Halachic literature contains much discussion about the permissibility of a woman serving in a communal leadership position.
The Rambam's Ruling
Those who claim that women may not serve in leadership positions cite the Rambam's famous ruling (Hilchot Melachim 1:5):
We do not appoint a woman as king, as the Torah states (Devarim 17:15), "Appoint a king" - but not a queen. Similarly, a woman may not be appointed to any leadership position among the Jewish people.
The Kesef Mishnah and Radbaz, both commenting on the Rambam, cite the Sifrei's comment on the aforementioned verse as the source for this ruling. However, the Sifrei merely states that a woman may not be appointed to the position of queen. The Rambam has no explicit source for his extension of the Sifrei's ruling to all leadership roles.
Rav Moshe Feinstein (Teshuvot Igrot Moshe, Yoreh De'ah 2:44-45) suggests a basis for the Rambam's ruling. The Gemara (Yevamot 45a) states that not only does the Torah forbid appointing a convert as king, but it also forbids appointing him to any position of authority. Rav Moshe posits that the Rambam derives from this passage that the position of king is the paradigm for every leadership position. Whoever may not be appointed king may not hold any leadership role. The Rambam thus concludes that, just as we may not appoint a queen, we may not appoint a woman to any position of leadership.
What Is Considered Authority?
Rav Moshe Feinstein defines the term "position of authority" based on the Gemara (Kiddushin 76b), which teaches that a convert should not be appointed as an inspector of weights and measures. Rav Moshe explains that even if a position is not prestigious, it can still be viewed as authoritative. The difference between a person with authority and a regular worker is that a regular worker is hired to do the bidding of his employer, whereas someone in a position of authority is hired to, at times, act against the will of his employer. The inspector of weights and measures is in a position of authority because he does not permit his employer to act dishonestly, regardless of the employer's wishes. The inspector, unlike other workers, is hired to enforce an outside set of rules on his employer.
Elsewhere (Teshuvot Igrot Moshe, Yoreh De'ah 4:26), in a somewhat cryptic responsum, Rav Moshe adds that a job's purpose must be to enforce rules upon the employer in order for it to constitute authority.1It is our hope that Rav Moshe's responsum is presented here accurately, as the responsum is extremely cryptic. If, however, a position contains a certain measure of power over others (such as subordinates), this is insufficient to exclude converts from that position. For example, Rav Moshe discusses the possibility of appointing a convert as a rosh yeshivah (dean) or mashgiach ruchani (spiritual guidance counselor of a Yeshivah). Although these jobs include certain administrative powers, such as the ability to reject and expel students, Rav Moshe nonetheless permits a convert to hold them. He explains that these powers are "merely like the authority of a boss over his workers," as opposed to an "appointment to act against the will of the employer, empowered by the Torah law."2It is unclear if Rav Moshe would equate a woman with a convert.
According to the Rambam's position (which Rav Moshe is following), it is the same prohibition for both women and converts to hold positions of authority. However, Rav Moshe writes that he is inclined to rule leniently with a convert because the Torah requires us to act kindly toward converts, leaving open the possibility that, in a questionable case, he would rule more strictly with a woman.
According to his criteria, Rav Moshe concludes (Teshuvot Igrot Moshe, Y.D. 2:45) that a professional kashrut supervisor is in a position of authority. He is hired by a food company to enforce the laws of kashrut even when the company would like to disregard them. Rav Moshe adds, though, that there is a very easy way to permit a woman to supervise kashrut. The food company can hire a rabbi to supervise it, and the rabbi, in turn, may hire the woman to oversee the kashrut. The rabbi bears ultimate accountability for the food's kashrut, while the female supervisor is merely obeying the directions of her employer, the rabbi.
Not everyone agrees with Rav Moshe's definition of authority. For greater discussion of how to define authority, see Rav J. David Bleich's Contemporary Halakhic Problems (2:254-267).
Other Rishonim
Although the Rambam rules that a woman may not be appointed to a position of authority, Rav Moshe and Rav Chaim David Halevi (Techumin 10:118-123) note that many Rishonim appear to disagree with them.
One opinion cited in Tosafot (Nidah 50a s.v. Kol) permits a woman to serve as a dayan (rabbinical judge). This view clearly believes that a woman can be appointed to a position of authority.
