Academy of Mathematics and Systems Science, CAS
Colloquia & Seminars

Speaker:	牟晨琪副教授，北京航空航天大学
Inviter:
Title:	Grobner Bases and Combinatorial Tools for Determinantal Ideals
Language:	Chinese
Time & Venue:	2023.06.29 14:00-16:00 思源楼525
Abstract:	This talk consists of two parts guided by the same principle: I try to clarify the combinatorial background and tools for studying the Gr?bner bases of two kinds of determinantal ideals. In the first part, I will present how the Robinson-Schensted-Knuth (RSK) correspondence from combinatorics, combined with the straightening law, establishes a bijection between the standard monomials as products of comparable minors and the monomials as products of variables in the determinantal ring. Then I will explain the proof from Sturmfels in 1990 for that all the r-minors form the Groebner basis of the determinantal ideal generated by them with respect to any diagonal term order by using the RSK correspondence. In the second part, our focus switches to the Schubert determinantal ideal, a fundamental algebraic concept in the study of Schubert calculus. For a permutation w, its Schubert determinantal ideal I_w is an ideal generated by the so-called Fulton generators specified by the essential set of w. The Fulton generators are identified as the Gr?bner basis of I_w with respect to any anti-diagonal term order in the influential paper of Knutson and Miller in 2005, where deep connections are also established between the initial ideal of I_w and the Stanley-Reisner simplicial complex, the double Schubert polynomial of w, and the reduced pipe dreams of w. I will review the above fundamental theories concerning Schubert determinantal ideals and then present our recent works on their reduced Gr?bner bases by using techniques similar to the elusive minors introduced by Gao and Yong in 2022.