How did Umar (may Allah be pleased with him) divide the wealth equally among full siblings and maternal half-siblings, even though maternal half-siblings inherit and full siblings do not inherit in this particular issue? And was this an independent legal judgment despite the existence of a specific legal text concerning inheritance?

Inheritance is a matter whose division Allah Himself undertook. Some issues are agreed upon, while others are points of disagreement among jurists.

The mentioned case is called "Al-Musharakah (definition)," "Al-Himariyyah," or "Al-Hijariyyah." It occurs when a woman dies, leaving behind a husband, a mother or grandmother, uterine brothers (from the mother's side), and full brothers (from both parents).

The Quran specifies the husband's share (half), the mother's or grandmother's share (one-sixth), and the uterine brothers' share (one-third). The dispute arises concerning the inheritance of the full brothers.

One group holds that the full brothers inherit nothing because they are 'asabah (agnatic heirs), and the prescribed shares have exhausted the inheritance. This is the view of Ali, Ibn Mas'ud, Abu Hanifah, and Ahmad.

The other group believes that the full brothers share the one-third with the uterine brothers because they share the same mother. This is the view of Umar, Uthman, Zayd ibn Thabit, Malik, and Al-Shafi'i.

This issue is an example of ijtihad (definition) (independent reasoning) matters in inheritance, over which the Companions and those who came after them differed, and the texts accommodate both interpretations. The principle "no ijtihad with a text" means no independent reasoning is permissible when there is a clear text that bears no other meaning. However, when a text is open to interpretation or is contradicted by another text, then ijtihad is necessary to understand it or to reconcile the texts.