Subgroup averages and character twists

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

The embedded subgroups \(K_v\cong\mathbb F_\ell^2\) of the glued group have averages \(P_v\), idempotents in the group algebra. Weighted by the matrices \(x_vx_v^{\mathsf T}\), these averages produce a rectangular matrix identity \(UT=I_m\): an \(m\times t\) matrix with a right inverse, with \(t>m\). A rectangular one-sided inverse is not yet a counterexample; to make the matrices square, the \(t\) intermediate coordinates are packed into \(m\) slots, using idempotents attached to different characters of a cyclic group. This lesson builds the group in which these characters can be changed by conjugation, and proves the identities that the packing in Packing and the reverse defect uses.

Throughout, \(\ell\), \(K\), \(\zeta\), \(\kappa=\ell^2\), \((V,\mathcal F)\), \(t\), \(m\) and the vectors \(x_v\) are as in Theorem 1.1 of The glued group and planar balance, and \(N\), \(K_v\), \(L_f\) are the glued group and its subgroups from Section 4 there. The integers \(\ell\) and \(\kappa\) are odd, so they are invertible in \(K\); in characteristic two they are even equal to \(1\), but we keep them visible.

1. Averages over the local groups

For \(v\in V\) put \[ P_v=\kappa^{-1}\sum_{u\in K_v}u\in K[N] . \]

Lemma 1.1. \(P_v^2=P_v\), and \(uP_v=P_vu=P_v\) for \(u\in K_v\).

Proof. Multiplication by \(u\in K_v\) permutes \(K_v\), so \(u\sum_{w\in K_v}w=\sum_{w\in K_v}w\), and the same on the right. Hence \(P_v^2=\kappa^{-1}\sum_{u\in K_v}uP_v=\kappa^{-1}\kappa P_v=P_v\). \(\square\)

For a line \(f\) write \(\sigma_f=\sum_{u\in L_f}u\in K[N]\). This element is the same for every point \(v\in f\), because \(L_f\) is one subgroup of \(N\).

Lemma 1.2. For every point \(v\), \[ \sum_{u\in K_v}u=1+\sum_{f\ni v}\bigl(\sigma_f-1\bigr) . \]

Proof. The \(\ell+1\) subgroups \(L_f\), \(f\ni v\), cover \(K_v\) and meet pairwise only in \(1\). So every \(u\ne1\) of \(K_v\) occurs in exactly one \(\sigma_f-1\), and \(1\) occurs only in the first term. \(\square\)

The identity takes place inside \(K_v\); nothing about intersections of different \(K_v\) is used.

2. The rectangular identity

Write \(W_v=x_vx_v^{\mathsf T}\), an \(m\times m\) matrix over \(K\), and fix an ordering of \(V\). Define matrices over \(K[N]\): \[ \begin{gathered} U\in\operatorname{Mat}_{m\times t}(K[N]),\\ \text{column }v\text{ of }U=-\kappa\,x_vP_v,\\ T\in\operatorname{Mat}_{t\times m}(K[N]),\\ \text{row }v\text{ of }T=P_v\,x_v^{\mathsf T}, \end{gathered} \] and \(P=\operatorname{diag}_{v\in V}(P_v)\in\operatorname{Mat}_t(K[N])\). The entries of \(x_v\) are scalars and commute with every element of \(K[N]\).

Proposition 2.1. \[ UP=U,\qquad PT=T,\qquad UT=I_m . \]

Proof. The first two identities follow from \(P_v^2=P_v\). For the third, \[ \begin{aligned} UT&=\sum_{v\in V}(-\kappa\,x_vP_v)(P_vx_v^{\mathsf T})\\ &=-\kappa\sum_vW_vP_v=-\sum_vW_v\sum_{u\in K_v}u . \end{aligned} \] By Lemma 1.2 this equals \[ -\sum_vW_v-\sum_v\sum_{f\ni v}W_v(\sigma_f-1) . \] In the double sum each pair \(v\in f\) occurs once, and \(\sigma_f-1\) does not depend on \(v\), so it equals \(\sum_f\bigl(\sum_{v\in f}W_v\bigr)(\sigma_f-1)\), which is zero by the line identities \(\sum_{v\in f}W_v=0\). The total identity \(\sum_vW_v=-I_m\) gives \(UT=I_m\). \(\square\)

So the \(m\times t\) matrix \(U\) has a right inverse. Over a field, such a matrix with \(t>m\) has no left inverse, and the corresponding defect \(TU-P\) is what the counterexample is made of. The regrouping of the double sum is where the glued group matters: the same element \(\sigma_f\) appears at every point of \(f\).

3. A twisted extension

Let \(z\) generate a cyclic group of order \(\ell\), and put \(H=N\times\langle z\rangle\); the element \(z\) is central in \(H\). Choose a point \(v_*\in V\) and write \(K_*=K_{v_*}\). Each \(K_v\) carries the coordinates of \(\mathbb F_\ell^2\) transported by \(\theta_v\), and \(\varphi_v:K_v\to K_*\) denotes the isomorphism that keeps the coordinates. A linear form \(\alpha:K_*\to\mathbb F_\ell\) is a homomorphism to the additive group of \(\mathbb F_\ell\); there are exactly \(\kappa=\ell^2\) of them. Exponents in \(\mathbb F_\ell\) are read in \(\mathbb Z/\ell\).

