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Let there be a character. Let the character's name be, for example, Alexander.

Now, Alexander's parents call him 'Sasha'. His friends call him 'Xander'. His girlfriend calls him 'Alex'. In formal circumstances, he's 'Alexander, son of Philipp'. To himself, he is all of those - they are, after all, variations on a theme.

What should the omniscient narrator call him? That is, if the omniscient narrator sits on the parents' shoulder, Alexander is called 'Sasha'. If the omniscient narrator sits on the girlfriend's shoulder, Alexander is called 'Alex'. But what if the narrator is sitting on Alexander's shoulder, or on no shoulder at all?

I have a character named Alexander and the name form he thinks of himself as at any one time reflects his mood and the relationship of those around him. To his sister, he is Alex or Xander, rarely Xan - though when angry she calls him Alexander Nicolaus.

When he is with her, he thinks of himself as Alexander or Alex and is called such by me. When he is with a lifelong friend, he sometimes becomes Xan.

When he is thinking professionally and becomes more formal, he refers to himself by his surname.

Use of the various forms will inform the reader both of the depth of the relationship and his attitude at the moment. I trust them not to be bewildered and think he is really five people.

I have situations where he starts out with his surname but starts to relax and thinks of himself as Alexander. A little later he considers himself Alex and is getting rather mellow.

A friend told me I had to change his name because there is already a famous character named Alex. I told this friend it is a relatively common name and mine won’t be mistaken for him, though if he could think of a name that suited him as well and has the flexibility of Alexander - go for it.

My omniscient narrator changes the form of his name to suit the situation. It works.

I would think Alexander would think of himself with the name he first learned and responded to as a child; likely this was what his mother called him on a daily basis, I would guess that is where the "Sasha" comes from.

However, were I writing, the narrator would use "Alex", for brevity in reading and being slightly less intimate than "Sasha". I feel that distance is important: I don't feel like a narrator should present like his friend, to me ruling out "Sasha" and "Xander", but then again the narrator will be talking about him a great deal, and would likely resort to a shortened version of his full first name; hence "Alex" instead of "Alexander" (or something even longer and more formal).

(You picked a great example name; and it makes me wonder if all writers know an Alexander who likes to be called Sasha)

I have a friend named Alexander, who I call Sasha and his answer has always been "it depends on who I'm talking to." If I were to write a book about him and I were me, the narrator; I'd call him Sasha because that's who he is to me. If most people in the book are going to call him Alex, I'm going to let them call him Alex. I'm going to use both names, it might be worth having Alex say at to someone new he is meeting, "Oh, they call me Alex, but my friends and family call me Sasha," which is how he explained it to me after I knew him by the name Sasha for about a year and he finally introduce me to someone who knew him as Alex.

This can be confusing for readers if they aren't familiar with such names, so it's important to stick to one for a bit (unless you need to vary early) and then explain it quickly or make it clear that Sasha is a nickname and this is just accepted. Prose can get away from you and be confusing anyways, so try something heavy and go lighter if your alpha/beta readers complain.