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We now recall several standard methods for constructing modules for wreath products, as described in [21, Section 4.3] and [6, Section 3]. Recall that we are using right modules.

Firstly, let G be a subgroup of Sm, and let X be a kG-module. We define

Xne

to be the k(GoSn)-module obtained by equipping the k-vector space X⊗n

(that is, the tensor product over k of n copies of X) with the action given by the formula

(x1⊗ · · · ⊗ xn)(σ; α1, . . . , αn) = (x(1)σ−1α₁)⊗ · · · ⊗ (x_(n)σ−1α_n)

for x1, . . . , xn∈ X, α1, . . . , αn ∈ G, σ ∈ Sn.

More generally, let X1, . . . , Xt be kG-modules, and γ = (γ1, . . . , γt) a

composition of n of length t. We form a k(G_oSγ)-module by equipping the

k-vector space X⊗γ1

formula

(x1⊗ · · · ⊗ xn)(σ; α1, . . . , αn) = (x(1)σ−1α₁)⊗ · · · ⊗ (x_(n)σ−1α_n)

where each xi lies in the appropriate Xj, α1, . . . , αn ∈ G, and σ ∈ Sγ. We

denote this module by X1, . . . , Xt

γe

, and we note that Xne

is the special case of this construction where γ has an n in one place and all the other parts are 0.

Now let G be a subgroup of Sm, H be a subgroup of Sn, and Y a kH-

module. It is easy to check that we may make Y into a k(GoH)-module via the formula

y(σ; α1, . . . , αn) = yσ (4.3.1)

for y ∈ Y , α1, . . . , αn∈ G, and σ ∈ H. This module may be understood by

noting that G_{oH is the semidirect product of the normal subgroup consisting} of all elements (e; α1, . . . , αn) for α1, . . . , αn ∈ G with the subgroup consisting

of all elements (σ; e, . . . , e) for σ _{∈ H. This latter subgroup is canonically} isomorphic to H, and hence we see that the module obtained from Y via (4.3.1) is the inflation of Y from H to G_{oH with respect to the semidirect} product structure. Hence we shall denote this module by InfGoH_H Y . We shall be particularly interested in the case where H is Sn or a Young subgroup Sγ

of Sn and G is Sm or a Young subgroup Sλ of Sm, and in accordance with our

notational conventions, we shall write these modules as, for example, Infm_nonY or Infλ_γoγY .

Now let H be a subgroup of Sn, G be a subgroup of Sm, Y be a kH-module,

and further let Z be a k(GoH)-module. Then we define a k(GoH)-module Z_{Y as follows: the underlying k-vector space is Z⊗Y , and the action is} given by the formula

for z ∈ Z, y ∈ Y , α1, . . . , αn ∈ G, σ ∈ H. Thus we see that we have an

equality of k(G_oH)-modules

ZY = Z ⊗ InfGoHH Y (4.3.2)

where the module on the right-hand side is the internal tensor product of the k(GoH)-modules Z and InfGoHH Y .

We can combine the above constructions as follows: if G is a subgroup of Sm, X1, . . . , Xt are kG-modules and Y is a kSγ-module for γ a composition

of n, then we obtain a k(G_oSγ)-module

X1, . . . , Xt

γe

Y with underlying vector space X⊗γ1

1
⊗ X2⊗γ2
⊗ · · · ⊗ Xt⊗γt
⊗Y and action

given by the formula

(x1⊗ · · · ⊗xn⊗y)(σ; α1, . . . , αn) =

(x(1)σ−1α₁)⊗ · · · ⊗ (x_(n)σ−1α_n)⊗(yσ) (4.3.3)

for xi ∈ X, αi ∈ G, y ∈ Y , σ ∈ Sγ. Further, the k(GoSγ)-module

X1, . . . , Xt

γe

Y is exactly the inner tensor product

X1, . . . , Xt

γe

⊗ InfGoSγ

Sγ Y

of k(GoSγ)-modules, as explained above. We shall often be interested (for

the case G = Sm) in inducing such modules from SmoSγ to the full wreath

product SmoSn, that is, in modules

h X1, . . . , Xt
γe Yix  mon moγ.

