Uncertainty principles and sum complexes

Abstract

Let $p$ be a prime and let $A$ be a nonempty subset of the cyclic group $C_p$. For a field ${\mathbb F}$ and an element $f$ in the group algebra ${\mathbb F}[C_p]$ let $T_f$ be the endomorphism of ${\mathbb F}[C_p]$ given by $T_f(g)=fg$. The uncertainty number $u_{{\mathbb F}}(A)$ is the minimal rank of $T_f$ over all nonzero $f \in {\mathbb F}[C_p]$ such that $\mathrm{supp}(f) \subset A$. The following topological characterization of uncertainty numbers is established. For $1 \le k \le p$ define the sum complex $X_{A,k}$ as the $(k-1)$-dimensional complex on the vertex set $C_p$ with a full $(k-2)$-skeleton whose $(k-1)$-faces are all $\sigma \subset C_p$ such that $|\sigma |=k$ and $\prod _{x \in \sigma }x \in A$. It is shown that if ${\mathbb F}$ is algebraically closed then

$$\begin{aligned} u_{{\mathbb F}}(A)=p-\max \{k :\tilde{H}_{k-1}(X_{A,k};{\mathbb F}) \ne 0\}. \end{aligned}$$

The main ingredient in the proof is the determination of the homology groups of $X_{A,k}$ with field coefficients. In particular it is shown that if $|A| \le k$ then $\tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)\!=\!0.$

1 Introduction

Uncertainty type inequalities reflect various quantitative aspects of the general principle that a nonzero function and its Fourier transform cannot both be sharply localized. The first such result is the Fourier theoretic version of the Heisenberg quantum-mechanical uncertainty principle: If $f \in L^2({\mathbb R})$ satisfies $||f||_2=1$, then

$$\begin{aligned} \Vert xf\Vert _2 \, \Vert \xi \widehat{f}\Vert _2 \ge \frac{1}{4 \pi }~~. \end{aligned}$$

(1)

This classical inequality and its numerous extensions (see e.g. [3]) have major applications in diverse areas ranging from mathematical physics and differential equations to signal recovery and number theory. Here, we are concerned with discrete versions of the uncertainty principle. Let $G$ be a finite abelian group and let ${\mathbb F}[G]$ be the group algebra of $G$ over the field ${\mathbb F}$. For an element $f \in {\mathbb F}[G]$ let $T_f:{\mathbb F}[G] \rightarrow {\mathbb F}[G]$ be given by $T_f g=fg$. Let $A$ be a subset of $G$. Here and throughout the paper, we assume that $A \ne \emptyset$. The uncertainty number of $A \subset G$ is defined by

$$\begin{aligned} u_{{\mathbb F}}(A)=\min \{ \mathrm{rank}\, T_f~:~ \emptyset \ne \mathrm{supp}(f) \subset A\}. \end{aligned}$$

The motivation for this definition is as follows. Let $m$ be the exponent of $G$ and suppose ${\mathbb F}$ contains a primitive $m$-th root of unity. Let $\widehat{G}$ denote the group of ${\mathbb F}$-valued characters of $G$. Identifying ${\mathbb F}[G]$ with the space of ${\mathbb F}$-valued functions on $G$, the Fourier transform of a function $f \in {\mathbb F}[G]$ is the function $\widehat{f} \in {\mathbb F}[\widehat{G}]$ given by $\widehat{f}(\chi )=\sum _{x \in G} \chi (x^{-1}) f(x)$. The characters $\chi \in \widehat{G}$ are eigenfunctions of $T_f$ with eigenvalues $\widehat{f}(\chi )$, hence $\mathrm{rank}\, T_f=|\mathrm{supp}(\widehat{f})|$. Therefore, in the semisimple case

$$\begin{aligned} u_{{\mathbb F}}(A)=\min \{|\mathrm{supp}(\widehat{f})|~:~ \emptyset \ne \mathrm{supp}(f) \subset A\}. \end{aligned}$$

The discrete counterpart of (1) (see e.g. [1]) asserts that for any ${\mathbb F}$ and $A \subset G$

$$\begin{aligned} u_{{\mathbb F}}(A) \ge \frac{|G|}{|A|}. \end{aligned}$$

(2)

In the semisimple case (2) is equivalent to

$$\begin{aligned} |\mathrm{supp}(f)|\cdot |\mathrm{supp}(\widehat{f})| \ge |G| \end{aligned}$$

for all nonzero $f$’s. While (2) is sharp when $A$ is a coset of $G$, it can often be improved for particular choices of $G,A$, and ${\mathbb F}$. One such result (see [9]) states that if $p$ is prime and $A$ is a nonempty subset of the cyclic group $C_p$ then, $u_{{\mathbb C}}(A)=p-|A|+1$. See [7] for an extension to general abelian groups.

For a finite abelian group $G$, let $\Delta _{G}$ denote the $(|G|-1)$-dimensional simplex with vertex set $G$ and let $\Delta _{G}^{(j)}$ be the $j$-dimensional skeleton of $\Delta _{G}$. Let $A \subset G$ and let $1 \le k \le |G|$. The Sum Complex $X_{A,k}$ was defined in [6] by

$$\begin{aligned} X_{A,k}=\Delta _{G}^{(k-2)}~ \cup ~\{ \sigma \subset G~:~ |\sigma |=k~,~\prod _{x \in \sigma } x \in A \}. \end{aligned}$$

Here, we obtain the following topological characterization of uncertainty numbers of subsets of $C_p$.

Theorem 1.1

Let $A \subset C_p=G$. If ${\mathbb F}$ is algebraically closed, then

$$\begin{aligned} u_{{\mathbb F}}(A)=p-\max \{1 \le k \le p:\tilde{H}_{k-1}(X_{A,k};{\mathbb F}) \ne 0\}. \end{aligned}$$

(3)

If $\tilde{H}_{k-1}(X_{A,k};{\mathbb F})=0$ for all $1 \le k \le p$, then the right-hand side of (3) is defined as $p$.

Example: Let $p=7$ and $A=\{1,z,z^3\} \subset C_7=\langle z \rangle$. The sum complex $X_{A,3}$ is depicted in Figure (1b) where vertex label $\alpha$ corresponds to the element $z^{\alpha }$. Note that $X_{A,3}$ is obtained from a $7$-point triangulation of the real projective plane ${\mathbb R}{\mathbb P}^2$ (Figure 1a) by adding the faces $\{z^2,z^3,z^5\}$, $\{1,z^2,z^6\}$, and $\{z,z^2,z^4\}$. $X_{A,3}$ is clearly homotopy equivalent to ${\mathbb R}{\mathbb P}^2$, hence $H_2(X_{A,3};{\mathbb F}_2) \ne 0$. Theorem 1.1 then implies that $u_{\overline{{\mathbb F}_2}}(A) \le 4$. Together with the easy fact that $u_{{\mathbb F}}(B) \ge p- \max B$ for any ${\mathbb F}$ and $B \subset C_p$ it follows that $u_{\overline{{\mathbb F}_2}}(A)=4$. It can be checked that in fact $u_{{\mathbb F}_2}(A)=4$.

