Background.

Brown’s work on Euler characteristics introduced the poset \(\mathcal S_p(G)\) of all nonidentity \(p\)-subgroups and related its Euler characteristic to the order of a Sylow subgroup (Brown 1975). Quillen proved that the inclusion \(\mathcal A_p(G)\subseteq\mathcal S_p(G)\) is a homotopy equivalence (Quillen 1978, Proposition 2.1), so the two posets carry the same homological information. He formulated the contractibility conjecture in 1978 and established it for solvable groups, groups of elementary abelian \(p\)-rank at most two, and groups of Lie type in defining characteristic (Quillen 1978, Conjecture 2.9, Proposition 2.10, Theorem 3.1, and Corollary 12.2). For solvable groups with trivial \(p\)-core, his proof produces nonzero homology in the largest possible degree. This stronger conclusion, subsequently called the Quillen dimension property, became a central input to the component reductions.

The almost-simple case was established by Aschbacher and Kleidman (1990). Passing from simple components to an arbitrary finite group requires control of their extensions and of the homology that survives in the ambient group. Aschbacher and Smith (1993) developed this approach for \(p>5\), reducing the remaining problem to the dimension property for specified elementary extensions of unitary groups. Piterman and Smith (2025, Theorems 1.1 and 1.4) extended the reduction to all odd primes. Díaz Ramos (2026, Theorems B and C) proved the required unitary result and deduced the rational homological assertion for every finite group at odd primes. We retain a direct frame proof of the required unitary dimension property. Its local statement, Proposition 23, covers the indicated almost-simple groups with elementary abelian quotient, without assuming a splitting.

At two, successive component eliminations narrowed the remaining families. Piterman and Smith (2022, Corollary 1.3) showed that every component of a minimal counterexample must admit an outer automorphism of order two induced by the ambient group. Piterman (2024, Theorem 1.1 and Corollary 1.2) established the dimension property for elementary \(2\)-extensions of exceptional Lie-type groups in odd characteristic, apart from a finite list in characteristic three. In a minimal counterexample, Piterman (2026, Theorem A) then eliminated the remaining Lie-type components in characteristic two. Combining these results with the preceding reductions gives the classical components in characteristic at least five listed in Piterman (2026, Theorem B). We use this reduction in the precise form recalled in Section 2 and prove the remaining cases by our cycle and propagation constructions.

Several of the methods behind these reductions are also central to the present proof. Segev and Webb (1994) developed exact-sequence methods relating subgroup posets to a normal subgroup and its centralizers. Piterman (2021, secs. 3–5) developed full-chain homology propagation, maximal faithful elementary extensions, and deletion of overgroups, building on the propagation arguments of Aschbacher and Smith (1993). Our propagation proof spells out the coefficient and maximality conditions used by the constructions below. The constructible cycles of Díaz Ramos (2018) and the admissible collections of Díaz Ramos and Mazza (2022) likewise organize subgroup flags by separating nonzero simplex coefficients from vanishing boundary coefficients. The frame and decoration constructions here use this chain-level viewpoint, with quantitative estimates that are uniform as the dimension grows over a fixed field.

The cycle construction.

A frame is a decomposition of a finite-dimensional vector space into lines; when a form is present, the lines are nondegenerate and mutually orthogonal. Independent scalar signs on the lines give an elementary abelian projective subgroup. Its full subgroup flags support familiar apartment chains. We impose relations between frames that differ only inside a plane, so that the boundaries of their apartment chains cancel.

The main quantitative ingredient is a lower bound on the dimension of the resulting coefficient space, uniform as the dimension of the ambient space increases while the field stays fixed. This bound gives room to impose further equations: adjoining commuting elementary automorphisms creates new boundary faces, and their cancellation is a linear condition on the frame coefficients. Counting the possible completions of each partial choice bounds the number of such conditions. The comparison produces a nonzero cycle on full subgroup flags, rather than merely a nonzero collection of parameters describing apartments.

To obtain the dimension bound, we encode merger trees of frame lines by multilinear Lie superbrackets. A Poincaré–Birkhoff–Witt argument and a bound for the actual kernel of the generator–relation complex turn elementary plane counts into the estimate of Theorem 11. The supporting subspace and its isometry type are retained throughout, so the same estimate can be used after the descent imposed by specified commuting automorphisms. In the symmetric case with field parameter at least five, the retained fraction is at least \(0.68^h\) in dimension \(h\).

The geometric and algebraic ingredients have established precedents. The unitary decomposition poset is studied by Piterman and Welker (2024); the relation between merger trees and multilinear Lie representations appears in Stanley (1982; Robinson 2004). We identify the precise coefficient space needed here and retain the supporting-subspace grading in the PBW calculation. The generator–relation series method is related to that of Golod and Shafarevich (1964); the additional kernel inequality is proved in Section 4 before the logarithmic comparison is used.

From a coefficient to homology.

The constructions may first give a cycle in an automorphism group larger than the chosen extension \(H\) of a simple component \(L\). Intersecting arbitrary subgroup flags with \(H\) could collapse them or cancel their coefficients. Lemma 19 avoids this by selecting a supported elementary group with largest intersection index, taking a residue at a fixed complement, and then intersecting the remaining flag with \(H\). On that residue the operation is injective, so one nonzero full-flag coefficient survives.

At the prime two we work in a minimal counterexample and choose the last subgroup \(A\) of maximal rank in a family of elementary subgroups generating faithful component extensions. Proper-subgroup induction supplies a nonbounding cycle in the centralizer of \(H=LA\). Its product with the constructed cycle has a residue at the chosen flag that detects the centralizer class. The deletion argument in Lemma 9 ensures that this residue still detects nonbounding in the ambient group. This proves nonvanishing without requiring \(A\) to have the largest possible elementary rank in \(G\).

For odd primes the cited reduction requires a different conclusion: nonzero homology in the largest possible degree for certain unitary extensions. We establish the needed rank formula before constructing their cycles. This distinguishes the top-degree argument from the propagation argument used at two.

Two parts of the proof take shorter or different routes through these steps. The symplectic cycles are constructed directly in \(H\) and need no restriction from a larger group. In the last branch for \(\mathop{\mathrm{PSL}}_4(5)\), an equivalent-poset construction instead retains the join of the component and centralizer homology. Section 9 states the precise conditions that lead to this branch.

Organization.

Section 2 fixes conventions and states the reduction theorems. Section 3 proves the propagation lemma. Sections 4 and 5 develop frame cycles, their dimension bounds, and the decoration and restriction procedures. Sections 6–8 establish the required linear, unitary, symplectic, and orthogonal cases. Section 9 treats the remaining small orthogonal case, realized as \(\mathop{\mathrm{PSL}}_4(5)\), and Section 10 assembles the proof of Theorem 1.

Posets, chains, and elementary rank

For a finite poset \(P\), let \(\Delta(P)\) be its order complex and write \(\widetilde H_*(P)\) for its reduced homology. We use the augmented chain complex: there is a one-dimensional group in degree \(-1\), generated by the empty simplex, and the boundary of each vertex is that generator. Consequently \(\widetilde H_{-1}(\varnothing)=\mathbb Q\). For a nonempty finite complex, nonvanishing in degree \(-1\) is impossible.

A full flag in an elementary abelian group \(E\) of rank \(r\) is a chain \[E_1<E_2<\cdots<E_r=E, \qquad \mathop{\mathrm{rk}}E_i=i.\] We also call such a flag saturated. A coefficient at a full flag always means its coefficient as an oriented simplex, with the displayed increasing order. Write \[m_p(G)=\max\{\mathop{\mathrm{rk}}E:E\leq G\text{ an elementary abelian }p\text{-subgroup}\},\] allowing \(E=1\) in this maximum. The order complex of \(\mathcal A_p(G)\) has dimension \(m_p(G)-1\), including the empty case. We say that \(G\) has the Quillen dimension property at \(p\) if \[ \widetilde H_{m_p(G)-1}(\mathcal A_p(G))\ne0. \tag{1}\] A nonzero cycle in that degree cannot be a boundary, because the chain group in the next degree is zero.

The homological implication for \((G,p)\) is \[ O_p(G)=1\quad\Longrightarrow\quad \widetilde H_*(\mathcal A_p(G))\ne0. \tag{2}\] When arguing by minimality we fix \(p\) and minimize \(|G|\) among counterexamples to (2). Thus every group of smaller order satisfies (2), with no restriction on its composition factors. If \(p\nmid|G|\), its elementary poset is empty and it is not a counterexample under our convention.

We use two standard elementary facts about finite posets. Pointwise comparable order-preserving maps induce homotopic maps on order complexes. Also, deleting a vertex whose strict upper interval is contractible preserves homotopy type: its link is the join of its lower and upper intervals and is contractible. These facts also follow from the standard fiber and gluing lemmas for posets; see Quillen (1978; Piterman and Smith 2025).

Components and automorphisms

A component of a finite group is a subnormal quasisimple subgroup. The components supplied by the reduction below are simple. Distinct components commute; distinct conjugates of a simple component intersect trivially and generate their direct product. We use these standard component facts and the usual automorphism structure of finite classical simple groups; see Gorenstein et al. (1998).

If \(L\) is nonabelian simple, a subgroup containing \(L\) and acting faithfully on \(L\) by conjugation is identified with a subgroup of \(\mathop{\mathrm{Aut}}(L)\). An elementary \(p\)-extension of \(L\) means a split extension \(L\rtimes B\) in which \(B\) is elementary abelian and induces only outer automorphisms; \(B=1\) is allowed. In particular, an elementary quotient alone is not being taken as a splitting hypothesis. In our configurations, \(H=LA\) with \(A\) elementary and \(H\) faithful on \(L\); a vector-space complement to \(A\cap L\) in \(A\) supplies the required subgroup \(B\).

Every such faithful extension \(L\le H\le\mathop{\mathrm{Aut}}(L)\) has \(O_p(H)=1\). Indeed, for \(Q=O_p(H)\) simplicity gives \(Q\cap L=1\), and normality gives \([Q,L]\le Q\cap L\). Hence \(Q\le C_H(L)=1\). Thus these extensions meet the trivial-core condition in the conditional dimension-property convention of the cited reductions.

For the linear and unitary groups, the parameter \(q\) is the order of the base field: the natural unitary space is over \(\mathbb F_{q^2}\). In the orthogonal sections, the discriminant means the determinant square class of a Gram matrix. Its relation to the Witt sign includes the usual dimension-dependent factor. These two conventions will not be interchanged.

The established reductions

This is the part of Piterman (2026, Theorem B) that we require. The theorem there also supplies an extension failing the dimension property; our argument at two will instead use maximal faithful configurations and does not need that additional conclusion.

The rest of the paper proves the new assertions needed by these two theorems. We first establish the chain and propagation methods; no dimension property at two is assumed in those methods.

Table 1 records where these local tasks are completed. At two, a supported coefficient is transferred to the ambient group by the propagation lemma; it need not occur in the largest chain degree of the component extension. The final exceptional case also has a branch using an equivalent poset instead of propagation.

Centralizers and faithful configurations

We use two standard consequences of the component theorem: distinct components commute, and every component centralizes the Fitting subgroup (Gorenstein et al. 1998). In particular, distinct conjugates of a nonabelian simple component have trivial intersection. The following consequence does not require the component to be normal.

