Basic operations on P -schemes

Chapter VI: Constructing new P -schemes from old ones

6.1 Basic operations on P -schemes

In this section, we introduce some basic operations onP-schemes, including inverse right translation, restriction, and passing to quotient groups. We then use them to prove Lemma 4.12.

Inverse right translation ofP-schemes. LetP be a subgroup system over a finite groupG. For eachH ∈ P, the group Gacts onH\Gby inverse right translation

gHh=Hhg⁻¹. This action induces an action ofGon the set of partitions ofH\G, defined by^gP ={^gB :B ∈P}for a partitionP ofH\G. ThenGalso acts on the set ofP-collections by inverse right translation:

Definition 6.1. The action ofGon the set ofP-collections by inverse right transla- tion is defined as follows: for aP-collectionC ={C_H :H ∈ P}andg ∈G, define

gC ={^gC_H :H ∈ P}.

Lemma 6.1. For aP-schemeCandg ∈G, theP-collection^gCis also aP-scheme.

Moreover, ifC is antisymmetric (resp. strongly antisymmetric), so is^gC.

Proof. This follows in a straightforward manner fromG-equivariance of projections and conjugations (see Lemma 2.2).

SoGalso acts on the set ofP-schemes by inverse right translation, which preserves antisymmetry and strong antisymmetry.

Restriction to a subgroup. We define therestrictionof a subgroup systemPover Gand that ofP-collections to a subgroup ofG.

Definition 6.2(restriction). LetP be a subgroup system over a finite groupG. For a subgroupG⁰ofG, define

P|_G⁰ :={H ∈ P :H ⊆G⁰},

which is a subgroup system overG⁰, called therestrictionofP toG⁰.

LetC ={C_H :H ∈ P}be aP-collection. ForH ∈ P|_G⁰, regardH\G⁰ as a subset ofH\Gin the obvious way. Then the partition C_H ofH\Grestricts to a partition

ofH\G, denoted byC_H|_G⁰. Define

C|_G⁰ :={C_H|_G⁰ :H ∈ P|_G⁰} which is aP|_G⁰-collection, called therestrictionofC toG⁰.

Next we show that whenC is P-scheme, its restriction C|_G⁰ to a subgroup G⁰ is a P|_G⁰-scheme. Moreover, antisymmetry and strong antisymmetry are preserved by restriction.

Lemma 6.2. LetP be a subgroup system over a finite group G. For a subgroup G⁰ ofGand aP-schemeC, the restrictionC|_G⁰ is aP|_G⁰-scheme. Moreover, ifC is antisymmetric (resp. strongly antisymmetric), so isC|_G⁰.

Proof. We have projections π_H,H⁰ and conjugations c_H,g defined between coset spaces H\G for various subgroups H ⊆ G. And we also have projections and conjugations between coset spaces H\G⁰ for H ⊆ G⁰. We useπ_H,H⁰ 0 andc⁰_H,g for the latter maps to distinguish them from the former.

For eachH ∈ P|G⁰, we have a projectionπH,G⁰ :H\G→G⁰\G. This allows us to partitionH\Ginto “fibers” ofπH,G⁰, i.e., preimages of elements inG⁰\G:

H\G= a

y∈G⁰\G

π_H,G⁻¹ 0(y).

We say x ∈ H\Gis in they-fiber ifπ_H,G⁰(x) =y, andyis called the index of x. Note that the subsetH\G⁰ ⊆H\Gis precisely they-fiber withy=G⁰e∈G⁰\G. Consider H, H⁰ ∈ P|_G⁰ and a map τ : H\G → H⁰\G that is either a projection π_H,H⁰, or a conjugation c_H,g for someg ∈ G⁰ satisfyingH⁰ = gHg⁻¹. We claim π_H,G⁰ = π_H⁰_,G⁰ ◦τ, i.e., the mapτ preserves the indices of elements. This can be checked directly: for Hh ∈ H\G, we haveπ_H,G⁰(Hh) = G⁰h. If τ = π_H,H⁰, we haveπ_H⁰_,G⁰ ◦τ(Hh) =π_H⁰_,G⁰(H⁰h) =G⁰h. And ifτ =c_H,g withg ∈G⁰, we have π_H⁰_,G⁰ ◦τ(Hh) =π_H⁰_,G⁰(H⁰gh) = G⁰gh=G⁰h. So the claim holds.