Fig. 1

a A $7$-point triangulation of ${\mathbb R}{\mathbb P}^2$. b $X_{A,3}$ for $A=\{1,z,z^3\} \subset C_7$

Let $z$ be a fixed generator of $C_p$ and let $A=\{a_1,\ldots ,a_m\} \subset C_p$ where $a_i=z^{\alpha _i}$ and $\alpha _i$ is an element of the prime field ${\mathbb F}_p=\{0,\ldots ,p-1\}$. Let $\alpha =(\alpha _1,\ldots ,\alpha _m) \in {\mathbb F}_p^m$. The main ingredient in the proof of Theorem 1.1 is the computation of the homology of $X_{A,k}$ with arbitrary field coefficients. Let ${\mathbb F}$ be a field of characteristic $\ell$. First suppose that $\ell \ne p$ and let $\omega$ be a primitive $p$-th root of unity in the algebraic closure $\overline{{\mathbb F}}$. For $\beta =(\beta _1,\ldots ,\beta _k) \in {\mathbb F}_p^k$ let $M_{\beta ,\alpha }$ be the $k \times m$ matrix given by $M_{\beta ,\alpha }(i,j)=\omega ^{\beta _i \alpha _j}$. Let

$$\begin{aligned} {\fancyscript{B}}_k=\{(\,\beta _1,\ldots ,\beta _k):0 \le \beta _1 < \cdots < \beta _k \le p-1\}. \end{aligned}$$

The case $m=k$ of the following result is implicit in [6].

Theorem 1.2

Let $A=\{z^{\alpha _1},\ldots ,z^{\alpha _m}\} \subset C_p$ and let $\alpha =(\alpha _1,\ldots ,\alpha _m)$. If $\mathrm{char} \, {\mathbb F}\ne p$ then for $1 \le k \le p$

$$\begin{aligned} \dim \tilde{H}_{k-1}(X_{A,k};{\mathbb F})=\frac{m}{k}\left( {\begin{array}{c}p-1\\ k-1\end{array}}\right) -\frac{1}{p} \sum _{\beta \in {\fancyscript{B}}_k} \mathrm{rank}\, M_{\beta ,\alpha }. \end{aligned}$$

(4)

Our main result concerns the homology of $X_{A,k}$ with ${\mathbb F}_p$ coefficients.

Theorem 1.3

Let $A \subset C_p$ such that $|A|=m$. Then for $1 \le k \le p$

$$\begin{aligned} \dim \tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)= \left\{ \begin{array}{cl} 0 &{} ~\mathrm{if}~~m \le k \\ (\frac{m}{k}-1)\left( {\begin{array}{c}p-1\\ k-1\end{array}}\right)&~\mathrm{if}~~m > k. \end{array} \right. ~~ \end{aligned}$$

(5)

Remarks

• (1) The case $k=p$ of both Theorems 1.2 and 1.3 is straightforward. On one hand, since $X_{A,p}$ is a subcomplex of the $(p-1)$-simplex it follows that $\tilde{H}_{p-1}(X_{A,p};{\mathbb F})=0$ for any $A$ and ${\mathbb F}$. On the other hand, let $\beta =(0,1,\ldots ,p-1)$ be the unique element of ${\fancyscript{B}}_p$. Then, $M_{\beta ,\alpha }$ consists of $m$ full columns of the Fourier matrix, and thus has full rank $m$. Hence the right-hand side of (4) is also zero. In the sequel we will, therefore, assume that $1 \le k <p$.

• (2) The argument given in [6] for the case $m=k$ of Theorem 1.2 does not extend to the modular case. The approach here is different and is also utilized in the proof of our main result Theorem 1.3.

• (3) The $f$-vector of $X_{A,k}$ satisfies $f_i(X_{A,k})=\left( {\begin{array}{c}p\\ i+1\end{array}}\right)$ for $0 \le i \le k-2$ and $f_{k-1}(X_{A,k})=\frac{m}{p}\left( {\begin{array}{c}p\\ k\end{array}}\right) =\frac{m}{k}\left( {\begin{array}{c}p-1\\ k-1\end{array}}\right)$. The reduced Euler characteristic of $X_{A,k}$ is, therefore,

  $$\begin{aligned} \tilde{\chi }(X_{A,k})&= -1+\sum _{i=0}^{k-2}(-1)^i \left( {\begin{array}{c}p\\ i+1\end{array}}\right) +(-1)^{k-1}\frac{m}{p}\left( {\begin{array}{c}p\\ k\end{array}}\right) \nonumber \\&= (-1)^{k-1}(\frac{m}{k}-1)\left( {\begin{array}{c}p-1\\ k-1\end{array}}\right) . \end{aligned}$$

  (6)

  Since $\tilde{H}_i(X_{A,k};{\mathbb F}_p)=0$ for $0 \le i <k-2$ it follows that

  $$\begin{aligned} \dim \tilde{H}_{k-2}(X_{A,k};{\mathbb F}_p)&= \dim \tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)-(\frac{m}{k}-1)\left( {\begin{array}{c}p-1\\ k-1\end{array}}\right) N\\&= \left\{ \begin{array}{cl} (1-\frac{m}{k})\left( {\begin{array}{c}p-1\\ k-1\end{array}}\right) &{} ~\mathrm{if}~~ m \le k \nonumber \\ 0 &{} ~\mathrm{if}~~ m > k. \end{array} \right. ~~ \end{aligned}$$

  (7)

• (4) A classical result of Chebotarëv (see e.g. [8]) asserts that for ${\mathbb F}={\mathbb Q}$ all $M_{\beta ,\alpha }$’s have full rank. Theorem 1.2, therefore, implies that (5) and (7) remain true for $\tilde{H}_*(X_{A,k};{\mathbb Q})$.

• (5) Theorem 1.3 does not hold for all fields ${\mathbb F}$ and in fact $\tilde{H}_{k-1}(X_{A,k};{\mathbb F})$ may be nontrivial even if $k>|A|=m$. For example, using Theorem 1.2 it can be shown that if $A=\{1,z^3,z^4,z^5,z^8\}\subset C_{17}$ then $\tilde{H}_{7}(X_{A,8};{\mathbb F}_2) \ne 0$.

The paper is organized as follows. In Sect. 2 we consider $X_{A,k}$ where $A$ is a subset of an arbitrary abelian group $G$ and identify $\tilde{H}_{k-1}(X_{A,k};{\mathbb F})$ with a certain subspace ${\fancyscript{H}}(A)$ of skew-symmetric elements of the group algebra ${\mathbb F}[G^k]$. In Sect. 3 we use this characterization in the special case $G=C_p$ to prove Theorem 1.2. The proof of Theorem 1.3 given in Sect. 4 is more involved and depends additionally on some properties of generalized Vandermonde determinants over the group algebra ${\mathbb F}_p[C_p^k]$. Theorem 1.1 is derived in Sect. 5 as a direct consequence of Theorems 1.2 and 1.3. We conclude in Sect. 6 with some comments and open problems.

