How to convert between AB and Vega magnitudes? This might be a bit silly, but I'm really struggling with a conversion.
I have apparent AB magnitudes: $$m_{AB} = -2.5 \log_{10}(f_\nu [\mu Jy]) + 25$$ that I would like to convert to Vega magnitudes, because I am trying to make a figure similar to the one below from Stern et al. 2005. I used the conversions from this site: http://www.astronomy.ohio-state.edu/~martini/usefuldata.html, but it came out all wrong. Should I not use these? Because the axes contain colors, I suppose I don't have to worry about the units of $f_\nu$, because this will create an offset that cancels when you subtract two magnitudes, right? But I find that one color-conversion must be positive, because
$$[3.6]_{AB} - [4.5]_{AB} = [3.6]_{Vega} + 0.79 - [4.5]_{Vega} - (-0.09) \\= [3.6]_{Vega} - [4.5]_{Vega} + 0.88$$
while the other is negative, because
$$[5.8]_{AB} - [8.0]_{AB} = [5.8]_{Vega} +0.02 - [8.0]_{Vega} - 0.45\\
= [5.8]_{Vega} - [8.0]_{Vega} - 0.43$$
but if you look at this figure below, you see that you need two corrections with the same sign. Any thoughts on my attempt at conversion? Thank you.
A few things:

The conversion between magnitudes is dependent on the filters being used, you can see it on the tables from the website you linked. That being said, the conversions available in there will not be useful for you if you are trying to replicate Stern et al 2005, as they use redder filters (in the website with the conversions, the longest wavelength available is 2.19 microns, whereas the shortest in Stern et al 2005 is 3.6 microns). If you can find the proper relationships, you should get the expected results.
The paper states that they used IRAC data, so it is a good idea to start looking over there. In fact, you can find some relations in point 7 here and you could also take a look at point 3, might be necessary (I do not know what data you are working with). If you are not using IRAC data, you might as well look for a different source for the conversions.
Are you perhaps using the UltraVISTA catalog? If not, it might also be a good idea to check it out. Their README contains some details about the photometry and references about how they acquired the data (including IRAC), which could give you some more hints.