This means the mapτis also fibered overG⁰\Gsuch that its “y-fiber”τy :=τ|_π⁻¹

H,G0(y)

maps they-fiber ofH\Gto they-fiber ofH⁰\G. Settingy=G⁰egives us the map τ_y :H\G⁰ →H⁰\G⁰that is either the projectionπ_H,H⁰ 0, or the conjugationc⁰_H,g. From this observation it is easy to see that compatibility, invariance, or regularity ofC|_G⁰ follows from the corresponding property ofC: fixy =G⁰e. Assume to the

contrary thatC|_G⁰ does not satisfy compatibility. Then some projectionτ_y =π_H,H0

maps two elements in the same block ofC_H|_G⁰ into different blocks ofC_H⁰|_G⁰. But thenτ =π_H,H⁰ also maps these two elements that are in the same block ofC_H into different blocks ofC_H⁰, contradicting compatibility ofC. Invariance is proved in the same way except that we consider conjugations instead of projections. For regularity, note that for each projectionπ_H,H⁰ 0 : H\G⁰ →H⁰\G⁰ and blocksB ∈ C_H|_G⁰,B⁰ ∈ C_H⁰|_G⁰, we haveB = ˜B ∩(H\G⁰), B⁰ = ˜B⁰ ∩(H\G⁰)whereB˜ ∈C_H,B˜⁰ ∈C_H⁰. And forz ∈B⁰ we haveπ⁰⁻¹_H,H0(z)∩B =π_H,H⁻¹ 0(z)∩B˜∩(H\G⁰) = π⁻¹_H,H0(z)∩B˜. Regularity ofC|G⁰ then follows from regularity ofC.

Now assumeC|_G⁰ is not antisymmetric. Then for someH ∈ P|_G⁰ andg ∈N_G⁰(H), the mapc⁰_H,g restricts to a nontrivial permutation of some blockB ∈ C_H|_G⁰. Then we haveg ∈N_G(H)andc_H,g restricts to a nontrivial permutation ofB˜, whereB˜is the block ofC_H satisfyingB˜∩(H\G⁰) = B. SoC is not antisymmetric.

Finally, assume C|_G⁰ is not strongly antisymmetric. Then there exists a nontrivial permutation τ of a blockB ∈ CH|G⁰ for some subgroup H ∈ P|G⁰ such that τ is a composition of mapsσi : Bi−1 → Bi, i = 1. . . , k, where each Bi is a block of C_Hi|_G⁰,H_i ∈ P|_G⁰, andσ_iis of the formc⁰_Hi−1,g|_Bi−1 (whereg ∈G⁰),π⁰_Hi−1,H

i|_Bi−1, or(π_H⁰

i,Hi−1|_Bi)⁻¹ (see Definition 2.7). Each blockB_i is of the formB˜_i ∩(H_i\G⁰) for some B˜_i ∈ C_Hi. In the case that σ_i is of the form (π⁰_H

i,Hi−1|_Bi)⁻¹, we know

|π_H⁰⁻¹

i,Hi−1(z)∩B_i| = |π_H⁻¹

i,Hi−1(z)∩B˜_i| = 1 for all z ∈ Bi−1. So π_Hi,Hi−1|_Bi is well defined. Then τ = ˜τ|_B for the nontrivial permutationτ = σ_k· · · ◦σ₁ of the blockB˜ = ˜B₀ ∈ C_H, where each map σ˜_i is of the formc_Hi−1,g|B˜i−1, π_Hi−1,Hi|B˜i−1, or(π_Hi,Hi−1|B˜i)⁻¹. SoC is not strongly antisymmetric.

Next we describe the analogue of Definition 6.2 form-schemes.

