Construct a basis of the matrix elements in the space $\mathbb{C}[S_{3}]$

linear algebramatricesrepresentation-theory

Construct a basis of the matrix elements in the space $\mathbb{C}[S_{3}]$.

If the definition of the Spaces of Matrix Elements is as given below:

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And the answer of the question at the back of Vinberg book (Linear representations of groups) is given below:

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But I do not know how is this the answer, could anyone explain how one element of this table calculated? and what is the number 11 in the second subscript of T?

Best Answer

Note that the definition involves a representation of the group.

From context, $T_1$, $T_2$, and $T_3$ are the irreducible representations of $S_3$. The first two are one dimensional and the last is two dimensional. $T_{k,ij}(g)$ in the table gives the $(i,j)$ entry of the matrix of $T_k(g)$ written in the given basis.

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