Full-Rank Valuations and Toric Initial Ideals

Let $V(I)$ be a polarized projective variety or a subvariety of a product of projective spaces, and let $A$ be its (multi-)homogeneous coordinate ring. To a full-rank valuation ${\mathfrak{v}}$ on $A$ we associate a weight vector $w_{\mathfrak{v}}$. Our main result is that the value semi-group of ${\mathfrak{v}}$ is generated by the images of the generators of $A$ if and only if the initial ideal of $I$ with respect to $w_{\mathfrak{v}}$ is prime. As application, we prove a conjecture by [ 7] connecting the Minkowski property of string polytopes to the tropical flag variety. For Rietsch-Williams’ valuation for Grassmannians, we identify a class of plabic graphs with non-integral associated Newton–Okounkov polytope (for ${\operatorname *{Gr}}_k(\mathbb C^n)$ with $n\ge 6$ and $k\ge 3$).

The political idea of wilayat al-faqih throughout history: An analysis of specific historical models of the idea from the Safavid period to the present day

Ever since the emergence of Islam in the Arabian Peninsula, the question of the relationship between religion and government, or more specifically, the issue of the relationship between representatives of religious thought and life with authorities has always been highly topical. The Prophet of Islam already formed the first Islamic rule in Medina, as a model of concretization of a new form of social life. His successors tried to preserve this model even after his death, even though they were not most successful in that. The institution of the caliphate was formed very quickly, which was not a bad idea, but in practice it proved to be a far cry from the initial ideals of Islam and the Prophet. First the Umayyad and then the Abbasid caliphates, and after them many other dynasties in various ruling forms such as sultanates and ilkhanates, ruled according to the traditional models of their ancestors, trying, at least formally, to approach some Islamic principles. In such circumstances, the special duty of representatives of Muslim thought, philosophers, lawyers, theologians and all others, was to clarify various practical models according to which Islamic dogma will appear in the clearest and most concrete form in the ruling structure and its ideological background. Thus, different models of relationships between representatives of religious thought and rulers appeared, and consequently, between religion and government itself. One of the most famous theories that explains this relationship was the theory of wilayat al-faqih, which advocates the idea that in the period after the Prophet and his true successors, it is necessary that Muslim jurists and thinkers, i.e. ulama, be included in power. The level of their participation in government can be maximum or minimum, and will vary depending on the political will of the governing structure dominant in society at a given moment. This theory became especially prominent in recent times when it was very extensively analysed and expressed by Ayatollah Ruhulah Khomeini, who even founded the structure of the newly formed Islamic Republic of Iran on its foundations. The truth is, however, that this theory has a very long history, more than a few hundred years old. In this paper, we will try to present some of the main models that this political and social idea has given rise to throughout history, and especially since the formation of the Safavid dynasty (16th century).