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Let $G$ be a finite group and let $\mathcal{A} = \mathbb{C} G$ be the group Hopf algebra.
The comultiplication on $\mathcal{A}$ is well-known to be given by $\Delta(g) = g \otimes g$.

For $G=S_3$, the algebra structure of $\mathcal{A}$ is $\mathbb{C} \oplus \mathbb{C} \oplus M_2(\mathbb{C})$.
Let $\{ e_1,e_2,a_{11}, a_{12}, a_{21}, a_{22} \}$ be a matrix basis of $\mathcal{A}$.

Question: What are the formulas for the comultiplication computed on a matrix basis?

These formulas can be computed by using the irreducible representations of $G$, but there is certainly a reference in which this computation is already done.

I would also be interested by generic formulas for every finite group $G$.

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1 Answer 1

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The first component of $\mathcal{A}$ corresponds to the trivial representation, the second component, to the sign representation, and the third, to the standard representation (see here).

By choosing a specific matrix basis, we obtain:
identity $\to$ $(1) \oplus (1) \oplus \begin{pmatrix} 1& 0 \newline 0& 1 \end{pmatrix}$, $(123)\to$ $(1) \oplus (1) \oplus \begin{pmatrix} z& 0 \newline 0& \bar{z} \end{pmatrix}$,
$(132)\to$ $(1) \oplus (1) \oplus \begin{pmatrix} \bar{z}& 0 \newline 0& z \end{pmatrix}$, $(12) \to$ $(1) \oplus (-1) \oplus \begin{pmatrix} 0& 1 \newline 1& 0 \end{pmatrix}$,
$(23)\to$ $ (1) \oplus (-1) \oplus \begin{pmatrix} 0& \bar{z} \newline z& 0 \end{pmatrix}$, $(13)\to$ $ (1) \oplus (-1) \oplus \begin{pmatrix} 0& z \newline \bar{z}& 0 \end{pmatrix}$

with $z= \zeta_3 = e^{2i\pi/3}$

Now by using the following SageMath computation:

sage: F=matrix([[1,1,1,1,1,1],[1,1,1,-1,-1,-1],[1,z,1/z,0,0,0],[0,0,0,1,1/z,z],[0,0,0,1,z,1/z],[1,1/z,z,0,0,0]])
sage: FI=F^(-1)
sage: BF=[matrix([[sum([F[a][i]*F[b][i]*FI[i][c] for i in range(6)]) for a in range(6)] for b in range(6)]) for c in range(6)]
sage: BF
[
[  1   0   0   0   0   0]  [   0    1    0    0    0    0]
[  0   1   0   0   0   0]  [   1    0    0    0    0    0]
[  0   0   0   0   0 1/2]  [   0    0    0    0    0  1/2]
[  0   0   0   0 1/2   0]  [   0    0    0    0 -1/2    0]
[  0   0   0 1/2   0   0]  [   0    0    0 -1/2    0    0]
[  0   0 1/2   0   0   0], [   0    0  1/2    0    0    0],

[0 0 1 0 0 0]  [ 0  0  0  1  0  0]  [ 0  0  0  0  1  0]  [0 0 0 0 0 1]
[0 0 1 0 0 0]  [ 0  0  0 -1  0  0]  [ 0  0  0  0 -1  0]  [0 0 0 0 0 1]
[1 1 0 0 0 0]  [ 0  0  0  0  0  0]  [ 0  0  0  0  0  0]  [0 0 1 0 0 0]
[0 0 0 0 0 0]  [ 1 -1  0  0  0  0]  [ 0  0  0  1  0  0]  [0 0 0 0 0 0]
[0 0 0 0 0 0]  [ 0  0  0  0  1  0]  [ 1 -1  0  0  0  0]  [0 0 0 0 0 0]
[0 0 0 0 0 1], [ 0  0  0  0  0  0], [ 0  0  0  0  0  0], [1 1 0 0 0 0]
]

we obtain the following formulas for the comultiplication $\Delta$:

$ \Delta(e_{1})=e_{1} \otimes e_{1} + e_{2} \otimes e_{2} + \frac{1}{2}a_{11} \otimes a_{22} + \frac{1}{2}a_{12} \otimes a_{21} + \frac{1}{2}a_{21} \otimes a_{12} + \frac{1}{2}a_{22} \otimes a_{11}$ $\Delta(e_{2})=e_{1} \otimes e_{2} + e_{2} \otimes e_{1} + \frac{1}{2}a_{11} \otimes a_{22} - \frac{1}{2}a_{12} \otimes a_{21} - \frac{1}{2}a_{21} \otimes a_{12} + \frac{1}{2}a_{22} \otimes a_{11}$ $\Delta(a_{11})=e_{1} \otimes a_{11} + e_{2} \otimes a_{11} + a_{11} \otimes e_{1} + a_{11} \otimes e_{2} + a_{22} \otimes a_{22}$
$\Delta(a_{12})= e_{1} \otimes a_{12} - e_{2} \otimes a_{12} + a_{12} \otimes e_{1} - a_{12} \otimes e_{2} + a_{21} \otimes a_{21}$
$\Delta(a_{21})= e_{1} \otimes a_{21} - e_{2} \otimes a_{21} + a_{12} \otimes a_{12} + a_{21} \otimes e_{1} - a_{21} \otimes e_{2}$
$\Delta(a_{22})=e_{1} \otimes a_{22} + e_{2} \otimes a_{22} + a_{11} \otimes a_{11} + a_{22} \otimes e_{1} + a_{22} \otimes e_{2}$

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    $\begingroup$ I don't think there is a division by 6. I think some of the twos should be ones, and some should be halves. Look at e.g. $\Delta(\delta^e)=\delta^e\otimes\delta^e$. $\endgroup$ Sep 13, 2019 at 9:37
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    $\begingroup$ @JPMcCarthy: Thanks! Fixed (better late than never...). $\endgroup$ Oct 12, 2022 at 9:48

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