Since \(t\le m\kappa\), we can assign to each point \(v\) a slot \(s(v)\in\{1,\ldots,m\}\) and a linear form \(\alpha_v\) so that the pairs \((s(v),\alpha_v)\) are distinct. Points in the same slot then have distinct forms.

Definition 3.1. \(L\) is the group generated by \(H\) and new elements \(\tau_v\), \(v\in V\), subject to the relations of \(H\) and \[ \begin{gathered} \tau_v\,(uz^r)\,\tau_v^{-1}=\varphi_v(u)\,z^{r+\alpha_v(\varphi_v(u))}\\ (u\in K_v,\ r\in\mathbb Z/\ell). \end{gathered} \]

Lemma 3.2. For each \(v\), the map \(\psi_v:K_v\times\langle z\rangle\to K_*\times\langle z\rangle\), \(uz^r\mapsto\varphi_v(u)z^{r+\alpha_v(\varphi_v(u))}\), is a group isomorphism.

Proof. Since \(z\) is central, \(\varphi_v\) is a homomorphism and \(\alpha_v\circ\varphi_v\) is additive, \[ \begin{aligned} &\psi_v(uz^r)\psi_v(u'z^{r'})\\ &\quad=\varphi_v(uu')\,z^{r+r'+\alpha_v(\varphi_v(uu'))}\\ &\quad=\psi_v(uu'z^{r+r'}) . \end{aligned} \] The map \(wz^r\mapsto\varphi_v^{-1}(w)z^{r-\alpha_v(w)}\) is inverse to it. \(\square\)

Lemma 3.3 ([OpenAI-DF2, Section 4.2]). The natural map \(H\to L\) is injective. Consequently \(K[H]\) is a subalgebra of \(K[L]\).

Proof. We build an action of \(L\) on the set \(H\) that extends the action of \(H\) on itself by left multiplication, \(\lambda(h)x=hx\). Write \(A_v=K_v\times\langle z\rangle\) and \(A_*=K_*\times\langle z\rangle\), subgroups of order \(\ell^3\). Under left multiplication by \(A_v\), the set \(H\) splits into the right cosets \(A_vh\), each a copy of \(A_v\) with its regular action; the same holds for \(A_*\). The two sets of cosets have the same cardinality: both have \(|H|/\ell^3\) elements if \(H\) is finite, and both are countably infinite otherwise, because \(H\) is finitely generated, hence countable, and the subgroups are finite. Choose a bijection between them, that is, representatives \(h_i\) and \(k_i\) with cosets \(A_vh_i\) paired with \(A_*k_i\), and define \(F_v:H\to H\) by \[ F_v(ah_i)=\psi_v(a)\,k_i\qquad(a\in A_v). \] This is a bijection, and for \(b\in A_v\), \(F_v(bah_i)=\psi_v(b)\psi_v(a)k_i\), that is, \[ F_v\,\lambda(b)=\lambda(\psi_v(b))\,F_v . \] Send each element of \(H\) to its left multiplication and each \(\tau_v\) to \(F_v\). The relations of \(H\) hold because \(\lambda\) is a homomorphism, and the relations of Definition 3.1 hold by the display. Different stable letters occur in different relations, so the bijections \(F_v\) can be chosen independently. This defines a homomorphism from \(L\) to the permutations of \(H\) whose restriction to \(H\) is \(\lambda\), which is injective. So \(H\to L\) is injective, and so is \(K[H]\to K[L]\). \(\square\)

Definition 3.1 adds \(|V|\) generators and finitely many relations to a presentation of \(H\). The extension is a multiple HNN extension; Lemma 3.3 proves directly the embedding property that is usually derived from the normal form for such extensions.

4. Character idempotents

Taking \(u=1\) and \(r=1\) in Definition 3.1 gives \(\tau_vz\tau_v^{-1}=z\). So \(z\) is central in \(L\), and so is \[ e=\ell^{-1}\sum_{r=0}^{\ell-1}\zeta^{-r}z^r\in K[L] . \]

Lemma 4.1. \(e\) is a nonzero central idempotent and \(ez=\zeta e\).

Proof. The coefficient of \(z^a\) in \(e^2\) is \(\ell^{-2}\sum_{r+r'=a}\zeta^{-r}\zeta^{-r'}=\ell^{-2}\cdot\ell\,\zeta^{-a}\), the coefficient of \(z^a\) in \(e\). Reindexing \(ez=\ell^{-1}\sum_r\zeta^{-r}z^{r+1}\) gives \(\zeta e\). The coefficient of the identity in \(e\) is \(\ell^{-1}\ne0\), and the elements \(z^r\) are distinct in \(L\) by Lemma 3.3. \(\square\)

For a linear form \(\alpha\) on \(K_*\) put \[ Q_\alpha=\kappa^{-1}\sum_{u\in K_*}\zeta^{\alpha(u)}u\in K[L] . \]

Lemma 4.2. \(Q_\alpha Q_\beta=Q_\alpha\) if \(\alpha=\beta\), and \(Q_\alpha Q_\beta=0\) otherwise.