If \(O_p(G)=1\), every nonidentity elementary abelian subgroup \(A\le L\) gives a faithful configuration: here \(LA=L\), the simplicity of \(L\) gives \(Z(L)=1\), and Lemma 4 gives the last condition. This observation will provide initial members of the families used later.

The next observation separates faithfulness of an elementary subgroup from faithfulness of the extension it generates with \(L\).

Deletion and augmented chains

For a finite poset \(X\), write \(X_{>x}\) and \(X_{<x}\) for its strict upper and lower intervals. All chains below use rational coefficients and the augmented simplicial boundary. In particular, the empty chain \([\,]\) has degree \(-1\), and the boundary of a vertex is \([\,]\).

We shall use the following elementary deletion facts. If \(X_{>x}\) is nonempty and contractible, then \[\Delta(X\setminus\{x\})\longrightarrow\Delta(X)\] is a homotopy equivalence. Indeed, the link of \(x\) is \(\Delta(X_{<x})*\Delta(X_{>x})\), which is contractible, also when \(X_{<x}\) is empty. The complex \(\Delta(X)\) is obtained by attaching the cone on this link to \(\Delta(X\setminus\{x\})\) along the link. Attaching a cone along a contractible subcomplex preserves homotopy type, by the homotopy extension property for simplicial complexes.

Also, if a finite group \(T\) has \(O_p(T)\ne1\), then \(\mathcal A_p(T)\) is contractible. Here is a verification that includes the elementary subgroup condition. Let \(\mathcal S_p(T)\) be the poset of all nonidentity \(p\)-subgroups of \(T\). Writing \(Q=O_p(T)\), the order maps \[P\longmapsto P,\qquad P\longmapsto PQ,\qquad P\longmapsto Q\] on \(\mathcal S_p(T)\) satisfy \(P\le PQ\ge Q\), so \(\mathcal S_p(T)\) is contractible. For any \(R\in\mathcal S_p(T)\), the inverse image of \((\mathcal S_p(T))_{\le R}\) under the inclusion \(\mathcal A_p(T)\hookrightarrow\mathcal S_p(T)\) is \(\mathcal A_p(R)\). Choose a central subgroup \(Z\le R\) of order \(p\). For \(E\in\mathcal A_p(R)\), the product \(EZ\) is elementary abelian, and the order maps \(E\le EZ\ge Z\) contract \(\mathcal A_p(R)\). The finite-poset fiber theorem (Quillen 1978), in the form asserting that contractible inverse images of all lower principal ideals give a homotopy equivalence, therefore applies to this inclusion. This proves the claim.

The chain operation that will detect the propagated cycle is a residue at an initial flag. Its compatibility with boundaries depends on including every possible rank in that flag.

Products of cycles

Suppose \(H,K\le G\) commute and \(H\cap K=1\). Set \(X=\mathcal A_p(H)\), \(Y=\mathcal A_p(K)\), and adjoin an identity subgroup to each, writing \(X^+=X\cup\{1\}\) and \(Y^+=Y\cup\{1\}\). The poset \[\mathcal P(H,K) =\{UV:U\in X^+,\ V\in Y^+,\ (U,V)\ne(1,1)\}\] is identified with \((X^+\times Y^+)\setminus\{(1,1)\}\). Indeed, \(UV\cap H=U\) and \(UV\cap K=V\), so both the subgroup and its inclusion relations recover the two coordinates.

The propagation lemma

We can now formulate the precise local input needed from the later classical-group constructions. The two maximality conditions in the statement will ensure that all faithful elementary overgroups can be deleted, while products with the centralizer remain.

Choosing a configuration of maximal rank

In the applications, the cycle may be supported at a different configuration from the one first selected. The following criterion records exactly what must be retained when making that replacement.

For example, fix a nonempty collection of nonidentity elementary abelian subgroups of \(L\), and take the configurations containing at least one of them. This family has the required closure property, and Lemma 4 proves its nonemptiness when \(O_p(G)=1\). If further restrictions are imposed on a family, its closure under qualifying overgroups must be verified as well. After constructing a cycle in \(H=LA\), it is sufficient to find a supported subgroup \(A'\in\mathcal F\) with \(\mathop{\mathrm{rk}}(A')=r\) and \(LA'=H\). These conditions ensure that the cycle lies in the required extension and that the maximality argument applies to its actual nonzero coefficient. No assertion that \(r\) is the full \(p\)-rank of \(H\) or of \(G\) is needed.

Frames, sign flags, and the statement

Fix one of the following structures:

1. vector spaces over \(\mathbb F_u\), with all lines allowed;

2. nondegenerate hermitian spaces over \(\mathbb F_{u^2}\), with all nondegenerate lines allowed;

3. nondegenerate symmetric spaces over \(\mathbb F_u\), where \(u\) is odd, with both square classes of nondegenerate lines allowed;

4. nondegenerate symmetric spaces over \(\mathbb F_u\), where \(u\) is odd, with only one specified square class of nondegenerate lines allowed.

All direct sums in a form space are required to be orthogonal. In the fourth case an admissible space means a space isometric to a sum of allowed lines. In the other cases every space with the indicated structure is admissible. A frame of an admissible space \(W\) is an unordered decomposition \(\mathcal D=\{L_1,\ldots,L_h\}\) into allowed lines, where \(h=\dim W\). Write \(\mathfrak F(W)\) for its set of frames. An ordered frame will also be called a word, and written \(L_1\cdots L_h\).

For any prime \(p\), attach to a frame its abstract projective sign space \[S(\mathcal D)=\mathbb F_p^{\mathcal D}/\langle(1,\ldots,1)\rangle.\] It is an elementary abelian group of rank \(h-1\). A partition \(\pi\) of \(\mathcal D\) specifies the subgroup on which the coordinates belonging to each block of \(\pi\) are equal. This subgroup has rank \(|\pi|-1\). Two partitions give distinct subgroups: two coordinates belong to the same block precisely when their difference vanishes identically on the subgroup. For coordinates in distinct blocks, assign the values \(0\) and \(1\) to those blocks to distinguish them. This argument works also for \(p=2\) and survives the quotient by simultaneous signs.

We make the identifications between different frames explicit. Let \(\mathcal D(W)\) be the poset of decompositions of \(W\) into at least two nonzero admissible subspaces. Put \(\mathcal E\leq\mathcal E'\) when \(\mathcal E'\) refines \(\mathcal E\). Merging blocks in a frame therefore gives the same vertex whenever it gives the same decomposition of \(W\). The flags from two blocks through a full frame have \(h-1\) vertices and dimension \(h-2\). At a fixed frame these are exactly the equal-sign flags just described.

In the hermitian case, this is the proper orthogonal-decomposition poset of Piterman and Welker (2024, sec. 2.1 and 7.2, arXiv version 2), with the order reversed. Their Proposition 7.3(i) relates eigenspace decompositions to elementary subgroups and credits the corresponding construction of Aschbacher and Smith (1993, 514–15). The partial-frame complex is a different object. Here the full coefficient space, and its dimension for each isometry type, are the objects required for the subsequent chain construction.

For an ordered frame \(w=L_1\cdots L_h\), let \(r=h-1\) and set \(\lambda_i=x_i-x_{i+1}\) for \(1\leq i\leq r\). These functionals form a basis of \(S(\mathcal D)^*\). For \(\sigma\in\mathfrak S_r\), put \[K_j(w,\sigma)=\bigcap_{a=1}^j\ker\lambda_{\sigma(a)} \quad(0\leq j\leq r),\qquad K_0(w,\sigma)=S(\mathcal D).\] The subgroups \(K_j\) are also vertices of \(\mathcal D(W)\) when \(j<r\). Define the coned apartment chain \[ a_w=\sum_{\sigma\in\mathfrak S_r}\mathop{\mathrm{sgn}}(\sigma) [K_{r-1}(w,\sigma)<\cdots<K_1(w,\sigma)<S(\mathcal D)] \in\widetilde C_{h-2}(\mathcal D(W);\mathbb Q). \tag{8}\] In dimension one the same convention gives the empty simplex in degree \(-1\). This convention will occur only when describing a boundary of a two-dimensional frame.

Let \(V_{\mathcal D}\) be the span of the actual flag-coefficient vectors of \(a_w\) as \(w\) runs over the orderings of \(\mathcal D\). Thus two linear combinations with identical coefficients define the same member of \(V_{\mathcal D}\). We will prove that \[ \dim V_{\mathcal D}=(h-1)!. \tag{9}\] An assignment is a member of \(\bigoplus_{\mathcal D\in\mathfrak F(W)} V_{\mathcal D}\). Summing its components embeds this direct sum into the chain space, since a full flag determines its top frame.

Here is the subspace of assignments that we use. Let \(f\) be a \(\mathbb Q\)-valued function on ordered frames. For every ordered decomposition \[L_1\oplus\cdots\oplus L_{i-1}\oplus Q\oplus L_{i+2}\oplus\cdots\oplus L_h=W, \qquad \dim Q=2,\] with the other summands allowed lines, require \[ \sum_{(A,B):\,Q=A\oplus B} f(L_1\cdots L_{i-1}AB L_{i+2}\cdots L_h)=0. \tag{10}\] The sum is over all ordered allowed frames of \(Q\), with orthogonality understood in the form cases. Define \(\mathcal Z(W)\) to be the image of all these functions under \(f\mapsto\sum_w f(w)a_w\), viewed as an assignment of actual coefficients. The following theorem is uniform in the isometry type of \(W\).

The last qualification states precisely what is required to carry the abstract construction into a group. Additional identifications of boundary faces cause no difficulty, because their coefficients already sum to zero. An identification of full flags could decrease the dimension; the applications will verify that this does not occur.

Boundary cancellation and coefficient detection

We first address the cycle assertion, then turn to coefficient detection.

The distinction between simplex coefficients and their boundary sums is central to the constructible cycles of Díaz Ramos (2018, Proposition 4.2 and Definition 4.3, arXiv version 1) and to Díaz Ramos and Mazza (2022, Proposition 3.2 and Theorems 3.3–3.5). The plane relations below impose this distinction on frame flags.

For a word \(w\) and a gap \(i\), denote by \(w/i\) the ordered decomposition obtained by replacing \(L_i,L_{i+1}\) by their sum. The definition (8) applies to this shorter list of blocks as well. Deleting an interior vertex of an apartment flag has two contributions whose permutations differ by transposing the two equations on either side of that deletion. Their signs are opposite. This includes deletion of the bottom vertex: in that case one transposes the last imposed equation and the one omitted equation. Only deletion of the cone vertex \(S(\mathcal D)\) remains. Grouping its contributions according to the first equation imposed gives the exact formula \[ \partial a_w=(-1)^{r-1}\sum_{i=1}^{r}(-1)^{i-1}a_{w/i}. \tag{13}\] Indeed, moving gap \(i\) to the first position contributes \((-1)^{i-1}\), and deletion of the final vertex of an increasing flag contributes \((-1)^{r-1}\). On \(\ker\lambda_i\), the remaining equations are the adjacent-difference equations of \(w/i\) in their original order. Formula (10) now cancels each term of (13) after summing with the weights \(f(w)\). If different coarsening descriptions give the same face, this only adds several zero sums. Consequently every member of \(\mathcal Z(W)\) is a cycle, including the augmented degree-zero assertion when \(h=2\).

To determine the rank of the coefficient map, regard a line as a formal letter of odd parity. If homogeneous words or brackets \(A,B\) have parities \(|A|,|B|\in\mathbb Z/2\), their supercommutator is \[ [A,B]=AB-(-1)^{|A||B|}BA. \tag{14}\] In particular \([L,M]=LM+ML\) for two line letters. A binary merger tree on a frame defines an iterated bracket by choosing an order for the two children of each internal vertex and using (14). The parity of a subtree is its number of leaves modulo two.

