Dear A.,
My kids are 10 years apart, so I definitely know about the "different interests" part. My daughter is the oldest and wasn't shy or afraid of anything or anybody. My son is far more mellow and casual about things. And very sensitive when it comes to other people's feelings, etc. He was not nearly as gregarious as his sister, but always had many friends. His first best friend was a little girl who lived a few houses down from us. We could see her and her mom out in the yard and he just ached to go play with her, but was far too shy. So, one day, I told him we were going on a walk and we introduced ourselves. They were only 3, but it was the cutest thing to watch them play and they just adored each other. He and a little boy buddied up their very first day of kindergarten. They'd never seen each other in their lives, but right away, they took to each other. They're 13 now and my son was at their house for dinner and a movie just last night. Special friendships really are important. But, you can't force them. They just kind of happen.
I had friends whose daughter was very introverted at school as well. The school is on a street close to the office we worked at and they would often drive by around lunch time and could see their daughter sitting out on the playground by the fence all by herself. It just broke their hearts. But, she had plenty of church and social friends.
Sometimes kids tend to pal up and group up. Interjecting herself might be something your daughter isn't especially interested in doing at school, especially if no one has really approached her. And, the other kids may be thinking she's too shy to approach.
Your daughter may never be the class social butterfly, but I bet that she will come into her own.
My best friend is a teacher and I know that pairing kids up to work on projects together is a great ice breaker. And, it's a way for the class to be involved with one another and interact with children who aren't necessarily the same ones they hang out with on the playground all the time.
She's only 6. Don't worry too much but encourage her to talk about her classmates. Who does she sit by in class? Who did she sit by for lunch? Does she know if any of them have pets at home? Are any of them allowed to walk or ride their bikes? You don't want to give her a zillion questions a day at a time, but it might at least get her talking about it. It might give you a clue as to why she prefers to keep her distance. If she just feels safer keeping it that way for now, that's okay too.