We now recall an elementary construction for producing kSγ-modules Y

for use in the above constructions. Indeed, for each i _{∈ {1, . . . , t}, let Y}i

be a right kSγi-module. Now recall that we have a canonical identification

of the group Sγ with the direct product Sγ1 × Sγ2 × · · · × Sγt of groups.

Thus any module for k(Sγ1×Sγ2× · · · ×Sγt) may be regarded as a kSγ-module

in a canonical way, and vice versa. In particular, if Yi is a kSγi-module

for each i, then the external tensor product Y1  Y2 · · ·  Yt, which is a

k (Sγ1 × Sγ2 × · · · × Sγt)-module, may be regarded as a kSγ-module.

Now recall further that we have a canonical isomorphism between GoSγand

(G_oSγ1)× (GoSγ2)× · · · × (GoSγt), and hence we have a canonical identification

of algebras

k(G_oSγ) = k (GoSγ1)⊗ k (GoSγ2)⊗ · · · ⊗ k (GoSγt) . (4.3.4)

Suppose that we have for each i = 1, . . . , t a k(Go Sγi)-module Zi. Thus with

modules Yi as above, we see that for each i, Zi  Yi is a k(Go Sγi)-module.

Hence via the identification (4.3.4), we see that both Z1  · · ·  Zt and

(Z1 Y1) · · ·  (Zt Yt) may be considered to be k(GoSγ)-modules. It is

now easy to see that we have an isomorphism of k(Go Sγ)-modules

(Z1· · ·  Zt) (Y1· · ·  Yt) ∼= (Z1 Y1) · · ·  (Zt Yt). (4.3.5)

There is an important special case of the isomorphism (4.3.5). Indeed, if we have kG-modules X1, . . . , Xt, then we may form for each i the k(Go Sγi)-

module Xγe _i i . We have X1, . . . , Xt
γe = Xγe ₁ 1 · · ·  X e γt t

an isomorphism X1, . . . , Xt
γe  Y1 Y2 · · ·  Yt
∼ = Xγe 1 1  Y1
 X e γ2 2  Y2
 · · ·  X e γt t  Yt
(4.3.6) (this isomorphism was given in [6, Lemma 3.2 (1)]). In subsequent chapters

we shall be particularly interested in modules of the form h X1, . . . , Xt
γe  Y1 Y2· · ·  Yt
ix   mon moγ. (4.3.7)

In the next chapter (Section 5.2), we shall generalise the construction (4.3.7) to the wreath product of a k-algebra with a symmetric group, and moreover we shall develop a graphical representation of pure tensors in such modules using modified permutation diagrams, which gives a more intuitive way of understanding their structure.

We now give some basic properties of the above constructions.

Proposition 4.3.1. Let G be a subgroup of Sm and H a subgroup of Sn. Let

W and Y be kH-modules. Then we have an isomorphism of k(Go H)-modules InfG_HoH(W _{⊗ Y ) ∼}= InfG_HoH(W )_{⊗ Inf}G_HoH(Y ).

Proof. Both modules have underlying vector space W ⊗ Y , and it is easy to verify that the identity map on this space yields the required isomorphism of group modules.

Proposition 4.3.2. Let G be a subgroup of Sm and H a subgroup of Sn. Let

Z be a k(GoH)-module and Y a kH-module, so that ZY is a k(GoH)-module. Then we have an isomorphism of k(G_{o H)-modules}

Proof. We have

(Z_{Y )}∗ =Z_{⊗ Inf}G_HoHY∗ ∼

= Z∗⊗InfG_HoHY∗ (by (2.2.1))

And then by using the easily-verified fact thatInfG_HoHY∗ ∼= InfGHoH(Y∗) and

a second application of (2.2.1), the claim is established.

Proposition 4.3.3. Let G be a subgroup of Sm and let U, V be kG-modules.