This tree description is closely related to the multilinear super-Lie model of Robinson (2004, Proposition 3.4, Corollary 3.5, and Section 3.6); see also Stanley (1982, Theorem 7.3) for the partition representation. We prove the coefficient formula with our coning and orientation conventions. In particular, the chains here have degree \(h-2\), whereas the proper partition complex with both endpoints removed has degree \(h-3\).

Figure 1 illustrates a merger flag and its coefficient bracket for four lines.

The remaining task is quantitative. Plane relations connect different frames, so the local dimension \((h-1)!\) does not by itself give a lower bound on the space of cycles. We encode all these relations at once, while retaining enough information to distinguish every supporting subspace.

The supporting-subspace grading and PBW

Fix a finite ambient space \(W_0\) of one of the permitted structures. Form a commutative monoid with elements \(0\), all its nonzero admissible subspaces, and an additional element \(\bot\). For admissible subspaces \(A,B\), their sum in this monoid is the subspace \(A\oplus B\) if they form an allowed direct decomposition, and is \(\bot\) otherwise. Set \(\bot+A=\bot\), with \(0\) an identity. This operation is associative: an iterated product avoids \(\bot\) precisely when all its nonzero subspaces form a direct decomposition, orthogonal when required. Once directness or orthogonality fails, adjoining another summand cannot restore it. We call degrees other than \(\bot\) good degrees.

Let \(V\) be the rational vector space with one odd generator \(e_L\) for each allowed line \(L\leq W_0\), and give \(e_L\) degree \(L\). Also retain the usual word-length grading. In the tensor algebra \(T(V)\), a word has good degree \(X\) exactly when its letters are an ordered frame of \(X\); its length is then \(\dim X\). For each admissible plane \(Q\leq W_0\) define \[ r_Q=\sum_{(A,B):\,Q=A\oplus B}e_Ae_B =\sum_{\{A,B\}:\,Q=A\oplus B}[e_A,e_B]. \tag{16}\] The second sum is over unordered frames and is independent of their ordering, because the two generators are odd. Each \(r_Q\) is nonzero, even, and homogeneous of good degree \(Q\) and length \(2\).

Let \(\mathfrak L\) be the free Lie superalgebra on \(V\), and let \[P=\mathfrak L/(r_Q:Q\text{ an admissible plane})_{\mathop{\mathrm{Lie}}},\qquad U=T(V)/(r_Q:Q\text{ an admissible plane})_{\mathrm{ass}}.\] The subscripts mean the Lie ideal and the two-sided associative ideal, respectively. The universal properties show that \(U\) is the enveloping algebra of \(P\): an associative-algebra map out of either presentation is exactly a map on the line generators annihilating every \(r_Q\).

We use the characteristic-zero super PBW theorem (Scheunert 1979); the positive word-length graded form over \(\mathbb Q\) is also given by Milnor and Moore (1965, discussion following Proposition 5.2 and Theorem 5.16). We use it in the following explicit form. For a homogeneous ordered basis of a Lie superalgebra, its enveloping algebra has as a basis the ordered products of basis elements, with each odd basis element used at most once. Its associated graded algebra for the number-of-factors filtration is \(\mathop{\mathrm{Sym}}(P_{\mathrm{even}})\otimes\bigwedge(P_{\mathrm{odd}})\). The canonical map from the Lie superalgebra to its enveloping algebra is consequently injective. Recall why the parity qualification is necessary: the relation for homogeneous elements is \(xy-(-1)^{|x||y|}yx=[x,y]\), and for odd \(x\) it becomes \(2x^2=[x,x]\). Ordering adjacent factors and replacing odd squares therefore gives the stated normal forms. The overlap of three successive factors reduces to \[[x,[y,z]]-[[x,y],z]-(-1)^{|x||y|}[y,[x,z]]=0;\] the overlaps with repeated odd factors are its specializations, using \(2x^2=[x,x]\) and \([x,[x,x]]=0\). Thus these reductions are consistent by super Jacobi. Induction on factor length and then on the number of out-of-order pairs gives the PBW normal forms. This description also shows that PBW preserves any additional grading for which the bracket is homogeneous, including the supporting-subspace and length gradings above.

Write \(P(X)\) and \(U(X)\) for the good components on a subspace \(X\). These are finite-dimensional and intrinsic to \(X\). Indeed, a product of good degree \(X\) uses only supports contained in \(X\); an expression of bad degree can never contribute to it. Thus neither generators nor relations supported outside \(X\) affect that component. We may therefore form these spaces in any ambient space containing \(X\).

In a good PBW monomial, the supports of its nonzero factors are independent admissible subspaces. In particular no basis factor can occur twice, even when it is even. The distinction between symmetric and exterior multiplication changes signs but does not change the number of such monomials. We have proved a vector-space decomposition in good degree: \[ U(X)\ \cong\ \bigoplus_{k\geq0}\ \bigoplus_{X=X_1\oplus\cdots\oplus X_k\,/\,\mathfrak S_k} P(X_1)\otimes\cdots\otimes P(X_k), \tag{17}\] where all \(X_i\) are nonzero, and the tensor factors on the right may be ordered by any fixed order of their supports. The term \(k=0\) occurs only for \(X=0\).

Normalized series and the relation kernel

We now use formal series to count the good components in all dimensions. Let \(G_X\) be the full linear group of \(X\) in the no-form case and the full isometry group in the form cases, with \(|G_0|=1\). For any isometry-invariant collection \(M(X)\) of finite-dimensional vector spaces, define \[\mathsf M=\sum_{[X]}\frac{\dim M(X)}{|G_X|}\,[X].\] Here \([X]\) is a formal symbol for its isomorphism or isometry type, and \([X][Y]=[X\oplus Y]\). Series are completed by dimension, so each coefficient in a product is a finite sum. The coefficientwise order always refers to this type basis. If \[(M*N)(W)=\bigoplus_{W=A\oplus B}M(A)\otimes N(B),\] then its series is \(\mathsf M\mathsf N\). To verify this, for specified types \(A,B\) the group \(G_W\) acts transitively on the ordered decompositions of these types, with stabilizer \(G_A\times G_B\). Transitivity follows by taking the direct sum of chosen isomorphisms or isometries on the summands. The number of decompositions is \(|G_W|/(|G_A||G_B|)\), which is exactly the normalization asserted.

Write \(\mathsf P,\mathsf U\) for the series of \(P,U\). Dividing the ordered versions of (17) by \(k!\) gives \[ \mathsf U=\exp(\mathsf P). \tag{18}\] There is no stabilizer correction beyond \(k!\): the actual nonzero summands of a direct decomposition are distinct, even if some of their isometry types agree. An ordering of a decomposition therefore has exactly \(k!\) possibilities.

Let \(R\) be the vector space with one formal even basis element \(\mathbf r_Q\) for each admissible plane \(Q\), of degree \(Q\). Denote the series of \(V\) and \(R\) by \(v\) and \(w\), respectively. Split each relation just before its last letter to define the left \(U\)-module map \[d_2:U\otimes R\longrightarrow U\otimes V, \qquad d_2(u\otimes\mathbf r_Q) =\sum_{(A,B):Q=A\oplus B}u e_A\otimes e_B.\] Let \(d_1:U\otimes V\to U\) be multiplication and put \(C=\ker d_2\). We have an exact sequence \[ 0\longrightarrow C\longrightarrow U\otimes R \xrightarrow{d_2}U\otimes V\xrightarrow{d_1}U \longrightarrow\mathbb Q\longrightarrow0. \tag{19}\] Here is the only exactness assertion needing verification. Write \(T=T(V)\) and let \(I\) be the two-sided ideal generated by the \(r_Q\). Splitting the last letter identifies \(T_+\) with \(T\otimes V\) and \(U\otimes V\) with \(T_+/(IV)\). Hence the kernel of \(d_1\) identifies with \(I/(IV)\). Every element of \(I\) is a sum of terms \(a r_Q b\). When \(b\) has positive length this term lies in \(IV\), by separating the last letter of \(b\). The terms with \(b=1\) are exactly the images of \(d_2\). This proves exactness; \(d_1\) plainly has image the augmentation ideal. These maps preserve the supporting-subspace grading, so exactness holds in every good component.

Taking dimensions of (19) and normalizing as above gives, with \(\mathsf C\) the series of the good components of \(C\), \[ \mathsf U(1-v+w)=1+\mathsf C. \tag{20}\] This identity uses no assumption that the relations form a regular sequence. The possible dependencies between relations are recorded by the actual kernel \(C\).

Define the degree operator \(D\) on series by \(D[X]=(\dim X)[X]\). It is a derivation because dimensions add under direct sum. The next inequality is the step that turns the associative presentation into a bound on its Lie part.

The generator–relation series comparison has its classical antecedent in Golod and Shafarevich (1964, sec. 2, Lemmas 2–3). For the present coefficientwise bound, the actual kernel \(C\) must remain in the calculation. The following marked-factor argument is what permits the later logarithmic comparison.

We now return to the dimension bound. The remaining steps are a series comparison and the evaluation of its constants.

Set \(J=-\log(1-v+w)\), a well-defined formal series with zero constant term. From (18) and (20), formal differentiation gives \[(1+\mathsf C)D\mathsf P=(1+\mathsf C)DJ+D\mathsf C.\] Subtract \(\mathsf C D\mathsf P\) and apply Lemma 14. If \(DJ\) is coefficientwise nonnegative, then \[ D\mathsf P =(1+\mathsf C)DJ+ \bigl(D\mathsf C-\mathsf C D\mathsf P\bigr) \ \geq\ (1+\mathsf C)DJ\ \geq\ DJ. \tag{22}\] No division by a series with unknown coefficient signs is used here. It remains to compute \(DJ\) and compare its coefficients with the number of frames.

Plane counts and constants

Write \(\operatorname{coeff}_{[W]}(B)\) for the coefficient of the type \([W]\) in a series \(B\). In dimension \(h\), the frame count gives \[ \operatorname{coeff}_{[W]}(v^h) =\frac{h!\,|\mathfrak F(W)|}{|G_W|}. \tag{23}\] Thus, if the degree-\(h\) part of \(DJ\) is at least \(F_hv^h\) coefficientwise, (22) yields \[\dim P(W)\geq\frac{|G_W|}{h}\operatorname{coeff}_{[W]}(DJ) \geq F_h\,|\mathfrak F(W)|(h-1)!.\] Lemma 13 will then give the theorem.

First suppose that only one line type is allowed and that there is one resulting admissible type in each dimension. This includes the no-form and hermitian cases. Use a variable \(t\) for the line type. Then \(v=\alpha t\), \(w=\beta t^2\), where \(\alpha=|G_{\text{line}}|^{-1}\) and \(\beta=|G_{\text{plane}}|^{-1}\). The quotient \[\rho=\frac{\beta}{\alpha^2}\] is the reciprocal of the number of ordered allowed frames of an admissible plane, by the decomposition count already proved. When \(\rho\leq1/4\), set \(y=(1+\sqrt{1-4\rho})/2\). Factoring \(1-v+w=(1-yv)(1-(1-y)v)\) gives \[ DJ=\sum_{h\geq1}\bigl(y^h+(1-y)^h\bigr)v^h. \tag{24}\] All its coefficients are positive, including the double-root case \(y=1/2\).

