Implementing characters inside the function
I am new to Sage so probably the question is simple but web search and reading docs did not help so far to find answer. I want to compute Taylor series of certain function that contains character of certain representation inside. As a simple example let's say we want to compute:
$$e^{t*\chi_{Adj}(a)}$$
where $a$ is for example $SU(2)$ matrix. I would like to expand it in series of $t$ and compute coefficient of the singlet representation. I found SageMath WeylCharacterRing method to be impressively effective in computing representations decomposition in irreps and it seems a natural choice for this problem. To solve my problem I take an adjoint representation of SU(2)
G=WeylCharacterRing(['A',1],style="coroots")
ad = G.adjoint_representation()
and try to feed it into symbolic expression
exp(t*ad)
The result is TypeError. There are two problems both due to the TypeError that I see
1) While ad**n gives me write decomposition if I try the following
t = var('t')
g(t) = t**2
g(ad)
I get TypeError. So ad is of the wrong type to be inside symbolic expression as far as I understand.
2) The same happens if I just try to multiply ad with some symbolic variables
t*ad
The question is if there a way to make objects from WeylCharacterRing to work in symbolic expressions. All I need is to substitute representation characters into an expression containing other symbolic variables and powers of representations and expand in irreps.
