Even though you "share" custody at 50%, the IRS rules state that the parent who has the child more during the year is the custodial parent and that parent is the one who can claim Head of Household. Since there are 365 days/nights in the year, one of you must have one more night than the other (I'm just restating the rules.).
This means that both of you cannot be the "custodial parent" per the IRS rules. Only one of you can claim HOH on the basis of your daughter - and the right one to claim her is the person who is the custodial parent (the custodial parent i also the one to claim dependent and child care credit as well as EIC if applicable). The non-custodial parent may only claim the child if Form 8332 is signed by the custodial parent to give the non-custodial parent the right to claim the chid tax credit, additional child credit, and exemption for the child. If both of you claim her for the purposes of HOH, the one who doesn't have the right to claim (based on the IRS's ruling at a later date) will be subject to interest and penalty. If the IRS believes you intentionally committed fraud, they can charge you with more (and they can also go back as far as they want to inspect prior year returns - it is not subject to the 3 year "rule" of audits when fraud is suspected).
I don't believe that the system will let both of you claim HOH on the basis of the same child. You must still enter your daughter's SSN to e-file HOH (there is a area where you have to enter your qualifying person that you are not claiming on your return) and if your ex already claimed her and filed, your return is likely to be rejected.
IRS Pub 501 has a section regarding children of divorced/separated/never married parents that goes into more detail.
