Let $X$ be the "standard open simplex in $\mathbb{R}^n_{\geq 0}$",

$X = \{(x_1, \ldots, x_n) : \text{ all } x_i > 0 \text{ and } \sum x_i < 1 \}.$

And, for a subset $A \subseteq \{1, \ldots, n\}$, let $\tilde A : X \to \mathbb{C}$ denote the indicator function

$ \tilde A(x_1, \ldots, x_n) = \begin{cases} 1 & \text{ if } \sum_{i \notin A} x_i - \sum_{i \in A} x_i > \frac{1}{2}, \newline 0 & \text{ otherwise.} \end{cases} $

Finally, given a simplicial set $S$ on up to $n$ vertices, let $\tilde S : X \to \mathbb{C}$ be the product of the indicator functions for its facets, (so $\tilde S$ indicates solutions to the corresponding system of inequalities / linear program.)

The $\mathbb{C}$ algebra $\tilde R$ generated by the functions $\tilde A$, where $A \subseteq \{1, \ldots, n\}$, is spanned by the $\tilde S$ as a $\mathbb{C}$-vector space.

Some of these simplicial complexes lead to functions that are identically zero, such as $\{\{1,2\},\{2,3\}\}$ on $n=3$. Others are obviously feasible (for example, if some $i$ is not a vertex of $S$). So, I have two questions:

Is there a simple criterion for which simplicial complexes $S$ are 'feasible', i.e. the indicator function $\tilde S$ is not identically zero on $X$?

Are the remaining 'feasible' indicator functions $\tilde S$ $\mathbb{C}$-linearly independent? (The answer seems like it should be "yes". Certainly it is for small $n$).

I am trying to analyze a sum of such indicator functions of this form, that arises as the Fourier transform of a probability density related to zeros of a family of $L$-functions. (If it makes the answers easier/neater, the strict inequalities can become $\geq$ or $\leq.$ I only care up to sets of measure zero anyway.)

I can provide additional number theoretic or combinatorial background info if necessary. In any case, the above two questions have stymied me, so I'd be grateful for any help!

Thank you! --Jake

edit: (David Speyer has made some useful observations about Spec $\tilde R$, which I could reproduce here.)