In a two-dimensional space over \(\mathbb F_u\) without a form there are \(u+1\) possible first lines, and \(u\) complementary second lines, so there are \(u(u+1)\) ordered frames. In a hermitian plane over \(\mathbb F_{u^2}\) there are \(u^2+1\) lines, of which \(u+1\) are isotropic. For completeness, in an orthonormal basis an isotropic line has a representative \((a,1)\) with \(a^{u+1}=-1\); the norm map has \(u+1\) elements in each nonzero fiber. The other \(u(u-1)\) lines are nondegenerate, each with its unique orthogonal complement. These are therefore the respective ordered frame counts. For \(u\geq5\) both reciprocal counts are at most \(1/20\). It follows that \(y\geq(1+\sqrt{4/5})/2>17/25\), proving the asserted uniform bound in these two cases.

We record also the orthogonal plane count with its dependence on determinant. A plane with diagonal form \(\operatorname{diag}(a,b)\) has \[ |G_Q|=2\bigl(u-\chi(-ab)\bigr). \tag{25}\] Indeed, the number of vectors of norm \(a\) is the number of solutions of \(x^2+(b/a)y^2=1\), namely \(u-\chi(-ab)\). One way to verify this count is to use \(\sum_{y\in\mathbb F_u}\chi(y^2-c)=-1\) for \(c\ne0\). The latter identity follows by counting \((y,z)\) with \(y^2-z^2=c\): the invertible change of variables \((y,z)\mapsto(y-z,y+z)\) identifies these solutions with the \(u-1\) pairs of nonzero elements having product \(c\). For each first vector of norm \(a\), its orthogonal line has the square class of \(b\), since the determinant of the form changes by a square under a change of basis. There are exactly two vectors of norm \(b\) on that line. Ordered orthogonal bases with norms \((a,b)\) are acted on freely and transitively by \(G_Q\), giving (25).

For the same fixed norm type on both lines, \(ab\) is a square and an ordered line-frame stabilizer has order \(2\cdot2=4\). Hence the number of ordered allowed frames is \[\frac{u-\chi(-1)}2.\] This is at least \(4\) for every odd prime power \(u\geq7\): it is \(4\) at \(u=7\) and is at least \((u-1)/2\geq4\) for \(u\geq9\). Formula (24) therefore applies with \(\rho=2/(u-\chi(-1))\leq1/4\), proving the one-type assertion.

Finally allow both symmetric line types. The determinant square class is multiplicative under orthogonal direct sum. Write \(z\) for the nonsquare class, with \(z^2=1\), and let \(t\) record dimension. Every nondegenerate symmetric space of positive dimension over an odd finite field is determined by its dimension and determinant square class. Each line has isometry group of order \(2\). Formula (25) consequently gives \[ v=\frac{1+z}{2}\,t,\qquad w=(a+bz)t^2,\qquad \{a,b\}=\left\{\frac1{2(u-1)},\frac1{2(u+1)}\right\}. \tag{26}\] Which of \(a,b\) occurs at determinant square depends only on \(\chi(-1)\); the bound will be valid for either order.

Evaluation at \(z=1\) gives \(v=t\) and \(w=u t^2/(u^2-1)\). Since \(u/(u^2-1)<1/4\) for \(u\geq5\), the two-root computation gives \[[t^h]DJ\big|_{z=1}=A_h:=x^h+(1-x)^h, \qquad x=\frac{1+\sqrt{1-4u/(u^2-1)}}2.\] At \(z=-1\) we have \(v=0\) and \(w=\varepsilon t^2/(u^2-1)\) for \(\varepsilon\in\{1,-1\}\). Expanding \(-\log(1+w)\) therefore gives \[B_h:=[t^h]DJ\big|_{z=-1}=0\quad(h\text{ odd}), \qquad |B_h|=2(u^2-1)^{-h/2}\quad(h\text{ even}).\] The two actual determinant coefficients of \([t^h]DJ\) are \((A_h+B_h)/2\) and \((A_h-B_h)/2\). On the other hand \(v^h=(1+z)t^h/2\) for every \(h\geq1\). Thus each determinant coefficient of \(DJ\) is at least the corresponding coefficient of \(F_hv^h\), where \(F_h\) is (12). In particular this calculation keeps both types separately throughout the comparison.

We finish by proving positivity and the uniform numerical assertions. The function \(u/(u^2-1)\) decreases for \(u>1\), so \(x\) increases with \(u\). For each integer \(h\geq1\), the function \(s^h+(1-s)^h\) is nondecreasing on \(1/2\leq s\leq1\); its derivative is \(h(s^{h-1}-(1-s)^{h-1})\geq0\). The subtracted even-degree correction also decreases with \(u\). It suffices, therefore, to use \(u=5\). Here \[x=\frac{1+1/\sqrt6}{2}>\frac7{10}.\] For odd \(h\), \(F_h\geq x^h>(17/25)^h\). At \(h=2\) a direct calculation gives \(F_2=1/2>(17/25)^2\). For even \(h\geq4\), \[\frac{2\cdot24^{-h/2}}{(7/10)^h} =2\left(\frac{25}{294}\right)^{h/2} \leq2\left(\frac{25}{294}\right)^2<\frac1{50}.\] Consequently \[F_h>\frac{49}{50}\left(\frac7{10}\right)^h >\left(\frac{17}{25}\right)^h.\] For the last inequality, divide by \((7/10)^h\) and use \((34/35)^h\leq(34/35)^4<49/50\). Finally \[F_3=x^3+(1-x)^3 =1-\frac{3u}{u^2-1}\geq1-\frac{15}{24}=\frac38.\] These inequalities also prove that every coefficient of \(DJ\) used above is nonnegative. Applying (22), (23), and Lemma 13 proves Theorem 11 in all four cases.

Split chains and their boundary

Write \(\widetilde C_*(P)\) for the augmented rational chain complex of a finite poset \(P\), with increasing order as the orientation of every simplex. Let \(S\) and \(D\) be commuting elementary abelian \(p\)-subgroups of a finite group \(M\), and suppose \(S\cap D=1\). Given flags \[u=[U_1<\cdots<U_a],\qquad v=[V_1<\cdots<V_b]\] in \(S\) and \(D\), respectively, put \(U_0=V_0=1\). An \((a,b)\)-shuffle is a path \(\gamma\) from \((0,0)\) to \((a,b)\) whose steps are \((1,0)\) or \((0,1)\). Write \((i_k,j_k)\) for its vertex after \(k\) steps, and let \(\nu(\gamma)\) count pairs consisting of a \(D\)-step followed later by an \(S\)-step. Define \[ u\boxtimes v =\sum_{\gamma}(-1)^{\nu(\gamma)} [U_{i_1}V_{j_1}<\cdots<U_{i_{a+b}}V_{j_{a+b}}]. \tag{27}\] Each displayed inclusion is strict, because the two factors have trivial intersection. The origin is omitted. Empty flags are allowed: if \(a=0\) or \(b=0\), there is one path and this construction is the identity on the other flag. Extend (27) bilinearly to chains. A chain of degree \(a-1\) and a chain of degree \(b-1\) give a chain of degree \(a+b-1\).

This is the product shuffle of Lemma 8, specialized to elementary groups. That lemma gives \[ \partial(u\boxtimes v) =(\partial u)\boxtimes v+(-1)^a u\boxtimes(\partial v). \tag{28}\] Here and below the empty simplex has degree \(-1\) and zero boundary. Iterating the construction gives the sum over paths with any fixed number of kinds of steps. Its sign counts inversions between the ordered kinds, so the iteration is associative.

Suppose that \(D=D_1\times\cdots\times D_t\), where the \(D_i\) are labelled subgroups of order \(p\). Its coordinate apartment, coned at \(D\), is the chain \[ a(D_1,\ldots,D_t)= \sum_{\pi\in\mathfrak S_t}\mathop{\mathrm{sgn}}(\pi) [D_{\pi(1)}<D_{\pi(1)}D_{\pi(2)}<\cdots<D]. \tag{29}\] Set \(a(\varnothing)=[\,]\). Interior faces cancel in pairs, and the remaining faces give the exact formula \[ \partial a(D_1,\ldots,D_t) =\sum_{i=1}^t(-1)^{i-1} a(D_1,\ldots,\widehat D_i,\ldots,D_t). \tag{30}\] For example, when \(t=2\) the boundary is \([D_2]-[D_1]\). If \(x\) is a coned apartment in \(S\), then \(x\boxtimes a(D_1,\ldots,D_t)\) is the apartment for the union of its basis with the decoration basis, up to the fixed choice of orientation of \(x\). More generally, (27) defines this operation on the coefficients of any chain \(x\), without choosing an expression of \(x\) as a sum of apartments.

The coefficient spaces to be counted

Fix an integer \(h\geq2\), and put \[r=h-1,\qquad q_h=(h-1)!.\] A frame \(\mathcal F\) with \(h\) summands has a projective sign group \(S_{\mathcal F}\) of rank \(r\). Denote by \(V_{\mathcal F}\) the space of actual coned frame-chain coefficients at \(S_{\mathcal F}\) from Section 4. Thus \(V_{\mathcal F}\) is a subspace of the chain group on full flags ending at \(S_{\mathcal F}\), every member has no boundary face retaining \(S_{\mathcal F}\), and \[ \dim V_{\mathcal F}\leq q_h. \tag{31}\] In the abstract frame model equality holds. The inequality is the only local bound needed here.

The precise output we use from Theorem 11 is the following. For the set \(\mathfrak F\) of all allowed frames of one of the spaces specified there, it provides a subspace \[ Z\ \subseteq\ \bigoplus_{\mathcal F\in\mathfrak F}V_{\mathcal F}, \qquad \dim Z\geq F\,|\mathfrak F|q_h, \qquad \sum_{\mathcal F\in\mathfrak F}\partial x_{\mathcal F}=0 \quad(x\in Z). \tag{32}\] The last equality uses the coarsening identifications of that theorem. Whenever the frame model is realized by elementary subgroups, we require these identifications to be valid subgroup identifications. A disjoint union of compatible spaces is also allowed, provided the same lower bound \(F\) holds for every space. The direct sum of their spaces \(Z\) then satisfies the same estimate. For example, the all-type estimates with field parameter at least five permit \(F=(17/25)^h\); for symmetric \(h=3\) they permit \(F=3/8\).

Equation (32) counts the image on coned flag coefficients, not the dimension of the word functions that produce it. It still retains the frame labels in the direct sum. If different full data can give the same subgroup flag after decorations are adjoined, their possible coincidence must be checked separately. The next lemma states this last requirement as an injectivity hypothesis.

Here is a useful way to check the injectivity assumption in Lemma 16. Suppose a full group \(E\) determines \(S_{\mathcal F}\) and its frame \(\mathcal F\). Suppose also that \(D=D_1\cdots D_t\) determines the labelled axes, for instance because they lift a fixed ordered basis of a quotient on which \(D\) maps isomorphically. Inspect a split full flag which first grows through \(D_1<D_1D_2<\cdots<D\) and then through \[DU_1<\cdots<DU_r=E\] for a sign flag \([U_1<\cdots<U_r=S_{\mathcal F}]\). If another datum contributes to this flag, its full group gives the same sign group and frame. At the vertex \(D\), its split form is \(U V\) with \(U\leq S_{\mathcal F}\) and \(V\) contained in its decoration complement. Since \(D\cap S_{\mathcal F}=1\), necessarily \(U=1\). The rank of \(D\) then forces the whole alternative complement to equal \(D\). The labelled axes agree by hypothesis. The coefficient of the inspected flag is therefore the original sign coefficient, up to the nonzero sign of the unique shuffle taking all decoration steps first. These flags detect every coordinate in the direct sum. The applications will verify the recovery of \(S_{\mathcal F}\), \(\mathcal F\), and the labelled axes in their particular groups.

