M.P.
This is normal. Sibs don't always get along.
I do think that you can help this by insisting that the girls stay apart for as long as it takes for the older one to be OK being with her little sister. School is hard work and she is telling you she needs alone time. That's a reasonable request that should be honored. I'd be glad that she recognizes her needs.
Yes, the younger girl wants to be with her big sister. Tell her she can be with her after her sister has some quiet time. Help her understand that being together and fighting is not fun but that they can have fun a bit later. Arrange an activity for the younger one.
It may not seem fair because they both have needs but at the same time it's the older one who has an actual physical need. The younger one wants to be with her sister but her need can be postponed much easier.
This reminds me of couples who allow for down time after work. Each does something else until they get together for dinner. Without the time apart the couple fights and the evening is spoiled.
Find a way for both girls to have something to do apart from the other one.