2 A characterization of cycles

Let $A$ be a nonempty subset of a finite abelian group $G$ and let ${\mathbb F}$ be a field. In this section, we provide a characterization (Claim 2.1) of the homology group $\tilde{H}_{k-1}(X_{A,k};{\mathbb F})$ in terms of the group algebra $F[G^k]$. A simplified version valid under the assumption $gcd(|G|,k)=1$ is given in Claim 2.2.

We first introduce some terminology. Fix $1 \le k \le |G|$. For an element $g \in G$ and $1 \le i \le k$ let $e_i(g)=(1,\ldots ,1,g,1, \ldots ,1) \in G^k$ with $g$ appearing in the $i$-th coordinate. An element

$$\begin{aligned} s=\sum _{(g_1,\ldots ,g_k) \in G^k} s(g_1,\ldots ,g_k) (g_1,\ldots ,g_k) \in {\mathbb F}[G^k] \end{aligned}$$

is skew-symmetric if $s(g_{\sigma ^{-1}(1)},\ldots ,g_{\sigma ^{-1}(k)})=\mathrm{sgn}(\sigma )s(g_1,\ldots ,g_k)$ for all $(g_1,\ldots ,g_k) \in G^k$ and $\sigma$ in the symmetric group $S_k$. If $\mathrm{char} \,{\mathbb F}=2$ then $s$ is additionally required to satisfy $s(g_1,\ldots ,g_k)=0$ if $g_i=g_j$ for some $i \ne j$. Let ${\fancyscript{S}}$ denote the space of skew-symmetric elements of ${\mathbb F}[G^k]$. For $h \in G$ let $W_h=\{(g_1,\ldots ,g_k) \in G^k: \prod _{i=1}^k g_i=h\}$ and let ${\fancyscript{S}}_h=\{s \in {\fancyscript{S}}: \mathrm{supp}(s) \subset W_h\}$. Let $\rho _h$ denote the projection from ${\fancyscript{S}}$ onto ${\fancyscript{S}}_h$ given by $\rho _h(\sum _{u \in G^k} s(u) u)=\sum _{u \in W_h} s(u) u$.

Let $Y$ be a $(k-1)$-dimensional simplicial complex on the vertex set $V$ and let $Y(\ell )$ denote the set of its (unordered) $\ell$-simplices. Let $C_{k-1}(Y;{\mathbb F})$ denote the space of ${\mathbb F}$-valued $(k-1)$-chains of $Y$. Recall that $\phi \in C_{k-1}(Y;{\mathbb F})$ is a skew-symmetric ${\mathbb F}$-valued function on the ordered $(k-1)$-simplices of $Y$. A $(k-1)$-chain $\phi \in C_{k-1}(Y;{\mathbb F})$ is a reduced $(k-1)$-cycle if for all $\{v_1,\ldots ,v_{k-1}\} \in Y(k-2)$ it holds that

$$\begin{aligned} \sum _{\{v_k \in V: \{v_1,\ldots ,v_{k-1},v_k\} \in Y(k-1)\} } \phi (v_1,\ldots ,v_{k-1},v_k)=0. \end{aligned}$$

(8)

Specializing to the case $Y=X_{A,k}$ it is clear that

$$\begin{aligned} C_{k-1}(X_{A,k},{\mathbb F})=\{s \in {\fancyscript{S}}: \mathrm{supp}(s) \subset \cup _{a \in A} W_a\}=\bigoplus _{a \in A} {\fancyscript{S}}_a. \end{aligned}$$

By (8), $s \in C_{k-1}(X_{A,k},{\mathbb F})$ is a reduced $(k-1)$-cycle of $X_{A,k}$ if for all $(g_1,\ldots ,g_{k-1}) \in G^{k-1}$

$$\begin{aligned} \sum _{a \in A} s(g_1,\ldots ,g_{k-1},a \prod _{j=1}^{k-1} g_j^{-1})=0. \end{aligned}$$

(9)

Let

$$\begin{aligned} {\fancyscript{H}}(A)=\{s \in \bigoplus _{a \in A} {\fancyscript{S}}_a~:~\sum _{a \in A}e_i(a^{-1}) \rho _a(s)=0 \mathrm{~~for~all~~} 1 \le i \le k\}. \end{aligned}$$

The homology space $\tilde{H}_{k-1}(X_{A,k};{\mathbb F})$ is characterized by the following

claim 2.1

$$\begin{aligned} \tilde{H}_{k-1}(X_{A,k};{\mathbb F})={\fancyscript{H}}(A). \end{aligned}$$

Proof

First note that by skew symmetry

$$\begin{aligned} {\fancyscript{H}}(A)=\{s \in \bigoplus _{a \in A} {\fancyscript{S}}_a~:~\sum _{a \in A}e_k(a^{-1}) \rho _a(s)=0\}. \end{aligned}$$

(10)

Now let $s \in C_{k-1}(X_{A,k};{\mathbb F})=\bigoplus _{a \in A} {\fancyscript{S}}_a$. Then

$$\begin{aligned}&\sum _{a \in A} e_k(a^{-1}) \rho _a(s) \nonumber \\&\quad =\sum _{a \in A} e_k(a^{-1}) \sum _{\left( g_1,\ldots ,g_{k-1}\right) \in G^{k-1}} s\left( g_1,\ldots ,g_{k-1},a\prod _{j=1}^{k-1} g_j^{-1}\right) \left( g_1,\ldots ,g_{k-1},a \prod _{j=1}^{k-1} g_j^{-1}\right) \nonumber \\&\quad =\sum _{(g_1,\ldots ,g_{k-1}) \in G^{k-1}}\left( \sum _{a \in A}s(g_1,\ldots ,g_{k-1},a\prod _{j=1}^{k-1} g_j^{-1}) \right) \left( g_1,\ldots ,g_{k-1},\prod _{j=1}^{k-1} g_j^{-1}\right) . \end{aligned}$$

Therefore $\sum _{a \in A} e_k(a^{-1}) \rho _a(s)=0$ iff

$$\begin{aligned} \sum _{a \in A}s\left( g_1,\ldots ,g_{k-1},a\prod _{j=1}^{k-1} g_j^{-1}\right) =0 \end{aligned}$$

(11)

for all $(g_1,\ldots ,g_{k-1}) \in G^{k-1}$. Hence the Claim follows from (9) and (10).$\square$

Let ${\fancyscript{S}}^A=\bigoplus _{a \in A} {\fancyscript{S}}=\{(r_a)_{a \in A}: r_a \in {\fancyscript{S}}\}$ and let

$$\begin{aligned} {\fancyscript{R}}(A)=\{(r_a)_{a \in A} \in {\fancyscript{S}}^A~:~ \sum _{a \in A}e_i({a^{-1}})r_a=0 \mathrm{~~for~all~~} 1 \le i \le k\}. \end{aligned}$$

For $g \in G$ let $e(g)=\prod _{i=1}^k e_i(g)=(g,\ldots ,g) \in G^k$. In Section 3 we shall need the following

claim 2.2

Assume that $gcd(|G|,k)=1$. Then, the mapping $\Psi :{\mathbb F}[G] \otimes _{{\mathbb F}} {\fancyscript{H}}(A) \rightarrow {\fancyscript{R}}(A)$ given by

$$\begin{aligned} \Psi (g \otimes s)=(e(g) \, \rho _a(s))_{a \in A} \end{aligned}$$

is an isomorphism.

