Uncertainty principles and sum complexes

Let $$p$$p be a prime and let $$A$$A be a nonempty subset of the cyclic group $$C_p$$Cp. For a field $${\mathbb F}$$F and an element $$f$$f in the group algebra $${\mathbb F}[C_p]$$F[Cp] let $$T_f$$Tf be the endomorphism of $${\mathbb F}[C_p]$$F[Cp] given by $$T_f(g)=fg$$Tf(g)=fg. The uncertainty number$$u_{{\mathbb F}}(A)$$uF(A) is the minimal rank of $$T_f$$Tf over all nonzero $$f \in {\mathbb F}[C_p]$$f∈F[Cp] such that $$\mathrm{supp}(f) \subset A$$supp(f)⊂A. The following topological characterization of uncertainty numbers is established. For $$1 \le k \le p$$1≤k≤p define the sum complex$$X_{A,k}$$XA,k as the $$(k-1)$$(k-1)-dimensional complex on the vertex set $$C_p$$Cp with a full $$(k-2)$$(k-2)-skeleton whose $$(k-1)$$(k-1)-faces are all $$\sigma \subset C_p$$σ⊂Cp such that $$|\sigma |=k$$|σ|=k and $$\prod _{x \in \sigma }x \in A$$∏x∈σx∈A. It is shown that if $${\mathbb F}$$F is algebraically closed then $$\begin{aligned} u_{{\mathbb F}}(A)=p-\max \{k :\tilde{H}_{k-1}(X_{A,k};{\mathbb F}) \ne 0\}. \end{aligned}$$uF(A)=p-max{k:H~k-1(XA,k;F)≠0}.The main ingredient in the proof is the determination of the homology groups of $$X_{A,k}$$XA,k with field coefficients. In particular it is shown that if $$|A| \le k$$|A|≤k then $$\tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)\!=\!0.$$H~k-1(XA,k;Fp)=0.