A rank-two decoration with three lines

For the symplectic construction the additional elementary group is \(T\cong C_2^2\). Its three order-two subgroups are not supplied with a preferred pair of coordinate axes. Instead we use its full two-dimensional coefficient space \[ Q_T= \left\{\sum_{\ell<T,\ |\ell|=2}c_\ell[\ell<T]: \sum_\ell c_\ell=0\right\}. \tag{36}\] For \(q=\sum c_\ell[\ell<T]\in Q_T\), direct calculation gives \[ \partial q=-\sum_\ell c_\ell[\ell]. \tag{37}\] The two-dimensional space in (36) is spanned by differences of its three flags, equivalently by the apartments from pairs of independent lines. The name in the next lemma refers to its later realization by a projective quaternion group; the chain statement needs only \(T\cong C_2^2\).

The use of two parameters at \(T\) is essential to this count. There are three retained-line boundary conditions but only one sign coefficient vector at each such partial datum. Dividing the resulting loss by the initial factor two gives \(3/(2k_T)\). To verify the detection hypothesis, the applications inspect split flags through \(T\) and each of its three lines; when \(B\) is present they begin at \(B\) and then pass through \(B\ell<BT\). These flags recover the coefficients in \(Q_T\) as well as the sign-chain coefficients, once the full data are recovered from the corresponding subgroups.

Opposite chambers and a nonzero restricted coefficient

We now prove the restriction statement. Its input is a homogeneous cycle on full flags of one fixed rank. This condition will let us take a local building cycle at a specified top group. The following elementary building fact supplies a complement on a supported flag.

Its conclusion has the classical apartment-basis interpretation of Solomon (1969, Theorem 1) and Björner (1984, Proposition 4.5 and equation (4.6)): coefficients on opposite chambers detect a top cycle. We retain a direct panel argument, including the augmented low-rank cases, before proving the restriction statement that uses the resulting complement.

The maximal-index hypothesis is used only in (45). A constant index over all supported top groups is therefore sufficient, but is not required. Neither \(m\) nor \(m-c\) needs to be the maximum elementary rank of its ambient group unless the final top-degree conclusion is invoked. In the applications at two, the nonzero saturated coefficient supplied by Lemma 19 is instead used in Lemma 9.

Figure 2 illustrates the fixed-complement step when \(0<c<m\). Its middle and bottom rows are two descriptions of the same suffix flag, which is why the intersection operation preserves the chosen coefficient.

Weights, signs, and projective commutators

We first record two elementary observations about weights. If \(\Lambda\) is a finite subset of an affine space over \(\mathbb F_p\), and translation by a nonzero vector \(a\) preserves \(\Lambda\), then \[ \dim\operatorname{aff}(\Lambda)\le \frac{|\Lambda|}{p}. \tag{47}\] Indeed, the translation has orbits of length \(p\). If there are \(t\) orbits, choose representatives \(\lambda_1,\ldots,\lambda_t\); the affine span is generated by \(a\) and the \(t-1\) differences \(\lambda_i-\lambda_1\).

For the second observation, let an elementary abelian \(p\)-group \(D\) act faithfully projectively on a vector space in characteristic different from \(p\). Suppose that its matrix representatives commute. Over an algebraic closure, rescale representatives of a basis of \(D\) to have order \(p\). They give a linear representation of \(D\) with a set of characters \(\Lambda\subseteq D^*\). The differences of these characters span \(D^*\): an element annihilated by every difference acts by a scalar and is therefore trivial in \(D\). Thus \[ \dim\operatorname{aff}(\Lambda)=\mathop{\mathrm{rk}}D. \tag{48}\] A transformation centralizing the projective action, and acting trivially on the \(p\)th roots of unity, permutes these characters by a common translation. To see this, conjugating each chosen representative changes it by a scalar; the scalars, which are \(p\)th roots of unity after normalization, form a character of \(D\).

Here is the complementary estimate when the representatives do not commute. It will also be useful in the exceptional case of Section 9.

Let \(L=\mathop{\mathrm{PSL}}_n(q)\) or \(\mathop{\mathrm{PSU}}_n(q)\), and put \(M=\mathop{\mathrm{PGL}}_n(q)\) or \(\mathop{\mathrm{PGU}}_n(q)\), respectively. In the unitary notation, \(\mathop{\mathrm{GU}}_n(q)\) denotes the group of isometries of a nondegenerate hermitian form over \(\mathbb F_{q^2}\); its projective image also contains every projective similitude, since the norm map onto \(\mathbb F_q^\times\) is surjective. In the linear case set \(C=\mathbb F_q^\times\), and in the unitary case set \(C=\mu_{q+1}\subseteq\mathbb F_{q^2}^\times\). Determinant induces \[ M/L\cong C/C^n. \tag{50}\] In particular this quotient is cyclic.

A line frame \(\mathcal D\) is an unordered direct-sum decomposition into \(n\) lines; in the unitary case the lines are required to be orthogonal and nondegenerate. For a prime \(p\mid |C|\), define \(S(\mathcal D)\) to be the projective images of the diagonal operators with entries in \(\mu_p\). Thus \(S(\mathcal D)\cong\mathbb F_p^n/\langle(1,\ldots,1)\rangle\) has rank \(n-1\). For any fixed \(L\le H\le\mathop{\mathrm{Aut}}(L)\), \[ h_0:=\mathop{\mathrm{rk}}(S(\mathcal D)\cap H)\in\{n-2,n-1\} \tag{51}\] is independent of \(\mathcal D\). Indeed all these frames are conjugate under \(M\), and their images in the abelian group \(M/L\) are consequently the same subgroup, of order at most \(p\).

We use the standard automorphism description for these groups in dimension \(n\ge5\); see Steinberg (1960, sec. 2 and statements 3.2–3.6) and Gorenstein et al. (1998). If \(q=r^f\), the quotient \(\mathop{\mathrm{Aut}}(\mathop{\mathrm{PSL}}_n(q))/\mathop{\mathrm{PGL}}_n(q)\) is the product of the field group of order \(f\) and the diagram group of order two. The diagram operation is \(\gamma:g\mapsto(g^{-1})^{\mathsf t}\). The quotient \(\mathop{\mathrm{Aut}}(\mathop{\mathrm{PSU}}_n(q))/\mathop{\mathrm{PGU}}_n(q)\) is the cyclic field group of order \(2f\) on \(\mathbb F_{q^2}\). Semilinear isometries represent the latter automorphisms. Thus an elementary subgroup has at most two external directions at two in the linear case, at most one in the unitary case, and at most one at odd primes in either case.

A correlation here is a linear or semilinear identification with the dual space, acting on the group through inverse transpose. A dual frame consists of the lines of linear functionals vanishing on all but one summand of the original frame. A frame is matched with its dual frame when each of its lines is matched with the corresponding dual line under this identification.

Compatible frames for involutory decorations

We next identify the frame spaces to which Theorem 11 will be applied. A decoration is an involution outside \(M\). A frame is compatible with it if every line is stable, with matching to the dual frame in the correlation case. For several decorations we require them to commute and require compatibility with each of them. In what follows we consider only tuples having at least one compatible frame.

We use the following fixed-vector form of Galois descent. Let \(K/k\) be a cyclic finite-field extension of degree \(d\), with generator \(\sigma\), and let \(B\) be an additive map on a finite-dimensional \(K\)-space \(V\) satisfying \(B(av)=\sigma(a)B(v)\) and \(B^d=\mathop{\mathrm{id}}\). The powers of \(B\) define a semilinear Galois action, and Milne (2024, Proposition 5.5) gives \[K\otimes_k V^B\longrightarrow V,\qquad a\otimes v\longmapsto av,\] as an isomorphism. In particular \(\dim_k V^B=\dim_K V\). Every \(B\)-stable \(K\)-subspace descends by the same assertion applied to its restriction; see also Milne (2024, Proposition 3.5). Thus a stable line has a fixed nonzero generator.

After a form \(f\) has been made \(\sigma\)-equivariant, meaning \(f(Bv,Bw)=\sigma(f(v,w))\), its Gram matrix in a fixed-vector basis has entries in \(k\). Its restriction consequently extends back to the original form and preserves nondegeneracy. For a hermitian form with field involution \(\tau\) commuting with \(\sigma\), the restricted form has involution \(\tau|_k\); it is symmetric when that restriction is the identity. All field involutions used here commute. The normalizations giving \(B^d=\mathop{\mathrm{id}}\) and equivariance of the form are checked separately below; scalar Hilbert 90, as in Milne (2022, Corollary 5.25), handles only the scalar part of those checks.

In the linear case, an external field involution requires \(q=s^2\); write \(\sigma:x\mapsto x^s\). On a compatible frame a representative is \(T=\mathop{\mathrm{diag}}(a_1,\ldots,a_n)\sigma\), and \(T^2=cI\) implies \(a_i a_i^\sigma=c\) for every \(i\). Multiplying \(T\) by \(a_1^{-1}\) makes \(T^2=I\). Semilinear descent gives an \(n\)-dimensional fixed space \(V_0\) over \(\mathbb F_s\), with \(V=V_0\otimes_{\mathbb F_s}\mathbb F_q\). A stable line has a fixed generator by the one-dimensional norm-one equation, so compatible frames correspond exactly to line frames of \(V_0\).

A correlation without field action is represented by a nonsingular bilinear form. The involutory condition says that its transpose is a scalar multiple of itself. In coordinates from a compatible frame its matrix is diagonal and nonsingular; the scalar is therefore one. The form is symmetric, and the compatible frames are its orthogonal nondegenerate line frames. For a diagram-field involution, the same argument uses a \(\sigma\)-sesquilinear form. Its conjugate transpose is a scalar multiple of itself; scaling the form, using the norm-one scalar equation, makes it hermitian. Its compatible frames are hermitian frames with parameter \(s\). Alternating forms have no compatible nondegenerate line frame and do not occur in either assertion.

If there are two independent external involutions, choose their external basis to be field and diagram-field. Descend the first one to \(\sigma\). Choose generators for the lines of one compatible frame in the descended space. In these coordinates write the other involution as \(\operatorname{Int}(D)\gamma\sigma\), where \(D\) is nonsingular diagonal and \(\operatorname{Int}(D)(g)=DgD^{-1}\). Projective commutation with \(\sigma\) gives \(D^\sigma=cD\). Dividing by any diagonal entry makes every entry of \(D\) lie in \(\mathbb F_s\). The corresponding correlation form on \(V_0\) has Gram matrix \(D^{-1}\). Indeed, for \(g\in\mathop{\mathrm{GL}}(V_0)\) and \(\lambda\in\mathbb F_s^\times\), \[gDg^{\mathsf T}=\lambda D \quad\Longleftrightarrow\quad g^{\mathsf T}D^{-1}g=\lambda D^{-1}.\] This form is symmetric and nondegenerate over \(\mathbb F_s\). A frame is compatible with both involutions precisely when it is the scalar extension of an orthogonal frame for this symmetric form: the first condition descends its lines, and the second is orthogonality for that form.

