= Diagonal trace Morse function on a complex Grassmannian
{title2=$f(P)=\operatorname{tr}(CP),\quad\operatorname{ind}P_I=2\sum_{a=1}^k(i_a-a)$}
For real $c_1<\cdots<c_n$ and $C=\operatorname{diag}(c_1,\ldots,c_n)$, the <critical points> are the coordinate projections $P_I$, indexed by $k$-subsets $I=\{i_1<\cdots<i_k\}$. Indeed $df([A,P])=\operatorname{tr}([P,C]A)$, which vanishes for every skew-Hermitian $A$ exactly when $[P,C]=0$. In the graph chart at $P_I$,
$$
f(Z)=\sum_{i\in I}c_i+\sum_{i\in I,\ j\notin I}(c_j-c_i)|z_{ji}|^2+O(\|Z\|^4).
$$
Every coefficient is nonzero, so the <Hessian matrix> is nondegenerate. Each pair $j<i$ supplies two negative real directions, giving the displayed <Morse index>. The function is a <Morse function> even if distinct critical points happen to have equal critical values.
Back to article page