Definition 6.3. Let Π = {P₁, . . . , P_m} be an m-scheme on a finite set S. For (x₁, . . . , x_k)∈S^(k) wherek < m, define the(m−k)-collection

Π|_x1,...,xk :={P₁⁰, . . . , P_m−k⁰ }

on the setS− {x1, . . . , xk}, whereP_i⁰ is the partition ofS⁽ⁱ⁾such that two elements (y1, . . . , yi),(y⁰₁, . . . , y⁰_i)are in the same block ofS⁽ⁱ⁾iff(x1, . . . , xk, y1, . . . , yi)and (x₁, . . . , x_k, y₁⁰, . . . , y_i⁰)are in the same block ofS^(i+k).

We also have the analogue of Lemma 6.2 form-schemes.

Lemma 6.3. The(m−k)-collectionΠ|_x1,...,xkin Definition 6.2 is an(m−k)-scheme.

Moreover, ifΠis antisymmetric (resp. strongly antisymmetric), so isΠ|_x1,...,xk. And ifΠdoes not have a matching, neither doesΠ|_x1,...,xk.

The proof is straightforward by definition. Indeed, if we viewΠas aP-scheme via Definition 2.12 and Definition 2.13, whereP is the system of stabilizers of depthm with respect to the natural action ofG = Sym(S)on S. ThenΠ|_x1,...,xk is simply the restriction of thisP-scheme to the subgroup G_x1,...,xk. We leave the details to the reader.

Passing to quotient groups. LetGbe a finite group and letN be a normal in G.

WriteG¯forG/N andφfor the quotient mapG→G¯.

For a subgroup H ⊆ G¯, the group G acts on H\G¯ by inverse right translation (through its quotient group G¯). The stabilizer of He ∈ H\G¯ is φ⁻¹(H). So by Lemma 2.1, we have an equivalence between the action ofGonH\G¯ and that on φ⁻¹(H)\G, given by the bijection λ_He : H\G¯ → φ⁻¹(H)\G sending Hφ(g) to φ⁻¹(H)gforg ∈G.

LetP be a subgroup system over G¯. DefineP˜ = {φ⁻¹(H) : H ∈ P}, which is a subgroup system overG. By identifyingH\G¯withφ⁻¹(H)\Gviaλ_HeforH ∈ P, we see that a P-scheme overG¯ is equivalent to aP˜-scheme overG. This is made formal by the following lemma.

Lemma 6.4. LetP and P˜ be as above. For aP-collection C = {C_H : H ∈ P}, define theP-collection˜ C⁰ ={C_φ⁰−1(H) :H ∈ P}by choosing

C_φ⁰−1(H)={λHe(B) :B ∈CH}.

ThenC 7→ C⁰ is a one-to-one correspondence betweenP-schemes overG¯ andP˜- schemes over G. Moreover, C is antisymmetric (resp. strongly antisymmetric) iff C⁰ is antisymmetric (resp. strongly antisymmetric). And C is homogeneous (resp.

discrete) on a subgroupH ∈ P iffC⁰ is homogeneous (resp. discrete) onφ⁻¹(H).

Proof. We check that the mapsλ_He commute with conjugations and projections:

write π_H,H⁰ and c_H,g for conjugations and projections between coset spaces of G¯ and writeπ_H,H⁰ 0 andc⁰_H,g for those between coset spaces ofG. Then we always have

λ_H⁰_e◦π_H,H⁰ =π_φ⁰−1(H),φ⁻¹(H⁰)◦λ_He

forH, H ∈ P,H ⊆H, and

λ_H⁰_e◦c_H,φ(g) =c⁰_φ−1(H),g◦λ_He.

for H, H⁰ ∈ P, g ∈ G, H⁰ = φ(g)Hφ(g)⁻¹. Also note that the maps λ_He are bijections. The lemma then follows easily by definition.

We conclude this section by proving Lemma 4.12 using the results developed above.

First we prove the following lemma.

Lemma 6.5. Letk ∈ N⁺ andG_k ⊆ Gk−1 ⊆ · · · ⊆ G₁ ⊆ G₀ be a chain of finite groups. LetP be a subgroup system overG₀. We have:

1. If for alli ∈ [k], all strongly antisymmetric P|_Gi−1-schemes are discrete on G_i, then all strongly antisymmetricP-schemes are discrete onG_k.

