Dimension of a direct sum of characters (example with $S_3$)

Question

Here is the character table of $S_3$:

I was wondering how one can determine the dimension of for example the sign character $sgn$. Could we get it from the character table?

Also, if we define $A$ to be $A=sgn\oplus V$, where $V$ is the standard character. What would be the dimension of $A$ be?

Thank you for your time

Edit: I think it may be given in the first column of the character table... Since these are the degrees of the irreducible representations/characters

Answer 1

The degree of a representation is another name for its dimension, so yes, the (initial) column for the identity element gives the dimension. (This is because the trace of an identity operator is the dimension of the space it acts upon.) And the dimension of a direct sum of representations is the sum of the dimension of the summands, like for any direct sum.