Partial orders on standard Young tableaux( Gizem Karaali, Pomona)

Young diagrams are all possible arrangements of n boxes into rows and columns, with the number of boxes in each subsequent row weakly decreasing. For a partition λ of n, a standard Young tableau S of shape λ is built from the Young diagram of shape λ by filling it with the numbers 1 to n, each occurring exactly once in such a way that the numbers are strictly increasing across rows (left to right) and down columns. Young diagrams with n cells are in one-to-one correspondence with the irreducible representations of the symmetric group S_n,; the standard Young tableaux count the dimensions of these irreps and thus are some of the most essential objects of combinatorial representation theory and algebraic combinatorics. In this talk, based on joint work with Isabella Senturia (PO’20) and Müge Taskin, I will describe a handful of partial orders already defined on SYT_n, the set of all standard Young tableaux with n cells, and propose a new one.