The foundational ruling is the Rambam's, who states that we do not appoint a woman to kingship, deriving from the pasuk "appoint a king" (Mishneh Torah, Melachim and Wars 1:5) that a queen is excluded, and extends this to all positions of communal appointment in Israel, a position Gray Matter I, Modesty, Women and Leadership Positions identifies as the classic source cited against women serving in communal leadership roles.
This disqualification is echoed in the judicial sphere: the Rambam rules that only wise and understanding men may be appointed to the Sanhedrin, whether great or small (Mishneh Torah, The Sanhedrin 2:1), and the Sefer HaChinukh 77:5 likewise states that the mitzvah of appointing judges applies to men but not women, since women do not judge.
The Arukh HaShulchan explains the reasoning behind this disqualification from judging and from all appointments, rooting it in the same derivation from "from among your brothers" and not "from among your sisters" (Arukh HaShulchan, Choshen Mishpat 7:4), and he addresses the case of Devorah by explaining that she did not actually render judgment but only taught the people how to judge, or that the law was conveyed through divine word, or that all Israel accepted her authority upon themselves — the same approach taken by Sefer HaChinukh 77:5 regarding the pasuk describing Devorah as having judged Israel (Shoftim 4:4-5).
Separately, the sources address whether a woman may even study or teach Torah, a related but distinct question: the Shulchan Arukh rules that a father should not teach his daughter Torah, though a woman who studies nonetheless receives reward, and that she is obligated to learn laws pertaining to her (Shulchan Arukh, Yoreh De'ah 246:6), a position the Arukh HaShulchan restates, limiting the concern to Oral Torah and affirming the Rema's obligation that she learn laws relevant to her (Arukh HaShulchan, Yoreh De'ah 246:19).
Modern poskim have moved this question of women's Torah study forward: Rav Mordechai Eliyahu rules that in our times it is permitted for a woman to learn both Written and Oral Torah, and even to be taught by a teacher, since women are no longer secluded at home as in the past and it is better that they study sacred texts than secular or idle material (Shut Maamar Mordechai, Chelek 1, יורה דעה 11), while B'Mareh HaBazak Volume VII 71:4 raises the question of what Oral Torah may be taught to women in our generation without resolving it in the cited passage.
Maayan Omer (Maayan Omer 12:2:12) addresses whether a woman may deliver words of Torah in the presence of men at an annual thanksgiving meal, and rules that she may do so.
Tzitz Eliezer (Tzitz Eliezer 9:3) addresses the view that women are prohibited from studying the Oral Torah even for themselves, while noting that women nonetheless recite the blessing over Torah study and citing the Taz's position on teaching Torah to one's daughter.
Other sources on the sheet bear only indirectly: the Gemara's exclusion of women from testimony is derived from the masculine form "two men" in the pasuk (Shevuot 30a:4-5), the Chatam Sofer's responsum concerns a rabbi imposed on a community without its consent rather than the sex of the appointee (Responsa Chatam Sofer, Choshen Mishpat 19:2-3), and Contemporary Halakhic Problems, Vol III, Part I, Chapter V Women frames the broader principle that halacha assigns differing roles to men and women without denigrating women's spiritual capacities, while Vayikra 19:14 does not itself speak to the question of appointment or leadership.