Proof. Write the group \(K_*\) additively in its coordinates. The coefficient of \(w\) in \(Q_\alpha Q_\beta\) is \[ \begin{aligned} &\kappa^{-2}\sum_{u\in K_*}\zeta^{\alpha(u)+\beta(w-u)}\\ &\quad=\kappa^{-2}\zeta^{\beta(w)}\sum_{u}\zeta^{(\alpha-\beta)(u)} . \end{aligned} \] If \(\alpha=\beta\), the last sum is \(\kappa\), and the coefficient is that of \(w\) in \(Q_\alpha\). Otherwise choose \(u_0\) with \(\zeta^{(\alpha-\beta)(u_0)}\ne1\); replacing \(u\) by \(u+u_0\) multiplies the sum by this number without changing it, so the sum is zero. \(\square\)

Lemma 4.3 (twisted averages). For every point \(v\), \[ \tau_v\,eP_v\,\tau_v^{-1}=e\,Q_{\alpha_v} . \]

Proof. \(e\) is central, and conjugation by \(\tau_v\) maps \(u\in K_v\) to \(\varphi_v(u)z^{\alpha_v(\varphi_v(u))}\). So, with \(w=\varphi_v(u)\), \[ \begin{aligned} \tau_veP_v\tau_v^{-1}&=\kappa^{-1}\sum_{w\in K_*}e\,z^{\alpha_v(w)}w\\ &=\kappa^{-1}\sum_{w\in K_*}\zeta^{\alpha_v(w)}e\,w=eQ_{\alpha_v}, \end{aligned} \] using \(ez^r=\zeta^re\). \(\square\)

So conjugation by \(\tau_v\) carries the average \(eP_v\) over \(K_v\) to the character idempotent \(eQ_{\alpha_v}\) over the single group \(K_*\). Two points in the same slot have distinct forms, so their twisted averages are orthogonal by Lemma 4.2. This is what lets them share a slot in Packing and the reverse defect.

5. Exercises

5.1 (easy). Show that in characteristic two \(\kappa=1\) and \(\ell=1\) in \(K\), so that \(P_v=\sum_{u\in K_v}u\) and \(e=\sum_r\zeta^{-r}z^r\).

5.2 (easy). Show that \(\sum_\alpha Q_\alpha=1\), the sum over all \(\kappa\) linear forms on \(K_*\).

5.3 (medium). Show that a slot assignment with distinct pairs \((s(v),\alpha_v)\) exists exactly when \(t\le m\kappa\), and that if two points \(v\ne w\) in the same slot had the same form, then \(eQ_{\alpha_v}\cdot eQ_{\alpha_w}=eQ_{\alpha_v}\ne0\).

5.4 (medium). Let \(\ell=3\) and \(K=\mathbb F_4\), with \(\zeta\) a root of \(\zeta^2+\zeta+1\). Write out \(e\) and check \(e^2=e\) and \(ez=\zeta e\) directly in \(K[\langle z\rangle]\).

6. Solutions

5.1. In characteristic two, an odd integer is \(1\) in \(K\), and \(\kappa=\ell^2\) and \(\ell\) are odd.

5.2. The coefficient of \(u\) in \(\sum_\alpha Q_\alpha\) is \(\kappa^{-1}\sum_\alpha\zeta^{\alpha(u)}\). For \(u=1\) (the zero vector) every term is \(1\), giving \(1\). For \(u\ne1\) there is a form \(\beta\) with \(\beta(u)\ne0\); replacing \(\alpha\) by \(\alpha+\beta\) multiplies the sum by \(\zeta^{\beta(u)}\ne1\) without changing it, so the sum is zero.

5.3. There are \(m\kappa\) pairs, so \(t\) distinct pairs exist exactly when \(t\le m\kappa\). If \(\alpha_v=\alpha_w\), then \(eQ_{\alpha_v}eQ_{\alpha_w}=eQ_{\alpha_v}^2=eQ_{\alpha_v}\), using that \(e\) is a central idempotent, and \(eQ_{\alpha_v}\ne0\) because its coefficient at the identity is \(\ell^{-1}\kappa^{-1}\ne0\) (the elements \(z^ru\) are distinct in \(H\)). The cross term would then not vanish.

5.4. Here \(\ell^{-1}=1\) and \(\zeta^{-1}=\zeta^2=\zeta+1\), so \(e=1+\zeta^2z+\zeta z^2\). Then \[ e^2=1+\zeta^4z^2+\zeta^2z^4=1+\zeta z^2+\zeta^2z , \] since squaring is additive in characteristic two, \(\zeta^3=1\) and \(z^3=1\); this is \(e\). Also \(ez=z+\zeta^2z^2+\zeta\) and \(\zeta e=\zeta+z+\zeta^2z^2\), which agree.

References