For a finite dimensional algebra $A$, the 2-term silting complexes of $A$ give a simplicial complex $\Delta(A)$, fan $\sigma(A)$ and polytope $P(A)$, which are called the g-simplicial complex, g-fan and g-polytope, respectively. We study several properties of these objects from the viewpoint of the representation theory and combinatorics.
In particular, we give a tilting theoretic interpretation of the $h$-vectors and the Dehn-Sommerville equations of $\Delta(A)$.
Moreover, we give a characterization of the convexity of $P(A)$ and we explain a representation theoretic interpretation of the dual polytope of $P(A)$ in terms of simple minded collections.
This is joint work with Aoki-Higashitani-Iyama-Kase (arXiv:2203.15213).

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