This question comes from Exercise 4.1 of the lectures on geometric constructions, Kamnitzer-arXiv-Link, and is a consequence of a question that I have asked here that deals with Part 1 of the exercise. This question is part 2.

We get that in this exercise $ X: mathbb {C} ^ N to mathbb {C} ^ N $ is a not potent matrix with $ X ^ n = 0 $. The partition is connected to it $ mu = ( mu_1, dots, mu_n) $ With $$ mu_i = dim ker (X ^ i) – dim ker (X ^ {i-1}). $$

To $ X $ We can also map the partition $ nu = ( nu_1, dots, nu_m) $ where everyone $ nu_i $ is the size of the $ i $-th Jordan Block from $ X $Placing an order to make the 1st Jordan block the largest size, and so on. The young diagram of $ nu $ has a conjugate partition $ lambda $where everyone $ lambda_i $ is the number of $ j $ so that $ nu_j geq i $ (i.e. it is the number of Jordan blocks one size larger or equal $ i $).

The first part shows that for everyone $ k $, $$ mu_1 + dots + mu_k leq lambda_1 + dots + lambda_k. $$ Now I have to show that as $ GL_n $ Weights, $ lambda geq mu $.

We have $$ lambda – mu = (( lambda_1- mu_1), dots, ( lambda_n- mu_n)), $$ what I want to express as a sum $$ k_1 a_1 + dots + k_ {n-1} a_ {n-1}, $$ Where $ k_i $ are not negative integers and $ a_1 = (1, -1.0, points, 0), points, a_ {n-1} = (0, points, 0, 1, -1) $. In other words, the setup will $$ (( lambda_1- mu_1), dots, ( lambda_n- mu_n)) = (k_1, k_2-k_1, dots, k_ {n-1} -k_ {n-2}, -k_ { n-1}). $$

Starting from the left, we get an inductive equation $$ k_i = ( lambda_1 + dots + lambda_i) – ( mu_1 + dots + mu_i), $$ but I can't show that $$ – k_ {n-1} = lambda_n- mu_n tag {*}. $$ I'm pretty sure my calculations are correct, and the only place we use the part 1 inequality is to show that the constants $ k_i $ are not negative, but that's all right – the only part I'm sticking to is showing (*).