Proof

We first show injectivity. Let $w \in \ker \Psi$ and write $w=\sum _{g \in G} g \otimes s_g$ where $s_g \in {\fancyscript{H}}(A)$. Then for all $a \in A$

$$\begin{aligned} \sum _{g \in G} e(g) \rho _a(s_g)=0. \end{aligned}$$

Since $e(g) \rho _a(s_g) \in {\fancyscript{S}}_{ag^k}$ and $ag^k \ne ah^k$ for $g \ne h$ (by the assumption $gcd(|G|,k)=1$), it follows that $e(g) \rho _a(s_g)=0$ and hence $\rho _a(s_g)=0$ for all $g \in G$ and $a \in A$. Therefore $w=0$. To show surjectivity let $(r_a)_{a \in A} \in {\fancyscript{R}}(A)$. For $g \in G$ let

$$\begin{aligned} s_g=e(g^{-1}) \sum _{a \in A} \rho _{ag^k}(r_a) \in \bigoplus _{a \in A} {\fancyscript{S}}_a. \end{aligned}$$

We first show that $s_g \in {\fancyscript{H}}(A)$. For $1 \le i \le k$ and $g \in G$ let

$$\begin{aligned} t_{i,g}=\sum _{a \in A} e_i(a^{-1}) \rho _{a g^k}(r_a) \in {\fancyscript{S}}_{g^k}. \end{aligned}$$

Then

$$\begin{aligned} \sum _{g \in G} t_{i,g}&= \sum _{g \in G} \sum _{a \in A} e_i(a^{-1})\rho _{ag^k}(r_a) \\&= \sum _{a \in A} e_i(a^{-1}) \left( \sum _{g \in G} \rho _{a g^k}(r_a)\right) =\sum _{a \in A}e_i(a^{-1})r_a =0. \end{aligned}$$

It follows that $t_{i,g}=0$ for all $1 \le i \le k$ and $g \in G$. Therefore, for $1 \le i \le k$ and $g \in G$

$$\begin{aligned} \sum _{a \in A} e_i(a^{-1}) \rho _a(s_g)&= \sum _{a \in A}e_i(a^{-1}) \rho _a\left( e(g^{-1}) \sum _{a' \in A} \rho _{a'g^k}(r_{a'}) \right) \\&= \sum _{a \in A} e_i(a^{-1})e(g^{-1}) \rho _{a g^k}(r_a) =e(g^{-1})t_{i,g}=0. \end{aligned}$$

Hence $s_g \in {\fancyscript{H}}(A)$ and thus $w=\sum _{g \in G}g \otimes s_g \in {\mathbb F}[G] \bigotimes _{{\mathbb F}} {\fancyscript{H}}(A)$. Finally, for all $a \in A$

$$\begin{aligned} \sum _{g \in G}e(g) \rho _a(s_g)=\sum _{g \in G}e(g) \left( e(g^{-1}) \rho _{ag^k}(r_a) \right) = \sum _{g \in G} \rho _{ag^k}(r_a)=r_a \end{aligned}$$

and therefore, $\Psi (w)=(r_a)_{a \in A}$. $\square$

Corollary 2.3

If $gcd(|G|,k)=1$ then

$$\begin{aligned} \dim \tilde{H}_{k-1}(X_{A,k};{\mathbb F})=\frac{\dim {\fancyscript{R}}(A)}{|G|}. \end{aligned}$$

$\square$

3 The semisimple case

Let $G$ be the cyclic group of prime order $C_p=\langle z \rangle$ and let $A=\{a_1,\ldots ,a_m\}\subset C_p$ where $a_j=z^{\alpha _j}$. Let $\alpha =(\alpha _1,\ldots ,\alpha _k) \in {\mathbb F}_p^k$. In this section, we compute $\dim \tilde{H}_{k-1}(X_{A,k};{\mathbb F})$ when $char \, {\mathbb F}\ne p$. We may assume that ${\mathbb F}$ is algebraically closed. Recall that $\omega$ is a primitive $p$-th root of unity in ${\mathbb F}=\overline{{\mathbb F}}$. The character group $\widehat{C_p}$ consists of all characters $\eta _{u}$ where $u \in {\mathbb F}_p$ and $\eta _{u}(z)=\omega ^{u}$. Similarly, $\widehat{C_p^k}=\{\chi _{\beta }: \beta =(\beta _1,\ldots ,\beta _k) \in {\mathbb F}_p^k\}$ where for $\gamma =(\gamma _1,\ldots ,\gamma _k) \in {\mathbb F}_p^k$

$$\begin{aligned} \chi _{\beta }(z^{\gamma _1},\ldots ,z^{\gamma _k})= (\eta _{\beta _1} \times \ldots \times \eta _{\beta _k})(z^{\gamma _1},\ldots ,z^{\gamma _k})=\omega ^{\beta \gamma } \end{aligned}$$

and $\beta \gamma =\sum _{i=1}^k \beta _i \gamma _i$ is the standard inner product in ${\mathbb F}_p^k$. The Fourier transform of $f \in {\mathbb F}[C_p^k]$ is thus, $\widehat{f} \in {\mathbb F}[\widehat{C_p^k}]$ given by

$$\begin{aligned} \widehat{f}(\chi _{\beta })=\sum _{\gamma =(\gamma _1,\ldots ,\gamma _k) \in {\mathbb F}_p^k} f(z^{\gamma _1},\ldots ,z^{\gamma _k}) \omega ^{-\beta \gamma }. \end{aligned}$$

As already remarked, in proving Theorem 1.2 we may assume that $k<p$. Corollary 2.3 then implies that $\dim \tilde{H}_{k-1}(X_{A,k};{\mathbb F})=\frac{\dim {\fancyscript{R}}(A)}{p}.$ Theorem 1.2 will thus follow from

Proposition 3.1

$$\begin{aligned} \dim {\fancyscript{R}}(A)=m\left( {\begin{array}{c}p\\ k\end{array}}\right) -\sum _{\beta \in {\fancyscript{B}}_k} \mathrm{rank}\, M_{\beta ,\alpha }. \end{aligned}$$

Proof

Define an ${\mathbb F}$-linear mapping

$$\begin{aligned} \Phi : {\fancyscript{S}}^A \rightarrow \bigoplus _{\beta \in {\fancyscript{B}}_k} {\mathbb F}^m \end{aligned}$$

as follows. For $r=(r_{a_j})_{j=1}^m \in {\fancyscript{S}}^A$ let

$$\begin{aligned} \Phi (r)=\left( \left[ \begin{array}{c} \widehat{r_{a_1}}(\chi _{\beta }) \\ \vdots \\ \widehat{r_{a_m}}(\chi _{\beta }) \end{array} \right] ~:~ \beta \in {\fancyscript{B}}_k\right) . \end{aligned}$$

Note that since $r_{a_j}$ is a skew-symmetric element of ${\mathbb F}[C_p^k]$, it follows that $\widehat{r_{a_j}}$ is a skew symmetric element of ${\mathbb F}[\widehat{C_p^k}]$ and hence is determined by its values on ${\fancyscript{B}}_k$. Therefore, $\Phi$ is an isomorphism.$\square$

claim 3.2

$\Phi$ restricts to an isomorphism from ${\fancyscript{R}}(A)$ onto $\bigoplus _{\beta \in {\fancyscript{B}}_k} \ker M_{\beta ,\alpha }$.

