I'm studying null-cones right now, and am I trying to understand the null-cone for a representation $(W, \rho)$ of the torus $T_n = (\mathbb{C}^*)^n$.
Here are the things I understand:
If $f = \Sigma a_I m_I$ is $T_n$ invariant, and $m_I$ are monomials and $a_I$ scalars, then each $m_I$ is invariant.
A monomial $x^u$ is invariant iff $uA = 0$, where $A$ is the matrix of exponents in the characters of the action. (Explicitly, $\rho((t_1,\ldots,t_n))$ is a diagonal matrix with $ith$ entry given by $\pi_{j = 1}^n t_j^{a_{ij}}$. Then $A = (a_{ij})$.)
The notes I'm reading define the support of $w \in W$ to be the set of $\alpha_i \in \mathbb{Z}^n$ with $x_i(w) \not = 0$.
It's unclear what this means.
The theorem is then:
The null cone of this $T_n$ actions consists of the vectors $w$ so that $0$ is not in the convex hull of $supp(w)$.
So it seems clear from this theorem that the $supp(w)$ should refer to the action of the torus somehow.
Can someone clarify what the definition is?
(I'm reading: http://www.win.tue.nl/~jdraisma/teaching/invtheory0910/lecturenotes12.pdf and this is definition 7.0.13)
I found it a bit confusing me too. A bit earlier, the author took an explicit $W$, namely a direct sum of $1$-dim representation, determined by a sequence of vectors $\alpha_i \in \Bbb Z^m$. The representation is $(t,(x_1, \dots, x_m)) \mapsto (t^{\alpha_1}x_1, \dots, t^{\alpha_m}x_m)$ ( here $t \in (\Bbb C^*)^m)$.
Now, each $w \in W$ will be an element of $\Bbb C^m$, i.e $w = (w_1, \dots, w_m)$, so $\text{supp}(w)$ will be constitued of all $\alpha_i$ such that $w_i \neq 0$.