# Slopes of the Harder-Narasimhan filtration of a subbundle

Suppose $$(p_1, ..., p_n)$$ and $$(q_1, ..., q_n)$$ are two (non-strictly) decreasing sequences of positive real numbers. We say that $$(q_1, ..., q_n)$$ dominates $$(p_1, ..., p_n)$$ if $$q_n \geq p_n$$, $$q_n+q_{n-1} \geq p_n+p_{n-1}$$, ..., $$q_n + ... + q_2 \geq p_n + ... + q_2$$, $$q_n + ... + q_1 = p_n + ... + q_1$$.

For a vector bundle $$F$$ consider its slope $$\mu(F)=deg(F)/rk(F)$$. For a vector bundle $$E$$ with Harder-Narasimhan filtration $$0 \neq E_1 \subset E_2 \subset ... \subset E_m = E$$ we consider the sequence $$\mu(E_1),...,\mu(E_1), ..., \mu(E_m/E_{m-1}), ..., \mu(E_m/E_{m-1})$$, where $$\mu(E_i/E_{i-1})$$ is repeated $$rk(E_i/E_{i-1})$$ times, and call this sequence the degree of the Harder-Narasimhan filtration of $$E$$.

Suppose that $$A \subset B$$ are two vector bundles of the same rank $$n$$ such that $$deg B = deg A + 1$$. Assume that the degree of the Harder-Narasimhan filtration of $$B$$ is dominated by a (non-strictly) decreasing integer sequence $$(d_1, ..., d_n)$$ such that $$d_1>d_2$$. Is it true that then the degree of the Harder-Narasimhan filtration of $$A$$ is dominated by $$(d_1-1, d_2, ..., d_n)$$?

Sorry, I remembered how to answer my own question. Consider the degree of a HN stratum as a Young diagram with rational lengths of rows. It is enough to show that the Young diagram of A is a subset of the Young diagram of B because we can just omit the last inequality in the definition of domination. Suppose the Young diagram of A is not a subset of the Young diagram of B. Then there is a subbundle C of A consisting of several members of HN filtration of highest slope whose Young diagram is not inside the Young diagram of B. The bundle C maps by 0 to the quotient E of B consisting of several members of the HN filtration of lowest slope whose slope is smaller than the slope of the parts of C. So the bundle C is a subbundle of the kernel of the surjection from B to E. But since the Young diagram of C is not inside the Young diagram of B, the kernel of the map from B to E has smaller rank than the rank of C. Contradiction.

Please see a related question which I still need Slopes of the Harder-Narasimhan filtration of a limit of vector bundles