WEEKLY ACTIVITIES

Time: 9:30 -- 11:00, November 13, 2024

Venue: Room 612, A6, Institute of Mathematics-VAST

Abstract: A Young diagram (lambda = (lambda_1, ldots, lambda_n)) is a collection of boxes arranged in a left-justified shape with (n) rows, where the length of the (i)-th row is (lambda_i), and (lambda_1 geq lambda_2 geq cdots geq lambda_n). A skew Young diagram (lambda/mu) consists of the boxes in the Young diagram of (lambda) that do not belong to the Young diagram of (mu), where (mu_i leq lambda_i) for all (i).

A textit{filling} of a skew Young diagram is an assignment of positive integers ( omega(i,j) ) to each box ( (i,j) ) in the skew diagram. The {it skew tableau ideal} associated with a skew Young diagram (lambda/mu) with a given filling is an ideal in a standard graded polynomial ring (S = k[x_1, ldots, x_n, y_1, ldots, y_m]) over a field (k), where (m = lambda_1), and is defined as follows:

$$I_{lambda/mu}(mathbf{w}):= ((x_iy_j)^{omega(i,j)}mid 1le ile n, mu_i+1le j le lambda_i).$$

In this talk, we will provide a characterization for Cohen--Macaulayness of $I_{lambda/mu}(mathbf{w})$ and explore related problems. This work is based on a collaboration with Thanh Vu from the Vietnam Academy of Science and Technology. .

Program of Special Semester on Commutative Algebra