The minor sharp key in music is usually associated with the eponymous minor key. This is due only to the fact that a similar major is a gamut that is not used because of its initial complexity in terms of key signs, which will be considered a bit later.
Gamma in G Sharp Minor
With minor scale everything is quite simple. According to the principle of determining tonalities (quinto-quart circle), it is a parallel minor to the scale in C major and contains five key characters that apply to the entire gamut. These are the sharp signs fa / do / sol / re / la (standard sequence).
As it should be in this case, three major minor frets are built from the G sharp note: natural, harmonic and melodic minor. In accordance with the rules of solfeggio and harmony of music, the seventh step (F # (F-sharp) to the double-sharp of the same name (F ##)) is increased in a harmonic minor by half a tone. In the melodic minor, when playing gamma upwards, the sixth and seventh steps increase by half a ton (for mi it is a regular sharp (E #), for fa it is a double sharp (double) sharp (F ##)), and when the gamma goes down, the increases by half are canceled.
Gamma in G Sharp Major
Majority is not so simple. The fact is that, as mentioned above, this tonality in music is practically not used and is replaced by an enharmonic (equal in sound).
In this case, this is the usual A flat major. It and with signs are all easier.
But we will dwell separately on the major scale, built from the note of G sharp. In principle, it can be compared with the usual G major, in which all notes are raised by half a tone.
Following the rule of adding sharps or determining tonality by key characters, one could define the sharp order as follows: the usual order from fa to s, and then again increase by a half tone, but already a sharp sharp. Thus, it turns out that the key should contain a f-double-sharp.
It is clear that double sharps with a key are used extremely rarely. Nevertheless, speaking of such a complex range, key signs can be constructed in this order: F-sharp-sharp, and then the usual sequence from note to note s. As you can see, there are a lot of trouble with signs. That is why it is much easier to use the anharmonic flat major, because in its sound the notes in G sharp and A flat are absolutely equal.
The same goes for the parallel E-sharp minor. It is found almost exclusively in the theoretical solfeggio course.
Triads of the main steps
As for the main triads of scales, which are built on the I, III, and IV steps of the scale, for minor the tonic triad is a sequence of raised and pure notes: salt (G #) / pure si (H) / pe (D #), subdominant - to (C # ) / pure mi (E) / salt (G #), the dominant is re (D #) / fa (F ##) / la (A #).
For the major scale constructed from G-sharp, the tonic triad contains such notes increased by half a sharp with sharps: salt (G #) / si (H #) / pe (D #), subdominant - to (C #) / mi (E #) / salt (G #), the dominant is re (D #) / once again elevated fa (F ##) / la (A #).
Total
In conclusion, it remains to add that if there are difficulties with the definition of signs in the key for such complex keys as G sharp major, do not be scared. You just need to apply a clear rule for following sharps with the key one after another. That's all. And those who say that double sharps cannot be held with the key are wrong. Examples with the presence of such signs can be found just a huge number. Another thing is that such tonality remains unclaimed and is almost never used when writing musical works.