In a lecture note(written for physics graduate student) i found the following exercise:
Recall that $V=R^2$ is a $2$ dimensional representation of $D_3$(symmetries of triangle),hence $V \otimes V$ is also a representation of $D_3$,in particular consider the completely symmetric tensor product of $k$ copies of $V$.Each vector in $V$ is labeled by the coordinates $(x_1,x_2)$ of position vectors.
For $k=1$ we can take the the basis functions to be $x_1,x_2$ then for $k=2$ one can take the three basis functions $\phi_1=x_1^2,\phi_2=x_2^2,\phi_3= \sqrt2 x_1x_2$.Use $\phi_i$ to find $3$ dimensional matrix representation of $D_3$.
In my Representation theory course (for pure maths students)if i recall correctly i never heard about basis functions so i am wondering what are these basis functions and how do we use them to construct a matrix representation?
Physicists talk about math in a slightly different way than mathematicians, so I sympathize with your confusion. I think what's going on here is that the original representation $V$ is a vector space of functions. You're saying $V=\mathbb{R}^2$, but an isomorphic version would be the dual of $\mathbb{R}^2$, or linear functions on $\mathbb{R}^2$. It's spanned by the coordinate functions $x_1$ and $x_2$. The symmetric tensor powers of $V^*$ are polynomials in two variables.
So the author is using basis functions to mashup these structures of $S^kV^*$. The use of basis comes from its nature as a vector space; the use of function because they are in in fact functions on $\mathbb{R}^2$.