Proof

Note that if $1 \le i \le k$ and $\beta =(\beta _1,\ldots ,\beta _k) \in {\mathbb F}_p^k$ then

$$\begin{aligned} \widehat{e_i(a_j^{-1})}(\chi _{\beta })=\widehat{e_i(z^{-\alpha _j})}(\chi _{\beta })=\omega ^{\beta _i \alpha _j}. \end{aligned}$$

Let $r=(r_{a_j})_{j=1}^m \in {\fancyscript{S}}^A$ and fix $1 \le i \le k$ and $\beta =(\beta _1,\ldots ,\beta _k) \in {\mathbb F}_p^k$. Evaluating the Fourier transform of the element $\sum _{j=1}^m e_i(a_j^{-1}) r_{a_j}$ at the character $\chi _{\beta }$ we obtain

$$\begin{aligned}&\left( \sum _{j=1}^m e_i(a_j^{-1}) r_{a_j}\right) \, \hat{\, } \, (\chi _{\beta }) \\&\quad = \sum _{j=1}^m \widehat{e_i(a_j^{-1})}(\chi _{\beta }) \widehat{r_{a_j}}(\chi _{\beta }) \\&\quad = \sum _{j=1}^m \omega ^{\beta _i \alpha _j}\widehat{r_{a_j}}(\chi _{\beta }) . \end{aligned}$$

It follows that $r=(r_{a_j})_{j=1}^m \in {\fancyscript{R}}(A)$ iff for all $\beta =(\beta _1,\ldots ,\beta _k) \in {\mathbb F}_p^k$

$$\begin{aligned} M_{\beta ,\alpha }\left[ \begin{array}{c} \widehat{r_{a_1}}(\chi _{\beta }) \\ \vdots \\ \widehat{r_{a_m}}(\chi _{\beta }) \end{array} \right] = \left[ \begin{array}{ccc} \omega ^{\beta _1 \alpha _1} &{} \cdots &{} \omega ^{\beta _1 \alpha _{m}} \\ \vdots &{} \ddots &{} \vdots \\ \omega ^{\beta _k \alpha _1} &{} \cdots &{} \omega ^{\beta _k \alpha _{m}} \end{array} \right] \left[ \begin{array}{c} \widehat{r_{a_1}}(\chi _{\beta }) \\ \vdots \\ \widehat{r_{a_m}}(\chi _{\beta }) \end{array} \right] =0. \end{aligned}$$

Therefore, $r=(r_{a_j})_{j=1}^m \in {\fancyscript{R}}(A)$ iff $\Phi (r) \in \bigoplus _{\beta \in {\fancyscript{B}}_k} \ker M_{\beta ,\alpha }$. The Claim now follows from the bijectivity of $\Phi$.$\square$

Proof of Proposition 3.1:

By Claim 3.2

$$\begin{aligned} \dim {\fancyscript{R}}(A)&= \sum _{\beta \in {\fancyscript{B}}_k} \dim \ker M_{\beta ,\alpha } = \sum _{\beta \in {\fancyscript{B}}_k} (m-\mathrm{rank}\, M_{\beta ,\alpha }) \\&= m\left( {\begin{array}{c}p\\ k\end{array}}\right) -\sum _{\beta \in {\fancyscript{B}}_k} \mathrm{rank}\, M_{\beta ,\alpha }. \end{aligned}$$

$\square$

4 The modular case

In subsections 4.1 and 4.2 we study certain properties of determinants of generalized Vandermonde matrices over the group algebra ${\mathbb F}_p[C_p^k]$. These results are then used in subsection 4.3 to prove Theorem 1.3.

4.1 A generalized Vandermonde

Recall that $z$ is a fixed generator of $C_p$ and let $1 \le k \le p$. For $1 \le i \le k$ let $x_i=e_i(z)$. Then $\{x_1,\ldots ,x_k\}$ is a generating set of $C_p^k$. Let $x=(x_1,\ldots ,x_k)$. For $\beta =(\beta _1,\ldots ,\beta _k) \in {\fancyscript{B}}_k$ let

$$\begin{aligned} N_{\beta }= \left[ \begin{array}{ccc} x_1^{-\beta _1} &{} \cdots &{} x_1^{-\beta _{k}} \\ \vdots &{} \ddots &{} \vdots \\ x_k^{-\beta _1} &{} \cdots &{} x_k^{-\beta _{k}} \end{array} \right] . \end{aligned}$$

Proposition 4.1

Let $1 \le k \le p$. Then

$$\begin{aligned} \det N_{\beta } =w_{\beta } \prod _{1 \le i< j \le k} (x_i-x_j), \end{aligned}$$

(12)

where $w_{\beta }$ is a unit of $\, {\mathbb F}_p[C_p^k]$.

Recall the definition of Schur polynomials (see e.g., [2]). Let $\xi =(\xi _1,\ldots ,\xi _k)$ be a vector of variables. For a partition $\lambda =(\lambda _1 \ge \cdots \ge \lambda _k)$ let

$$\begin{aligned} D_\lambda (\xi )=D_{\lambda }(\xi _1,\ldots ,\xi _k)=\det ([\xi _i^{\lambda _j+k-j}]_{i,j=1}^k) \in {\mathbb Z}[\xi _1,\ldots ,\xi _k]. \end{aligned}$$

Note that for the zero partition $0=(0,\ldots ,0)$

$$\begin{aligned} D_0(\xi )=\det \left[ \begin{array}{cccc} \xi _1^{k-1} &{} \xi _1^{k-2} &{} \cdots &{} 1 \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ \xi _k^{k-1} &{} \xi _k^{k-2} &{} \cdots &{} 1 \end{array} \right] = \prod _{1 \le i<j \le k} (\xi _i-\xi _j). \end{aligned}$$

The Schur polynomial associated with $\lambda$ is

$$\begin{aligned} {\mathbb S}_{\lambda }(\xi )=\frac{D_{\lambda }(\xi )}{D_0(\xi )} \in {\mathbb Z}[\xi _1,\ldots ,\xi _k]. \end{aligned}$$

The dimension formula (see e.g., Proposition 5.21.2 in [2]) asserts that

$$\begin{aligned} {\mathbb S}_{\lambda }(1,\ldots ,1)=\prod _{1 \le i < j \le k}\frac{\lambda _i-\lambda _j+j-i}{j-i}. \end{aligned}$$

(13)