2. If for somei∈[k], all strongly antisymmetricP|Gi−1-schemes are inhomoge- neous onG_i, then all strongly antisymmetricP-schemes are inhomogeneous onG_k.

The same holds if strong antisymmetry is replaced by antisymmetry.

Proof. Assume that there exists a strongly antisymmetric P-scheme C = {CH : H ∈ P} that is not discrete on G_k. Then there exist two different elements x, x⁰ ∈ G_k\G lying in the same block of C_Gk. Pick the greatest integer i ∈ [k]

satisfyingπ_Gk,Gi−1(x) =π_Gk,Gi−1(x⁰). Suchiexists asπ_Gk,G0(x) =π_Gk,G0(x⁰). Let y = π_Gk,Gi(x) andy⁰ = π_Gk,Gi(x⁰). Then (1) y 6= y⁰ by maximality of iand the fact thatx6=x⁰, (2)y, y⁰ are in the same block ofC_Gi by compatibility ofC and the fact thatx, x⁰are in the same block ofC_Gk, and (3)π_Gi,Gi−1(y) =π_Gi,Gi−1(y⁰)since π_Gi,Gi−1(y) =π_Gk,Gi−1(x)andπ_Gi,Gi−1(y⁰) = π_Gk,Gi−1(x⁰).

SupposeπGi,Gi−1(y) =πGi,Gi−1(y⁰) =Gi−1g. By replacingC with^gC (with respect to the action of G_k on the set of P-schemes by inverse right translation) and applying Lemma 6.1, we may assumeGi−1g =Gi−1e. Then we can writey=G_ih and y⁰ = G_ih⁰ for some h, h⁰ ∈ Gi−1. By Lemma 6.2, the restriction C|_Gi−1 = {C_H|_Gi−1 : H ∈ P|_Gi−1}is a strongly antisymmetricP|_Gi−1-scheme. Asy, y⁰ are in the same block ofC_Gi, they are also in the same block of C_Gi|_Gi−1. Asy 6= y⁰, we knowC|_Gi−1 is not discrete onG_i. This proves the first claim of the lemma.

For the second claim, assume to the contrary that it does not hold. Choosei ∈ [k]

such all strongly antisymmetric P|_Gi−1-schemes are inhomogeneous on G_i. Let C = {C_H : H ∈ P} be a strongly antisymmetricP-scheme that is homogeneous on G_k. By compatibility, we know C is homogeneous on G_i. Then C|_Gi−1 is also homogeneous onG_i. It is also strongly antisymmetric by Lemma 6.2, which contradicts the assumption.

The proof for antisymmetry is the same.

Now we are ready to prove Lemma 4.12. For convenience, we restate the lemma.

Lemma 6.6. Let k ∈ N⁺ and G_k ⊆ Gk−1 ⊆ · · · ⊆ G₁ ⊆ G₀ be a chain of finite groups. For i ∈ [k], let N_i be a subgroup of G_i that is normal in Gi−1, π_i : Gi−1 → Gi−1/N_i be the corresponding quotient map, andP_i be a subgroup system overGi−1/N_i that containsG_i/N_i. Define

P ={gπ_i⁻¹(H)g⁻¹ : 1≤i≤k, H ∈ P_i, g ∈G₀},

which is a subgroup system overG₀ and containsπ_i⁻¹(G_i/N_i) =G_i for alli∈[k].

Then we have

1. If for alli∈[k], all strongly antisymmetricP_i-schemes are discrete onG_i/N_i, then all strongly antisymmetricP-schemes are discrete onG_k.

2. If for somei∈[k], all strongly antisymmetricP_i-schemes are inhomogeneous onG_i/N_i, then all strongly antisymmetricP-schemes are inhomogeneous on G_k.

The same holds if strong antisymmetry is replaced by antisymmetry.

Proof. Fixi ∈[k]. By Lemma 6.4 and the definition ofP, if all strongly antisym- metricP_i-schemes are discrete (resp. inhomogeneous) onG_i/N_i, then all strongly antisymmetric P|_Gi−1-schemes are discrete (resp. inhomogeneous) on G_i. The same holds if strong antisymmetry is replaced by antisymmetry. The lemma now follows from Lemma 6.5.