It's very important to give children options. So much of their lives they ha e no control of yet and that's why saying no all the time and having tantrums tend to happen. I'm not sure why she can't wear her hair down or her mother insists on it. But how her pictures of different pony tails, braided, high on the head, low on the head etc and let her choose which one. If she still wants it to be down then all you can do is force her. But honestly over such a little thing as her hair style I won't force her. She's just now starting to have her own voice and ideas and opinions, we want our children to grow up and take action and be decisive. She is learning to be decisive now.
you ask her nicely and try to convince her nicely that pigtails look prettier on her. to convince her you can ask the help of other people in your house perhaps to agree with you. always maintain a calm sweet jolly voice. don't ever demand cause the next scene would be chaotic. if she agreed, you give her compliment like "wow! our baby is such a good girl" with matching clapping of your hands and the people around or a kiss on her cheek. if she would still insist then just let her wear her hair down. :)
I think parents get too hung up on what little kids wear. Their expression of themselves is more important than you think. They're people, and they don't like being oppressed. I actually remember being that age and wanting to wear mismatched, gaudy clothes and when my mom made me wear polka dots (which I hated) it actually made me resent her a little.
Pig tails are cute, but let her choose her hair style.