In the unitary case the unique external involution induces \(\tau:x\mapsto x^q\) on \(\mathbb F_{q^2}\). Take an orthonormal basis along a compatible frame. A unitary semilinear representative has the form \[T=\mathop{\mathrm{diag}}(t_1,\ldots,t_n)\tau, \qquad t_i^{q+1}=1.\] It follows that \(T^2=I\). This conclusion uses compatibility with a nondegenerate orthogonal frame. An arbitrary projective unitary semilinear involution can instead have square \(-I\), which cannot be removed by multiplying by a norm-one scalar. No such involution is included here. The fixed space of \(T\) has dimension \(n\) over \(\mathbb F_q\). The original hermitian form restricts to a nondegenerate symmetric form on this space: its values on fixed vectors are fixed by \(\tau\), and hermitian symmetry then becomes ordinary symmetry. Compatible unitary frames correspond exactly to orthogonal nondegenerate line frames in this fixed space.

These descriptions also identify all coarsenings of frames. A sum of lines descends to the corresponding sum of fixed lines, and the subgroup requiring equal signs on that sum is unchanged by the description. Adjoining the fixed decorations preserves these identifications. Thus the frame relations in Theorem 11 are actual subgroup face relations for the decorated groups.

The construction at two

Odd-prime unitary extensions

Plane signs and the quaternion action

Let \(\mathcal D=\{V_1,\ldots,V_n\}\) be an unordered orthogonal decomposition of \(V\) into nondegenerate planes. Its linear sign group and projective sign group are \[\widehat S(\mathcal D) =\{\mathop{\mathrm{diag}}(\epsilon_1\mathop{\mathrm{id}}_{V_1},\ldots, \epsilon_n\mathop{\mathrm{id}}_{V_n}):\epsilon_i\in\{1,-1\}\}, \qquad S(\mathcal D)=\widehat S(\mathcal D)/\langle-\mathop{\mathrm{id}}_V\rangle.\] All these signs are symplectic, and \(S(\mathcal D)\leq L\) has rank \(n-1\).

A quaternion four-group compatible with \(\mathcal D\) is a subgroup \(T\leq L\) generated by the projective classes of two operators \(I,J\in\mathop{\mathrm{Sp}}(V)\) such that \[ I^2=J^2=-\mathop{\mathrm{id}}_V,\qquad IJ=-JI, \tag{59}\] and both operators preserve every \(V_i\). On each plane these operators generate its full matrix algebra. Indeed, after extending scalars to an algebraic closure, \(I\) has two distinct eigenvalues and \(J\) interchanges the two eigenlines. The resulting diagonal and off-diagonal matrix units span the full matrix algebra; its dimension is therefore already four over \(\mathbb F_q\).

Such pairs exist over every finite field of odd order. On a plane choose \[I=\begin{pmatrix}0&1\\-1&0\end{pmatrix},\qquad J=\begin{pmatrix}a&b\\b&-a\end{pmatrix}, \qquad a^2+b^2=-1.\] The equation has a solution: if \(-1\) is a square this follows by factoring the left side over the field, and otherwise it is the surjectivity of the norm from its quadratic extension. These matrices have determinant one and satisfy (59). Repeating the construction on orthogonal planes gives the required operators on \(V\). Each of \(I,J,IJ\) is nonscalar on every plane, so \(T\cap S(\mathcal D)=1\). Consequently \[ A_0(\mathcal D,T)=S(\mathcal D)\times T\leq L, \qquad \mathop{\mathrm{rk}}A_0(\mathcal D,T)=n+1. \tag{60}\]

For fixed \(T\), the compatible decompositions have the form required by the symmetric case of Theorem 11. Here are the details of that identification. The algebra generated by \(I\) and \(J\) is \(\mathop{\mathrm{End}}(U)\cong M_2(\mathbb F_q)\), where \(U\) is its two-dimensional simple module. Complete reducibility, or the matrix units of this algebra, gives \[ V\cong U\otimes_{\mathbb F_q} W,\qquad \dim W=n. \tag{61}\] Choose a nonzero invariant alternating form \(\omega_U\) on \(U\). Invariant bilinear forms on \(U\) form a one-dimensional space: under the identification \(U\cong U^*\) supplied by \(\omega_U\), a second invariant form corresponds to an operator commuting with \(I,J\), and hence to a scalar. It follows that the original form is \[\omega=\omega_U\otimes B_W\] for a bilinear form \(B_W\) on \(W\). Alternation of \(\omega\) makes \(B_W\) symmetric, and nondegeneracy of \(\omega\) makes \(B_W\) nondegenerate. Submodules of \(U\otimes W\) are exactly the spaces \(U\otimes W'\) with \(W'\leq W\). In particular, the nondegenerate invariant planes are \(U\otimes w\), where \(w\) is a nondegenerate line of \(W\), and two such planes are orthogonal exactly when their lines are orthogonal for \(B_W\). Thus compatible plane decompositions are in bijection with all orthogonal nondegenerate line frames of the symmetric space \((W,B_W)\). Both line norm types are allowed.

Maximal configurations and field descent

Suppose, toward Proposition 24, that \(G\) is a minimal counterexample with the component \(L\). Consider all elementary abelian subgroups \(A\leq N_G(L)\) such that \[ A\text{ contains some }A_0(\mathcal D,T),\qquad LA\text{ acts faithfully on }L,\qquad O_2(C_G(LA))=1. \tag{62}\] This family is closed under elementary overgroups that are faithful configurations, since such an overgroup still contains the same seed. It is nonempty: take \(A=A_0\leq L\) and apply Lemma 4 to obtain \(O_2(C_G(L))=1\). Choose a member \(A\) of maximum rank, put \(m=\mathop{\mathrm{rk}}A\), and write \(H=LA\). Identify this faithful group with its image in \(\mathop{\mathrm{Aut}}(L)\).

Since \(A\) centralizes its subgroup \(A_0\), Lemma 25 gives \(A\cap M=A_0\). The quotient \(\mathop{\mathrm{Aut}}(L)/M\) is cyclic, so its elementary abelian subgroups have rank at most one. Therefore \[ m=n+1\quad\text{or}\quad m=n+2. \tag{63}\] In the first case \(A=A_0\) and \(H=L\). In the second, \[A=A_0\times\langle b\rangle, \qquad H/L\cong C_2,\qquad H\cap M=L,\] where \(b\) induces the involution \(\sigma\) of \(\mathbb F_q/\mathbb F_s\) and \(q=s^2\). The field parameter \(s\) is at least five.

The next normalization is essential for the construction. We will require the field axis to commute exactly with the quaternion lifts, not merely with their projective classes.

We now specify the finite family of data from which the cycle will be formed. If \(m=n+1\), a datum is any pair \((\mathcal D,T)\) as in (60); its full elementary group is \(E=S(\mathcal D)T\). If \(m=n+2\), a datum is a triple \((\mathcal D,T,b)\), where \((\mathcal D,T)\) has that meaning, \(b\in H\setminus L\) is an involution preserving each plane, and a semilinear representative of \(b\) commutes exactly with the lifts \(I,J\). The full elementary group is then \[ E=S(\mathcal D)\times T\times\langle b\rangle. \tag{64}\] The exact condition is independent of the scalar representative of \(b\) and of the choices of the signs of \(I,J\). It also holds for every element of their quaternion group once it holds for \(I,J\). Lemma 26 shows that this family is nonempty, since the original field axis may be shifted inside \(T\leq L\).

For each fixed decoration \(T\), or fixed pair \((T,b)\), we use every compatible frame. The preceding arguments identify these frames with the symmetric spaces in Theorem 11, over \[ u=q\quad\text{if }m=n+1,\qquad u=s\quad\text{if }m=n+2. \tag{65}\] We impose no restriction on the isometry type of the multiplicity space as the decorations vary. The frame bound is uniform in that type; this matters because changing a field axis can change the descended symmetric form. Coarsening two lines of the multiplicity space coarsens their two invariant planes, and the equal-sign subgroups are unchanged by the identification. Thus the face identifications required by the frame theorem hold in the actual elementary subgroup poset.

Completion counts

To apply Lemma 17, we need lower bounds for two kinds of completions: restoring a quaternion four-group from one of its lines, and restoring a field axis. We keep all the remaining data fixed in each count.

Detection and the strict dimension inequality

We must check that the parameters counted by the decoration lemma are coefficients of actual subgroup flags. This check precedes the dimension comparison, since different data can have the same full elementary group \(E\).

For any linear group \(K=S(\mathcal D)T\), choose symplectic lifts of its elements. Their commutators define an alternating bilinear pairing \[\beta_K:K\times K\longrightarrow\{1,-1\}, \qquad [\widehat x,\widehat y]=\beta_K(x,y)\mathop{\mathrm{id}}_V.\] The pairing is independent of the lifts. It vanishes on \(S(\mathcal D)\) and is nondegenerate on \(T\) by (59); hence its radical is exactly \(S(\mathcal D)\). Thus \(K\) intrinsically recovers \(S(\mathcal D)\), whose full symplectic preimage has joint eigenspaces precisely the planes of \(\mathcal D\). Without a field axis, \(E=K\); with one, \(K=E\cap M\) is recovered first. In either case the full group recovers its plane frame and sign group.

Write \(\mathcal L(T)\) for the three lines of \(T\). Its local coefficient space is \[ \left\{(c_t)_{t\in\mathcal L(T)}\in\mathbb Q^3: \sum_{t\in\mathcal L(T)}c_t=0\right\}. \tag{66}\] It has dimension two and is represented by the coned chains \(\sum_t c_t[t<T]\). The sum-zero condition cancels the face \([T]\); dropping the top \(T\) leaves one coefficient at each retained line. This is the quaternion input to Lemma 17.

Let \(S_1<\cdots<S_{n-1}=S(\mathcal D)\) be a full sign flag. Without a field axis, test the split construction on flags \[ t<T<TS_1<\cdots<TS_{n-1}=E, \qquad t\in\mathcal L(T). \tag{67}\] A second datum with the same full group has the same sign group \(S\). Each vertex in its split construction is a product of a subgroup of \(S\) and a subgroup of its chosen complement \(T'\). Since the rank-two vertex \(T\) in (67) is disjoint from \(S\), it can occur only if \(T'=T\). The flags below \(T\) then evaluate the three coordinates in (66), and intersecting the remaining vertices with \(S\) recovers the sign flag. The coefficient on the displayed flag is, up to its fixed shuffle sign, the product of those two coefficient evaluations. Such flags therefore detect the full tensor product of the local and frame parameter spaces.

With a field axis, write \(B_0=\langle b\rangle\) and instead test \[ B_0<B_0t<B_0T<B_0TS_1<\cdots<B_0TS_{n-1}=E. \tag{68}\] The first vertex is outside \(M\). In a competing split construction, the only rank-one vertex outside \(M\) is its selected field axis, so that axis must equal \(B_0\). Next, \((B_0T)\cap M=T\), which is disjoint from \(S\) and has rank two. The same argument as before forces its quaternion complement to equal \(T\). The choices of \(t\) and of the sign flag detect all the remaining parameters. Consequently there is no loss of dimension from coincident cone groups in either construction.