Then we have an isomorphism of k(G_{o S}n)-modules

Une

⊗ Vne ∼

= (U⊗ V )ne

. Proof. Recall that, as k-vector spaces, we have Une

⊗ Vne = U⊗n

⊗ V⊗n_and

(U ⊗ V )ne

= (U⊗ V )⊗n. It is a routine calculation to verify that the formula (u1⊗ · · · ⊗ un)⊗ (v1⊗ · · · ⊗ vn)7−→ (u1⊗ v1)⊗ · · · ⊗ (un⊗ vn)

where ui ∈ U and vi ∈ V yields a well-defined k-linear map U⊗n⊗ V⊗n7−→

(U _{⊗ V )}⊗n which is a homomorphism of k(G_{o S}n)-modules. This map is then

immediately seen to be onto, and hence must be an isomorphism because the two spaces in question clearly have the same dimension.

Proposition 4.3.4. Let G be a subgroup of Sm and let U be a kG-module.

Then we have an isomorphism of k(G_{o S}n)-modules

Une
∗ ∼

= (U∗)ne

.

Proof. Firstly, recall that Une
∗ is equal as a k-vector space to Hom

k(U⊗n, k)

while (U∗)ne

is Homk(U, k)

⊗n .

Now if we take g1, . . . , gn ∈ U∗ = Homk(U, k), then it is a routine calcula-

on pure tensors by the formula u1⊗ · · · ⊗ un 7−→ g1(u1)· · · gn(un). Another

routine calculation then shows us that we have a well-defined k-linear map from Homk(U, k)

⊗n

to Homk(U⊗n, k) which is defined by mapping the pure

tensor g1⊗ . . . ⊗ gn in Homk(U, k)

⊗n

to the map defined above. We shall denote this map by Φ, and it is a routine calculation to show that Φ is then a k(G_{o S}n)-module homomorphism from (U∗)

e

n _{to U}ne
∗. Now it is clear

that these two modules have the same k-dimension, and so to prove that Φ is an isomorphism, it suffices to prove that it is onto.

So let us fix a k-basis u1, . . . , ud for U . Then U∗ has a k-basis f1, . . . , fd,

where fi : U −→ k is defined by fi(uj) = δij (where δij is the Kronecker

delta). Now for any n-tuple τ = (t1, . . . , tn) over the set {1, . . . , d}, we define

uτ to be the pure tensor ut1 ⊗ · · · ⊗ utn ∈ U

e

n_{. We now see that U}ne has

k-basis

{uτ : τ ∈ {1, . . . , d}n}.

Thus Une has k-basis

{fτ : τ ∈ {1, . . . , d}n}

where fτ is the k-linear map defined by fτ(uθ) = δτ θ. But it is now clear

that if τ = (t1, . . . , tn), then Φ(ft1 ⊗ · · · ⊗ ftn) = fτ so that Φ is onto as

required.

Proposition 4.3.5. Let G1 ⊆ G2 be subgroups of Sm and X a kG2-module.

Then we have an isomorphism of k(G1o Sn)-modules

h Xne i  y G2oSn G1oSn ∼ = hX y G2 G1 ine

Proof. This is immediate from the definition of (₋₎ne .

Proposition 4.3.6. [6, Lemma 3.1] Let G be a subgroup of Sm. Let α =

(α1, . . . , αt) be a composition of n, and for i = 1, . . . , t let Xi be a k(Go Sαi)-

Then X1· · · Xt x  Gon Goα ∼= X(1)π· · · X(t)π x  Gon Go(π_α)

where πα represents the composition (α(1)π, . . . , α(t)π), and where as usual

the symbols n, α, and π_{α represent the subgroups S}

n, Sα, and Sπ_α of S_n,

respectively.

Proposition 4.3.7. [6, Lemma 3.2] Let G be a subgroup of Sm. Let α =

(α1, . . . , αt) be a composition of n and let V be a k(GoSn)-module, W be a

k(G_oSα)-module, X be a kSn-module and Y be a kSα-module. Then we have

module isomorphisms 1. _{V  X}y Gon Goα∼= V↓ Gon Goα
 (X↓ n α) 2. V _{(Y ↑}n_α) ∼=  V_↓G_Goαon
 Y x  Gon Goα 3. W↑GGonoα
 X ∼= W  (X↓nα) x  Gon Goα

where as usual the symbols n and α represent the subgroups Sn and Sα of Sn,

respectively.

In document UNIVERSIDAD MAYOR DE SAN ANDRÉS FACULTAD DE AGRONOMÍA (página 61-0)