ has two representation $V_1,V_2$.

Representation $V_1$ is given as follows: \(\begin{align*} G_1=\langle \begin{pmatrix} 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}, \begin{pmatrix} \zeta_5 & 0 & 0 & 0\\ 0 & \zeta_5^{-1} & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 \end{pmatrix}, \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & \zeta_4 & 0\\ 0 & 0 & 0 & \zeta_4^{-1} \end{pmatrix}, \rangle \end{align*}\)

Another representation $V_2$ is given as follows:

\[\begin{align*} G_2=\langle \begin{pmatrix} 0 & 1 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}, \begin{pmatrix} \zeta_5^2 & 0 & 0 & 0\\ 0 & \zeta_5^{-2} & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix}, \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 \end{pmatrix}, \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & \zeta_4 & 0\\ 0 & 0 & 0 & \zeta_4^{-1} \end{pmatrix}, \rangle \end{align*}\]

To compare these 2 Burnside classes in $BC_3(D_5 \times D_4)$:

  1. Copy all lines in Computes equivariant birational class in $\mathrm{Burn}_n(G)$ by De Concini-Procesi.
  2. Copy all lines in Compare 2 Burnside Symbols in BC.
  3. Run code in Magma code.

The Magma state is: Here