We can now apply the quantitative lemmas. Let \(N\) be the number of full data and set \(a_n=(n-1)!\). By Theorem 11, the initial space of actual coefficients, summed over the fixed decorations, has dimension at least \[2F a_n N, \qquad F\geq (17/25)^n, \qquad F\geq3/8\text{ when }n=3.\] Here the same \(F\) is a valid lower bound for each multiplicity-space isometry type. There are three retained-line incidences per full datum. Lemma 27 bounds the number of such partial data by \(3N/k_T\), and each costs at most \(a_n\) equations. In the field case the number of data with the field axis removed is at most \(N/k_b\), each costing at most \(2a_n\) equations. Thus Lemma 17 yields a nonzero cycle provided \[ F>\frac{3}{2k_T} +\begin{cases}0,&m=n+1,\\ 2^{1-n},&m=n+2.\end{cases} \tag{69}\] The factor two in the denominator of the quaternion term comes from the two local parameters in (66); each retained-line face has only one.

For \(n=3\) the worst completion bound occurs at \(u=5\), giving \(k_T=4^3/4=16\). Even with a field axis, \[ F\geq\frac38=\frac{12}{32} >\frac3{32}+\frac14=\frac{11}{32}. \tag{70}\] For \(n=4\) the weaker uniform bound already gives \[\left(\frac{17}{25}\right)^4 =\frac{83521}{390625} >\frac{19}{128} =\frac{3}{2\cdot4^3}+\frac18.\] When \(n\) increases by one, the first term on the right is multiplied by \(1/4\) and the second by \(1/2\), whereas the left side is multiplied by \(17/25>1/2\). Induction proves (69) for every \(n\geq4\), and omitting the field term only strengthens it.

We have therefore obtained a nonzero cycle \[\alpha\in\widetilde C_{m-1}(\mathcal A_2(H);\mathbb Q)\] supported on full flags ending at the rank-\(m\) groups \(E\) in our data family. All its vertices lie in \(H\): plane signs and quaternion generators lie in \(L\), and every field axis was chosen in \(H\). No restriction from a larger automorphism group is needed in this case.

Propagation from a supported full group

Choose a full flag with nonzero coefficient in \(\alpha\), and let \(A'\) be its last group. The construction gives \(A'\cap L\supseteq A_0(\mathcal D',T')\ne1\) for its datum, and \[LA'=H.\] Indeed this is immediate if \(H=L\); otherwise \(A'\) contains its chosen field involution in the nontrivial coset of \(H/L\). Thus \(LA'\) acts faithfully on \(L\), and \(O_2(C_G(LA'))=O_2(C_G(H))=1\). The group \(A'\) belongs to the family (62) and has its maximum rank \(m\).

The selected flag is saturated and has nonzero coefficient in the cycle \(\alpha\in\widetilde C_{m-1}(\mathcal A_2(LA');\mathbb Q)\). Corollary 10 therefore applies to \(A'\) and the family (62). It contradicts the assumed minimal counterexample, proving Proposition 24.

Sign groups, parity, and their centralizers

An orthogonal line frame is an unordered decomposition \(\mathcal D=\{\ell_1,\ldots,\ell_N\}\) into nondegenerate orthogonal lines. Color a line square or nonsquare according to the square class of its nonzero norms. Let \(I\) be the set of nonsquare positions and put \(k=|I|\). The group of independent signs on these lines, modulo their common sign, is \[S(\mathcal D)=\mathbb F_2^N/\langle j\rangle, \qquad j=(1,\ldots,1),\] embedded in \(\mathop{\mathrm{PO}}(V)\). We abbreviate its inner part to \(T(\mathcal D)=S(\mathcal D)\cap L\). On \(\mathbb F_2^N\) define \[t(x)=\sum_{i=1}^N x_i,\qquad c(x)=\sum_{i\in I}x_i.\] The corresponding diagonal isometry has determinant \((-1)^{t(x)}\) and spinor norm of square class \(c(x)\). Consequently the inverse image of \(T(\mathcal D)\) in \(\mathbb F_2^N\) is \[ U=(\ker t\cap\ker c)+\langle j\rangle, \qquad U^\perp=\langle t,c\rangle\cap j^\perp. \tag{71}\] The addition of \(\langle j\rangle\) accounts for the possibility of replacing an isometry by its negative before testing membership in \(\Omega(V)\).

Here are the sign groups we shall use. The homogeneous rows mean that all lines have the same color; in the odd-dimensional row our determinant normalization makes this the square color. Mixed rows require both colors to occur. \[ \begin{array}{c|c|c|c} \text{dimension and determinant}&\text{frame}&U^\perp&\mathop{\mathrm{rk}}T(\mathcal D)\\ \hline N\text{ even, nonsquare}&k\text{ odd}&\langle t\rangle&N-2\\ N\text{ odd, square}&\text{homogeneous}&0&N-1\\ N\text{ even, square}&\text{homogeneous}&\langle t\rangle&N-2\\ N\text{ odd, square}&k\text{ even, mixed}&\langle c\rangle&N-2\\ N\text{ even, square}&k\text{ even, mixed}&\langle t,c\rangle&N-3 \end{array} \tag{72}\] Indeed \(t(j)=N\bmod2\) and \(c(j)=k\bmod2\), so the annihilators follow directly from (71); the ranks are \(\dim U-1\). The discriminant of a frame is the product of its line norm classes, so its being square is equivalent to \(k\) being even.

We say that a projective sign subgroup distinguishes the positions if no two coordinate characters agree on its inverse image in \(\mathbb F_2^N\). Equality at positions \(i\ne j\) would mean \(e_i^*+e_j^*\in U^\perp\). Thus the first three rows of (72) distinguish all positions for the dimensions under consideration. The fourth row does so when \(k\ge4\). The fifth does so when \(k\ge4\) and \(N-k\ge4\), because its three nonzero annihilator vectors have supports \(I\), \(I^c\), and the entire set.

All applications below meet this lemma’s rank hypotheses. For the first three rows of (72), the minimum rank is \(N-2>N/2\) because \(N\ge5\). For the last row we use \(N\ge8\), when \(N-3>N/2\). The required color class in dimension eight always exists.

Naturality in dimension eight

The weight argument applies once an automorphism is known to act on the natural space. In split dimension eight, two inner sign changes supply that conclusion; in minus dimension eight all automorphisms have natural realizations.

Descent and the choice of a maximal configuration

We now choose a family for Corollary 10. Choose a nonempty set of allowed frames whose inner signs meet Lemma 29. Consider all elementary abelian \(A\le N_G(L)\) such that \(A\cap L\) contains \(T(\mathcal D)\) for some allowed frame and \[ LA\text{ acts faithfully on }L, \qquad O_2(C_G(LA))=1. \tag{73}\] This family is nonempty: take \(A=T(\mathcal D)\le L\), and use Lemma 4. It is closed under enlargement to faithful configurations, because an enlargement retains the contained inner seed. Choose \(A\) of maximum rank \(r\), put \(H=LA\), and identify \(H\) with its faithful image in \(\mathop{\mathrm{Aut}}(L)\).

For its chosen frame, the centralizer lemma shows that \(A\) preserves every line and that its linear part is in \(S=S(\mathcal D)\). The group \(A\) contains \(S\cap H\). Indeed \(S\) centralizes \(A\), so adjoining any element of \(S\cap H\) preserves \(H\), its centralizer, and the contained inner seed; maximal rank forbids an enlargement. Hence either \[ A=S\cap H, \tag{74}\] or there is an involution \(b\in A\) with nontrivial field action and \[ E=S\times\langle b\rangle,\qquad A=E\cap H=(S\cap H)\times\langle b\rangle. \tag{75}\] In the latter case \(H\cap\mathop{\mathrm{PO}}(V)\) is the entire linear part of \(H\): \(H=L A\) and \(A\) has linear kernel in \(S\le\mathop{\mathrm{PO}}(V)\).

Our remaining task is to produce a cycle in \(\mathcal A_2(H)\) with a nonzero full-flag coefficient at some \(A'\) of rank \(r\) containing an allowed seed and satisfying \(LA'=H\). Since \(LA'=H\), we have \(O_2(C_G(LA'))=O_2(C_G(H))=1\), and the contained inner seed gives \(A'\cap L\ne1\). Thus \(A'\) belongs to the same maximizing family. Corollary 10 then applies to this actual supported subgroup.

For subsequent index calculations, let \[P=\mathop{\mathrm{PO}}(V)/L,\qquad \pi:\mathop{\mathrm{PO}}(V)\longrightarrow P.\] Determinant and spinor norm identify this elementary abelian group with \[ P\cong\mathbb F_2^2/\langle(N\bmod2,k\bmod2)\rangle. \tag{76}\] Both invariants are attained because in dimension at least five the form represents both nonzero square classes. A reflection in a square line has image \((1,0)\), and one in a nonsquare line has image \((1,1)\). Thus every mixed frame has \(\pi(S)=P\). For homogeneous frames of a fixed color the image \(\pi(S)\) is a fixed subgroup of \(P\), independent of the frame. These observations will give constant indices except in the small-field argument, where they give a maximum index.

Nonsquare determinant and homogeneous frames

First suppose \(N\) is even and the determinant is nonsquare. Every orthogonal frame has an odd number of nonsquare lines and is mixed. Use all these frames in the maximizing family. Their inner signs have rank \(N-2\) and distinguish positions. Lemma 31 excludes a field axis, so (74) holds and \(H\le\mathop{\mathrm{PO}}(V)\). Put \(B=H/L\le P\). Every frame sign group surjects onto \(P\), whence \[[S(\mathcal D):S(\mathcal D)\cap H]=[P:B], \qquad L(S(\mathcal D)\cap H)=H.\] The all-color case of Theorem 11 supplies a nonzero cycle of full flags on sign groups of rank \(N-1\). These are actual coefficients: a full sign group recovers its frame as the common eigenspaces of its diagonal lifts. Lemma 19, at the constant index \([P:B]\), gives the required cycle and coefficient at an intersection \(A'=S(\mathcal D)\cap H\). The index exponent is at most two and is smaller than \(N-1\), as required. All such intersections have the original maximal rank and contain the allowed inner signs. Corollary 10 excludes this case. This argument includes nonsquare determinant in dimension six for every \(q\) under consideration.

We may now assume that the determinant is square. Suppose first that \(q\ge7\). The classification of nondegenerate symmetric forms over a finite field by dimension and determinant supplies homogeneous square frames: the diagonal form with all entries one has the required determinant. Fix this color and maximize over configurations containing its inner sign groups. These have the second or third rank in (72).

If there is no field axis, use the one-color case of Theorem 11. To see the strict positivity at the smallest parameter, an orthogonal plane generated by two lines of the chosen color has \[d_q=\frac{q-\chi_q(-1)}2\ge4\] ordered frames of that color, where \(\chi_q\) is the quadratic character of \(\mathbb F_q^*\). The one-color estimate has two roots \(\rho_+,\rho_-\) of \(X^2-X+d_q^{-1}\), and its degree-\(N\) fraction is \(\rho_+^N+\rho_-^N>0\). This also holds when \(d_q=4\), with the repeated root \(1/2\). Thus a nonzero sign cycle exists. If \(U=\pi(S(\mathcal D))\), then \(U\) is constant over these frames and \(B=H/L\le U\) by the initial configuration. Each intersection has index \([U:B]\) and surjects onto \(B\). The same coefficient detection and constant-index restriction give a supported member of the original maximal rank, so Corollary 10 applies.