Proof of Proposition 4.1

Let

$$\begin{aligned} \lambda =(\lambda _1,\ldots ,\lambda _k)=(p-\beta _1-k+1,p-\beta _2-k+2,\ldots ,p-\beta _k). \end{aligned}$$

Note that $\beta =(\beta _1,\cdots ,\beta _k) \in {\fancyscript{B}}_k$ implies that $\lambda _1 \ge \cdots \ge \lambda _k$. Then

$$\begin{aligned} \det N_{\beta }&= D_{\lambda }(x)={\mathbb S}_{\lambda }(x) D_0(x) \\&= {\mathbb S}_{\lambda }(x) \prod _{1 \le i< j \le k} (x_i-x_j). \end{aligned}$$

By (13) the image of ${\mathbb S}_{\lambda }(x)\in {\mathbb F}_p[C_p^k]$ under the augmentation map ${\mathbb F}_p[C_p^k] \rightarrow {\mathbb F}_p$ is

$$\begin{aligned} {\mathbb S}_{\lambda }(1,\ldots ,1)\, (\mathrm{mod} \, p)=\prod _{1 \le i<j \le k} \frac{\beta _j-\beta _i}{j-i}\, (\mathrm{mod} \, p) \ne ~0\, (\mathrm{mod} \, p). \end{aligned}$$

It follows that $w_{\beta }={\mathbb S}_{\lambda }(x)$ is invertible in ${\mathbb F}_p[C_p^k]$. $\square$

4.2 Skew-symmetric annihilators of $\mathbf {D_0(x)}$

Recall that ${\fancyscript{S}}$ is the space of skew-symmetric elements of ${\mathbb F}_p[C_p^k]$. In this subsection we show

Proposition 4.2

Assume that $1 \le k<p$ and let $s \in {\fancyscript{S}}$. If $D_0(x)s=0$ then $s=0$.

The proof of Proposition 4.2 depends on Proposition 4.3 below. Let ${\mathbb N}$ denote the nonnegative integers and for $a,b \in {\mathbb N}$ let $[a,b]=\{a,\ldots ,b\}$. Let

$$\begin{aligned} {\mathbb N}_k=\{(\mu _1,\ldots ,\mu _k) \in {\mathbb N}^k~:~ \mu _i \ne \mu _j ~~\mathrm{if}~~ i \ne j\}. \end{aligned}$$

For $\mu =(\mu _1,\ldots ,\mu _k)~,~ \nu =(\nu _1,\ldots ,\nu _k) \in {\mathbb N}_k$ write $\mu \preceq \nu$ if $\{\mu _1,\ldots ,\mu _k\}$ precedes $\{\nu _1,\ldots ,\nu _k\}$ in the lexicographic order on $k$-subsets of ${\mathbb N}$, i.e., if

$$\begin{aligned} \sum _{i=1}^k 2^{-\mu _i} \ge \sum _{i=1}^k 2^{-\nu _i}. \end{aligned}$$

Fix a $\mu =(\mu _1,\ldots ,\mu _k) \in {\mathbb N}_k$ such that $\mu _1<\cdots <\mu _k$ and let

$$\begin{aligned} L=\{1 \le i \le k-1: \mu _i+1<\mu _{i+1}\}. \end{aligned}$$

Write $L=\{\ell _1<\ldots <\ell _{t-1}\}$ and let $\ell _0=0~,~ \ell _t=k$. For $1 \le i \le t$ let $K_i=[\ell _{i-1}+1,\ell _i]$. Let

$$\begin{aligned} {\fancyscript{G}}_1(\mu )=\{(\gamma ,\sigma )\in {\mathbb N}_k \times S_k~:~\gamma \preceq \mu ~\mathrm{and}~ \gamma _j-\sigma (j)=\mu _j-j ~~\mathrm{for~all}~~ j\}. \end{aligned}$$

We will need the following characterization of ${\fancyscript{G}}_1(\mu )$. Let $S_K$ denote the symmetric group on a set $K$. Let $T=S_{K_1} \times \cdots \times S_{K_t}$ be the Young subgroup of $S_k$ corresponding to the partition $[k]= \cup _{i=1}^t K_i$. Let

$$\begin{aligned} {\fancyscript{G}}_2(\mu )=\{(\gamma ,\sigma )\in {\mathbb N}_k \times T~:~ \gamma _j=\mu _{\sigma (j)}~~\mathrm{for~all}~~j\}. \end{aligned}$$

Proposition 4.3

${\fancyscript{G}}_1(\mu )={\fancyscript{G}}_2(\mu )$

Proof

We first show that ${\fancyscript{G}}_2(\mu ) \subset {\fancyscript{G}}_1(\mu )$. Let $(\gamma ,\sigma ) \in {\fancyscript{G}}_2(\mu )$ and let $1 \le j \le k$. If $j \in K_i$ then $\sigma (j) \in K_i$ and hence $\mu _{\sigma (j)}-\mu _j=\sigma (j)-j$. Therefore,

$$\begin{aligned} \gamma _j-\sigma (j)= \mu _{\sigma (j)}-\sigma (j)=\mu _j-j \end{aligned}$$

and so $(\gamma ,\sigma ) \in {\fancyscript{G}}_1(\mu )$. For the other direction let $(\gamma ,\sigma )\in {\fancyscript{G}}_1(\mu )$. Write $\gamma =(\gamma _1,\ldots ,\gamma _k)$ and let $\pi \in S_k$ such that $\gamma _{\pi (1)}< \cdots <\gamma _{\pi (k)}$.

claim 4.4

For $1 \le i \le t$ and $j \in K_i$

• (a) $\sigma (\pi (j))=j$.

• (b) $\gamma _{\pi (j)}=\mu _j$.

• (c) $\pi (j) \in K_i$.

Proof

We argue by induction on $j$. Suppose (a),(b), and (c) hold for all $j'<j$. (a) implies that $\{\sigma (\pi (j')): j'<j\}=[j-1]$ and hence $\sigma (\pi (j)) \ge j$. Therefore,

$$\begin{aligned} \mu _j-j \ge \mu _j -\sigma (\pi (j)). \end{aligned}$$

(14)

Next note that by (b) $\gamma _{\pi (j')}=\mu _{j'}$ for all $j'<j$. As $\gamma \preceq \mu$ it follows that $\mu _j \ge \gamma _{\pi (j)}$, and therefore,

$$\begin{aligned} \mu _j -\sigma (\pi (j)) \ge \gamma _{\pi (j)}-\sigma (\pi (j))=\mu _{\pi (j)}-\pi (j). \end{aligned}$$

(15)

Finally (c) implies that $\{\pi (j'):1 \le j' \le \ell _{i-1}\}=[1,\ell _{i-1}]$ and therefore $\pi (j) \ge \ell _{i-1}+1$. Together with the assumption $j \in K_i$ it follows that

$$\begin{aligned} \mu _{\pi (j)}-\pi (j) \ge \mu _{\ell _{i-1}+1}-(\ell _{i-1}+1)=\mu _j-j. \end{aligned}$$

(16)

It follows that the three inequalities in (14),(15),(16) are in fact equalities. Therefore $\sigma (\pi (j))=j$, $\gamma _{\pi (j)}=\mu _j$ and $\mu _{\pi (j)}=\mu _j+(\pi (j)-j)$, respectively, establishing (a),(b), and (c) for $j$. $\square$

Claim 4.4 implies that $\sigma =\pi ^{-1} \in T$ and that $\gamma _{j}=\mu _{\sigma (j)}$ for all $1 \le j \le k$. Therefore $(\gamma ,\sigma ) \in {\fancyscript{G}}_2(\mu )$.

