On characteristic polynomials of subspace arrangements
Let \(V\) be a vector space over a field or division ring, and let \(\mathcal A\) be an arrangement of affine subspaces in \(V\). Let \(L({\mathcal A})\) be the set of nonempty intersections of subspaces in \(\mathcal A\), partially ordered by inclusion. The characteristic polynomial of \(\mathcal A\) is defined by \[ \chi({\mathcal A},q)=\sum_{X\in L} \mu(X,V)q^{\dim(X)}, \] where \(\mu\) denotes the Möbius function on \(L({\mathcal A})\). Depending on the nature of \(\mathcal A\), \(\chi({\mathcal A},q)\) and its evaluations have various enumerative properties, counting for instance the cardinality of \(V-\bigcup {\mathcal A}\), for \(V=({\mathbb F}_q)^n\), or the number of connected components in \(V-\bigcup {\mathcal A}\), for \({\mathcal A}\) an arrangement of hyperplanes in \({\mathbb R}^n\) and \(q=-1\). In this short note the author derives these and several other known properties of \(\chi({\mathcal A},q)\) from simple arguments involving characteristic functions, Möbius inversion, and valuations on Boolean algebras. The basis for all of the derivations is the following relationship among the characteristic functions \(I_S\) of subsets of \(V\) arising from \(L({\mathcal A})\). For \(X\in L({\mathcal A})\) let \(X^0=X-\bigcup_{Y\in L({\mathcal A}), Y<X} Y\). Then \[ I_{X^0}=\sum_{Y\in L({\mathcal A}), Y\leq X} \mu(Y,X)I_Y, \] easily derived by Möbius inversion. The enumeration results follow easily by applying various counting functions to both sides of this formula.
scientific article; zbMATH DE number 1474640
scientific article