It remains in this case to treat a maximizing \(A\) with field axis. Fix its group \(H\) and consider data \((\mathcal D,b)\), where \(\mathcal D\) is a homogeneous square frame and \(b\in H\) is an involution with the same nontrivial coset in \(H/(H\cap\mathop{\mathrm{PO}}(V))\), preserving its individual lines. The initial configuration supplies at least one such datum. For a fixed \(b\) which occurs, Lemma 31 identifies all its compatible frames with the orthogonal frames of a symmetric space over \(\mathbb F_s\), where \(q=s^2\) and \(s\ge5\). Every such frame has the original square color over \(\mathbb F_q\); hence our restriction to that color discards none of the descended frames. We may apply the uniform all-color fraction \((17/25)^N\) over \(\mathbb F_s\), including both determinant types of the descended space.

For a fixed frame, forgetting the axis has at least \[ |T(\mathcal D)|\ge 2^{N-2} \tag{77}\] completions. Indeed all \(bt\) with \(t\in T(\mathcal D)\) are distinct involutions in \(H\) with the stipulated coset, centralize the signs, and preserve the same frame. Compatibility and its color are therefore retained. The needed strict comparison is \[ \left(\frac{17}{25}\right)^N>2^{-(N-2)}\qquad(N\ge5). \tag{78}\] At \(N=5\) the product of the left side with \(2^{N-2}\) is greater than one, and each increase in \(N\) multiplies that product by \(34/25>1\).

We verify detection before invoking the decoration count. The group \(E=S(\mathcal D)\times\langle b\rangle\) recovers \(S(\mathcal D)\) as its kernel under the field map, and hence recovers \(\mathcal D\). In the split shuffle construction, flags beginning with \(\langle b\rangle\) and continuing with \(\langle b\rangle U_i\), for a sign flag \((U_i)\), recover the original sign coefficients. A different axis cannot contribute to such a flag with the same cone group: a split rank-one subgroup outside the linear kernel is its own chosen axis. Thus all initial frame parameters are detected in actual chains. Lemma 16, using (77) and (78), gives a nonzero homogeneous cycle with full flags ending at groups \(E\) of rank \(N\).

Finally put \(B=(H\cap\mathop{\mathrm{PO}}(V))/L\) and let \(U\) be the fixed image of a homogeneous square sign group in \(P\). The initial configuration shows \(B\le U\) and that its signs supply all of \(B\). For every new datum, \[E\cap H=(S(\mathcal D)\cap H)\times\langle b\rangle, \quad [E:E\cap H]=[U:B],\quad L(E\cap H)=H.\] Restriction therefore has constant index, loses at most two ranks, and returns an actual nonzero coefficient at a member of the maximizing family of rank \(r\). Corollary 10 again contradicts the choice of \(G\).

Forcing the required colors over the field of five elements

We have reduced the proof of Proposition 28 to \(q=5\) and square determinant. There is no field direction. For odd \(N\ge5\) call a frame admissible if its nonsquare count satisfies \[ k\text{ even},\qquad 4\le k\le N-1. \tag{79}\] For even \(N\ge8\) use \[ k\text{ even},\qquad 4\le k\le N-4. \tag{80}\] All these norm patterns occur by the classification by determinant. Their inner signs distinguish positions and have rank \(N-2\) in odd dimension and \(N-3\) in even dimension. Consequently they meet Lemma 29. Maximize configurations over these admissible frames. For the resulting \(H\), its initial group is \(A=S\cap H\), and every admissible sign group surjects onto \(P\).

The all-color frame theorem gives many cycles, but some of their coefficients could be on inadmissible frames. The following argument shows that the produced coefficient space cannot be supported entirely on those frames. This is the remaining reason for the color ranges (79)–(80).

We finish the restriction and propagation in this last case. The lemma supplies a cycle on the full sign groups, of rank \(m=N-1\), and a nonzero coefficient at an admissible frame \(\mathcal D\). With \(B=H/L\le P\), every such frame has \[[S(\mathcal D):S(\mathcal D)\cap H]=[P:B].\] An arbitrary frame \(\mathcal D'\) occurring elsewhere in the same cycle has \(\pi(S(\mathcal D'))\le P\), and hence \[[S(\mathcal D'):S(\mathcal D')\cap H] =[\pi(S(\mathcal D')):\pi(S(\mathcal D'))\cap B] \le[P:B].\] Our selected coefficient therefore has maximum intersection index among all supported groups, even though the cycle also uses other color counts. Its index exponent is at most two, so it is strictly less than \(m\). Lemma 19 applies at this specific coefficient and gives a cycle in \(\mathcal A_2(H)\) with a nonzero full flag ending at \[A'=S(\mathcal D)\cap H.\] It contains the admissible inner seed, has the original maximal rank, and satisfies \(LA'=H\) because its signs supply all of \(B\). Restriction has preserved an actual coefficient: in that lemma the fixed complement makes intersection with \(H\) injective on the residue’s flags. Corollary 10 now applies to this very supported subgroup \(A'\) and gives the contradiction.

This proves Proposition 28. The excluded case has \(N=6\), \(q=5\), and square determinant, hence is plus type because \(-1\) is a square in \(\mathbb F_5\). Its simple group is \(\mathop{\mathrm{PSL}}_4(5)\) and is treated in Section 9.

The two natural representations

Put \(L=\mathop{\mathrm{PSL}}_4(5)\) and \(V=\mathbb F_5^4\). The determinant identifies \(\mathop{\mathrm{PGL}}_4(5)/L\) with \(\mathbb F_5^\times\), since fourth powers in \(\mathbb F_5^\times\) are trivial. Let \(d\) denote the outer class of \(\mathop{\mathrm{diag}}(2,1,1,1)\), and let \(\gamma\) be inverse transpose. The automorphism theorem for \(L\) (Steinberg 1960, sec. 2 and statements 3.2–3.6) gives \[ \mathop{\mathrm{Aut}}(L)=\mathop{\mathrm{PGL}}_4(5)\rtimes\langle\gamma\rangle,\qquad \mathop{\mathrm{Out}}(L)=\langle d,\gamma\mid d^4=\gamma^2=1,\ \gamma d\gamma=d^{-1}\rangle. \tag{84}\] There is no field automorphism because the field is prime. Thus the outer group is dihedral of order eight. We need to identify precisely the subgroup coming from six-dimensional isometries.

Choose a basis \(e_1,\ldots,e_4\) of \(V\), write \(e_{ij}=e_i\wedge e_j\), choose a volume form, and put \(W=\bigwedge^2V\). Define the nondegenerate symmetric form \(\beta\) by \[u\wedge v=\beta(u,v)\,\mathrm{vol}\qquad(u,v\in W).\] The pairs of basis vectors \[(e_{12},e_{34}),\quad(e_{13},e_{24}),\quad(e_{14},e_{23})\] are hyperbolic pairs, with respective pairings \(1,-1,1\). Consequently \(W\) has plus type and square discriminant over \(\mathbb F_5\). For \(g\in\mathop{\mathrm{GL}}(V)\), \[ \beta((\bigwedge^2g)u,(\bigwedge^2g)v)=\det(g)\beta(u,v). \tag{85}\] The standard exterior-square covering \(\mathop{\mathrm{SL}}_4(5)\to\Omega_6^+(5)\) has kernel \(\{\pm I\}\) (Taylor 1992, Theorem 12.20 and Corollary 12.21); after projectivization it is the displayed isomorphism with \(L\). In particular, the two groups called \(L\) in these representations are identified.

Let \(\tau\) interchange complementary basis vectors with the exterior signs: \[e_{12}\leftrightarrow e_{34},\qquad e_{13}\leftrightarrow-e_{24},\qquad e_{14}\leftrightarrow e_{23}.\] It is an isometry, \(\tau^2=1\), and \(\det(\tau)=-1\). Directly from the wedge pairing, \[\tau(\bigwedge^2g)\tau^{-1} =\det(g)\bigwedge^2(g^{-T}).\] Thus \(\tau\) induces \(\gamma\) on the projective image of \(\mathop{\mathrm{SL}}_4(5)\). The exterior-square image of \(\mathop{\mathrm{PGL}}_4(5)\) and \(\tau\) together give the full projective similarity group of \(W\) (Taylor 1992, Theorem 12.18): the former has relative determinant \(\det(\bigwedge^2g)/\det(g)^3=1\), whereas \(\tau\) has relative determinant \(-1\); their order is that of this similarity group. Equivalently, this is the natural \(D_3=A_3\) realization of (84). We write \(\mathop{\mathrm{PO}}(W)\) for the projective image of the isometry group, a subgroup of \(\mathop{\mathrm{Aut}}(L)\).

A similarity can be rescaled to an isometry exactly when its multiplier is square. Equation (85) and the fact that \(\tau\) is an isometry therefore show that \[ T:=\mathop{\mathrm{PO}}(W)/L=\langle d^2,\gamma\rangle\cong C_2^2. \tag{86}\] Here \(L\) is the joint determinant–spinor-norm kernel in \(\mathop{\mathrm{PO}}(W)\). In this dimension \(-I\) has determinant and spinor norm one, so both characters descend to the projective group. In particular, there is no ambiguity from changing an isometry representative by \(-I\).

There are only two possibilities for an elementary abelian subgroup \(B\leq\mathop{\mathrm{Out}}(L)\) satisfying \(B\cap T=1\). Since \(T\) has index two, either \(B=1\), or \(B\) has order two and is generated by \(d\gamma\) or \(d^3\gamma\). These are precisely the graph cosets whose diagonal determinant class is nonsquare.

The remaining preparatory inputs concern centralizers and homology. They will distinguish the three branches of the final argument.

An inheritance fact for centralizers

The passage from an extension to one of its outer directions requires the following elementary observation.

Homology of the component

The final branch will use nonzero homology of \(L\) itself. We prove that fact directly, without applying the assertion under induction.

The equivalent poset used in the final branch

We record the precise specialization of the equivalent-poset theorem that will be needed. A product below includes the trivial subgroup in either coordinate, but excludes \((1,1)\).

This centerless case is supplied by Piterman and Smith (2025, Propositions 4.11–4.12). It is also the specialization of Piterman (2026, Definition 3.2 and Theorem 3.3), together with the equivalence preceding that definition, to \(B=1\) and \(\mathcal P_L=\mathcal A_p(L)\). Its component hypothesis is \(p\mid|L|\) and \(p\nmid|Z(L)|\); the centerless hypothesis here supplies the latter. The required condition \(B\cap LC_G(L)=1\) is automatic for \(B=1\). No normality of \(L\) in \(G\), and no defining-characteristic assumption, is required. Since \(Z(L)=1\), the product \(LK\) in the statement is the direct product \(L\times K\), so the products \(UV\) used in (88) are unambiguous.

Exclusion of a minimal counterexample

The preparatory results are now in place. The first two branches return to frame cycles; the third uses the equivalent poset. We use configuration in the sense of Definition 5: \(A\leq N_G(L)\) is elementary abelian, \(A\cap L\ne1\), the group \(H=LA\) acts faithfully on \(L\), and \(O_2(C_G(H))=1\).

The first branch starts from a six-dimensional sign configuration whose rank and distinct weights permit propagation. If none exists, Lemma 33 will exclude every faithful extension with trivial centralizer core whose outer image meets \(T\) nontrivially. The only possible remaining nontrivial outer image is then an odd-diagonal graph line, which gives the second branch. If that branch is absent as well, the equivalent-poset theorem and the component homology proved above give the final argument. These transitions are proved inside the proposition; in particular, the graph-line restriction is imposed only after the first branch has been excluded.