Proof of Proposition 4.2

Let $s \in {\fancyscript{S}}$ such that $D_0(x)s=0$. We have to show that $s=0$. For $\gamma =(\gamma _1,\ldots ,\gamma _k) \in {\mathbb F}_p^k$ we abbreviate $x^{\gamma }=\prod _{j=1}^k x_j^{\gamma _j}=(z^{\gamma _1},\dots ,z^{\gamma _k}) \in C_p^k$. Note that this notation is unambiguous since $x_j^p=1$. By assumption

$$\begin{aligned} 0 = D_0(x)s&= \sum _{\sigma \in S_k} \mathrm{sgn}(\sigma ) \prod _{j=1}^k x_j^{k-\sigma (j)} \sum _{\gamma =(\gamma _1,\ldots ,\gamma _k)\in {\mathbb F}_p^k} s(x^{\gamma }) \prod _{j=1}^k x_j^{\gamma _j} \nonumber \\&= \sum _{\gamma =(\gamma _1,\ldots ,\gamma _k)\in {\mathbb F}_p^k} \sum _{\sigma \in S_k} \mathrm{sgn}(\sigma )s(x^{\gamma }) \prod _{j=1}^k x_j^{\gamma _j+k-\sigma (j)}. \end{aligned}$$

(17)

Suppose for contradiction that $s \ne 0$ and let

$$\begin{aligned} \mu =(\mu _1,\ldots ,\mu _k)=\max \{\gamma \in {\fancyscript{B}}_k~:~s(x^{\gamma }) \ne 0\}, \end{aligned}$$

where the maximum is taken with respect to $\preceq$. Let $\lambda \in {\mathbb F}_p$ denote the coefficient of $\prod _{j=1}^k x_j^{\mu _j+k-j}$ in the expansion of $D_0(x)s~$ in the standard basis $\{x^{\beta }:\beta \in {\mathbb F}_p^k\}$ of ${\mathbb F}_p[C_p^k]$. Note that if

$$\begin{aligned} \prod _{j=1}^k x_j^{\mu _j+k-j}=\prod _{j=1}^k x_j^{\gamma _j+k-\sigma (j)} \end{aligned}$$

then for all $1 \le j \le k$

$$\begin{aligned} \mu _j-j = \gamma _j-\sigma (j) \, (\mathrm{mod} \, p). \end{aligned}$$

Since

$$\begin{aligned} -1 \le \mu _j-j \le p-1-k \end{aligned}$$

and

$$\begin{aligned} -k \le \gamma _j-\sigma (j) \le p-2 \end{aligned}$$

it follows that

$$\begin{aligned} \mu _j-j = \gamma _j-\sigma (j). \end{aligned}$$

Hence, Eq.(17) and Proposition 4.3 imply that

$$\begin{aligned} \lambda&= \sum _{(\gamma ,\sigma ) \in {\fancyscript{G}}_1(\mu )} \mathrm{sgn}(\sigma ) s(x^{\gamma })= \sum _{(\gamma ,\sigma ) \in {\fancyscript{G}}_2(\mu )} \mathrm{sgn}(\sigma ) s(x^{\gamma }) \\&= \sum _{(\gamma ,\sigma ) \in {\fancyscript{G}}_2(\mu )} \mathrm{sgn}(\sigma ) s(z^{\gamma _1},\ldots ,z^{\gamma _k}) \\&= \sum _{\sigma \in S_{K_1} \times \cdots \times S_{K_t}} \mathrm{sgn}(\sigma ) s(z^{\mu _{\sigma (1)}},\ldots ,z^{\mu _{\sigma (k)}}) \\&= |S_{K_1} \times \cdots \times S_{K_t}| \, s(z^{\mu _1},\ldots ,z^{\mu _k})= \prod _{i=1}^t (\ell _i-\ell _{i-1})! \, s(x^{\mu }). \end{aligned}$$

Since $\ell _t=k<p$ it follows that $\prod _{i=1}^t (\ell _i-\ell _{i-1})! \ne 0 (\mathrm{mod} \, p)$ and so $\lambda \ne 0$. Therefore, $D_0(x)s \ne 0$, a contradiction.$\square$

Remarks

Proposition 4.2 does not hold for $k=p$. Indeed, in this case $s=D_0(x)=\prod _{1 \le i < j \le p} (x_i-x_j)$ is a nonzero skew-symmetric element of ${\mathbb F}_p[C_p^p]$ and it can be checked that $D_0(x) s=D_0(x)^2=0$.

4.3 Homology of $\mathbf {X_{A,k}}$ over $\mathbf {{\mathbb F}_p}$

In this subsection, we prove Theorem 1.3. We first consider the case $m=k$.

Theorem 4.5

If $|A|=k$ then $\tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)=0.$

Proof

As already noted the case $k=p$ is trivial so we assume $k<p$. Let $A=\{a_1,\ldots ,a_k\}$ where $a_i=z^{\alpha _i}$ and $\alpha =(\alpha _1,\ldots ,\alpha _k) \in {\fancyscript{B}}_k$. Let $s \in \tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)$ then by Claim 2.1

$$\begin{aligned} N_{\alpha } \left[ \begin{array}{c} \rho _{a_1}(s) \\ \vdots \\ \rho _{a_{k}}(s) \end{array} \right]&= \left[ \begin{array}{ccc} x_1^{-\alpha _1} &{} \cdots &{} x_1^{-\alpha _{k}} \\ \vdots &{} \ddots &{} \vdots \\ x_k^{-\alpha _1} &{} \cdots &{} x_k^{-\alpha _{k}} \end{array} \right] \left[ \begin{array}{c} \rho _{a_1}(s) \\ \vdots \\ \rho _{a_{k}}(s) \end{array} \right] \\&= \left[ \begin{array}{ccc} e_1(a_1^{-1}) &{} \cdots &{} e_1(a_k^{-1}) \\ \vdots &{} \ddots &{} \vdots \\ e_k(a_1^{-1}) &{} \cdots &{} e_k(a_k^{-1}) \end{array} \right] \left[ \begin{array}{c} \rho _{a_1}(s) \\ \vdots \\ \rho _{a_{k}}(s) \end{array} \right] =0. \end{aligned}$$

Therefore, $\, \det N_{\alpha } \cdot \rho _{a_j}(s)=0$ for all $1 \le j \le k$. Proposition 4.1 then implies that $D_0(x)\rho _{a_j}(s)=0$. Hence $\rho _{a_j}(s)=0$ by Proposition 4.2. It follows that $s=0$ and so $\tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)=0$. $\square$