The Sefer Hachinuch (Mitzvah 497) cites the Sifrei, which states that a woman may not be appointed as king. However, the Sefer Hachinuch does not explicitly exclude a woman from any other position of authority. In the next mitzvah (the prohibition of appointing a convert to be king), the Sefer Hachinuch emphasizes that a convert may not hold any position of authority. Apparently, when the prohibition applies to all forms of leadership, the Sefer Hachinuch writes so explicitly. Rav Moshe thus deduces that the Sefer Hachinuch does not explicitly exclude women from leadership roles (besides being a queen) because he does not agree with the Rambam.
Rav Moshe also suggests that both Rashi (Kiddushin 76b s.v. Kol) and the Ran (Kiddushin 31b in pages of the Rif, s.v. Garsinan) disagree with the Rambam's opinion. They explain that the source for the rule that a convert may not be appointed to any position of authority is the redundancy of words from the root s.y.m. (verbs of appointing) in Devarim 17:15, the verse that requires the king to be a natural born Jew ("mikerev achecha"). The Torah is teaching that all leadership appointments must be mikerev achecha. Regarding women, however, the prohibition is derived from the word "melech" ("king" - and not queen) and bears no connection to the word "tasim" ("You shall appoint"). Rav Moshe explains that Rashi and the Ran view the prohibition against women as precluding the position of queen but permitting other leadership appointments.
Finally, Rav Moshe claims that the Ramban (Shevuot 30a s.v. Vechein) and the Rashba (Shevuot 30a s.v. Velo) may also disagree with the Rambam, based upon their analysis of the status of Devorah. Devorah both judged individual people and led the nation like a queen (see Shofetim 4:4-5). The Ramban and Rashba do not appear to view the prohibition against a queen as applicable to Devorah's judiciary role (although the prohibition raises a problem for Devorah's role as a national leader). They do assert that women are disqualified from serving as judges,3It is accepted as the Halachah in Shulchan Aruch, Choshen Mishpat 7:4, that a woman may not serve as a dayan. While the Pitchei Teshuvah (C.M. 7:5) cites a dissenting opinion, he personally accepts the Shulchan Aruch's view. It should be noted that the Pitchei Teshuvah also cites prominent authorities that permit a female expert in Halachah to rule on halachic questions. but they do not connect this with women's disqualification from serving as queen.
On the other hand, the Ritva (Shevu'ot 30a s.v. Shevu'at) does agree with the Rambam. He states unambiguously that the Sifrei's prohibition against appointing a queen applies to all leadership positions.
Rav Moshe Feinstein
Rav Moshe writes (in his teshuvot quoted above) that although many Rishonim disagree with the Rambam, the Rambam's view should nonetheless be followed in practice. He explains that this issue is a dispute regarding a biblical prohibition, and the Shulchan Aruch did not resolve it, so we must observe the strict opinion. Thus, a woman should not be appointed to the board of a synagogue. Rav Moshe does permit relying on the other Rishonim in a case of great need. For example, he permitted a poor widow to become a mashgichah (kashrut supervisor), despite the fact that it is a position of authority (similar to the supervisor of weights mentioned in Kiddushin 76b), so that she could make a living.
Rav Chaim David Halevi
Rav Chaim David Halevi questions the Rambam's view by pointing to the historical fact that the righteous Queen Shlomtzion ruled over the Jewish people for nine years during the Second Temple period. He notes that the Talmud never criticizes the fact that she served as leader of the Jewish people, and Chazal even say very positive things about her (see Vayikra Rabah 35:10).4It is not clear why Rav Halevi raises this question against the Rambam specifically, considering that everyone appears to agree that a woman cannot be appointed queen.
Rav Halevi does not reject the Rambam's view, but he suggests that even the Rambam would permit a woman to lead the nation if she was willingly accepted by the people. Rav Halevi claims that the prohibition is against a woman taking power over people, but it does not apply when the woman is elected democratically. Rav Halevi notes that this approach seems to correspond with the Ramban and Rashba's opinion (above) regarding Devorah - she could serve as a ruler because the nation accepted her rulings. Rav Halevi therefore rules that a woman may hold an elected position of authority.
Other Authorities
The position of Rav Halevi parallels the view of his mentor, Rav Ben-Zion Uzziel (Teshuvot Mishpetei Uzziel, Choshen Mishpat 6). Interestingly, Rav Yehuda Amital adopted this ruling in practice. This author asked Rav Amital how, in 1988, he allowed a woman to run on the parliamentary list of the political party Meimad, in light of the Rambam's ruling. He replied that the Rambam's ruling does not apply to a democratically elected position. On the other hand, Rav Avraham Yitzchak Kook (Ma'amarei Hare'iyah pp. 189-194) opposes any participation by women in elections.