Proof of Theorem 1.3

Let $|A|=m \ge k$ and let $A'$ be an arbitrary subset of $A$ of cardinality $k$. Theorem 4.5 and Eq. (6) imply that $\tilde{H}_*(X_{A',k};{\mathbb F}_p)=0.$ Hence by the exact sequence

$$\begin{aligned} 0&= \tilde{H}_{k-1}(X_{A',k};{\mathbb F}_p) \rightarrow \tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p) \rightarrow \\&\rightarrow \tilde{H}_{k-1}(X_{A,k},X_{A',k};{\mathbb F}_p) \rightarrow \tilde{H}_{k-2}(X_{A',k};{\mathbb F}_p)=0 \end{aligned}$$

it follows that

$$\begin{aligned} \dim \tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)&= \dim \tilde{H}_{k-1}(X_{A,k},X_{A',k};{\mathbb F}_p) \\&= f_{k-1}(X_{A,k})-f_{k-1}(X_{A',k})=(\frac{m}{k}-1)\left( {\begin{array}{c}p-1\\ k-1\end{array}}\right) . \end{aligned}$$

$\square$

5 Uncertainty numbers and homology

Proof of Theorem 1.1

Recall that $A=\{z^{\alpha _1},\ldots ,z^{\alpha _m}\} \subset C_p$. Let ${\mathbb F}$ be an algebraically closed field with $\mathrm{char} \, {\mathbb F}=\ell$. We consider two cases:

(i) The semisimple case $\ell \ne p$. Here, it suffices to show that for any fixed $1 \le k \le p$ the following three conditions are equivalent:

• (C1)  $\tilde{H}_{k-1}(X_{A,k};{\mathbb F}) \ne 0$.

• (C2)  There exists a $\beta \in {\fancyscript{B}}_k$ such that $\mathrm{rank}\, M_{\beta ,\alpha } <m$.

• (C3)  $u_{{\mathbb F}}(A) \le p-k$.

First note that Theorem 1.2 implies that $\tilde{H}_{k-1}(X_{A,k};{\mathbb F}) \ne 0$ iff

$$\begin{aligned} \frac{m}{p}\left( {\begin{array}{c}p\\ k\end{array}}\right) >\frac{1}{p} \sum _{\beta \in {\fancyscript{B}}_k} \mathrm{rank}\, M_{\beta ,\alpha }. \end{aligned}$$

This proves the equivalence of (C1) and (C2). Next, let $\lambda =(\lambda _1,\ldots ,\lambda _m) \in {\mathbb F}^m$ and let $f_{\lambda }=\sum _{j=1}^m \lambda _j z^{\alpha _j} \in {\mathbb F}[C_p]$. Then, $\mathrm{supp}(f_{\lambda }) \subset A$ and for $\beta =(\beta _1,\ldots ,\beta _k) \in {\mathbb F}_p^k$

$$\begin{aligned} M_{\beta ,\alpha } \lambda =(\sum _{j=1}^m \lambda _j \omega ^{\beta _i \alpha _j})_{i=1}^k= \left( \widehat{f_{\lambda }}(\eta _{-\beta _i})\right) _{i=1}^k. \end{aligned}$$

It follows that if $\beta =(\beta _1,\ldots ,\beta _k) \in {\fancyscript{B}}_k$ then $\mathrm{rank}\, M_{\beta ,\alpha } <m$ iff there exists a nonzero $f=f_{\lambda }\in {\mathbb F}[C_p]$ such that $\mathrm{supp}(f) \subset A$ and $\mathrm{supp}(\widehat{f}) \cap \{\eta _{-\beta _i}\}_{i=1}^k = \emptyset$. This proves the equivalence of (C2) and (C3).

(ii) The modular case $\ell =p$. Let ${\mathbb F}$ be a field of characteristic $p$. By Theorem 1.3

$$\begin{aligned} p-\max \{k:\tilde{H}_{k-1}(X_{A,k};{\mathbb F}) \ne 0\}=p-m+1. \end{aligned}$$

It thus suffices to show that $u_{{\mathbb F}}(A)=p-m+1$. Let $0 \ne f=\sum _{j=1}^m \lambda _j z^{\alpha _j} \in {\mathbb F}[C_p]$. Regarding $f=f(z)$ as an element of the polynomial ring ${\mathbb F}[z]$, the rank of $T_f$ is given by

$$\begin{aligned} \mathrm{rank}\, T_f&= p-\deg \,\mathrm{gcd}(f(z),z^p-1) \nonumber \\&= p-\deg \, \mathrm{gcd}(f(z),(z-1)^p)=p-\mu (f), \end{aligned}$$

(18)

where $\mu (f)$ is the multiplicity of $1$ as a root of $f(z)$. By a simple well-known result (see e.g., Lemma 2 in [4]), $\mu (f) \le m-1$ and hence $u_{{\mathbb F}}(A) \ge p-m+1$. For the other direction note that the ${\mathbb F}$-linear space

$$\begin{aligned} {\fancyscript{P}}=\{f(z) \in {\mathbb F}[C_p]:~\mu (f) \ge m-1\} \end{aligned}$$

satisfies $\dim _{{\mathbb F}} {\fancyscript{P}}= p-m+1$ and hence must contain a nonzero $f$ of the form $f(z)=\sum _{j=1}^m \lambda _j z^{\alpha _j}$. It follows from (18) that $f$ satisfies $\mathrm{rank}\,T_f \le p-m+1$. Therefore, $u_{{\mathbb F}}(A) \le p-m+1$. $\square$

6 Concluding remarks

We mention two problems related to the results of this paper.

• 1. Let $k \ge 2$ and let $X$ be a $(k-1)$-dimensional complex $X$ with $N=f_{k-1}(X)$ facets. It was observed by G. Kalai, S. Weinberger and the author that the torsion subgroup $H_{k-2}(X)_{tor}$ satisfies $|H_{k-2}(X)_{tor}| \le \sqrt{k}^N$. Kalai on the other hand showed [5] that there exist $X$’s with $|H_{k-2}(X)_{tor}| \ge \sqrt{k/e}^N$. Computer experiments indicate that the ${\mathbb Q}$-acyclic sum complexes obtained by taking $|A|=k$ often have large torsion. For example, $A=\{1,z,z^{19}\} \subset C_{83}$ satisfies $|H_1(X_{A,3})| > 1.17^N$ where $N=f_2(X_{A,3})=\left( {\begin{array}{c}82\\ 2\end{array}}\right)$. Note that the base of the exponent $1.17$ is slightly bigger than the constant $\sqrt{3/e}\doteq 1.05$ in Kalai’s lower bound. In view of this, it would be interesting to determine (or estimate) the maximum torsion of sum complexes.

• 2. Theorem 1.1 characterizes the uncertainty number $u_{{\mathbb F}}(A)$ with $A \subset G=C_p$ and ${\mathbb F}$ algebraically closed, in terms of the homology of $X_{A,k}$ over ${\mathbb F}$. It would be useful to find appropriate extensions of this characterization to general finite groups $G$ and arbitrary fields ${\mathbb F}$.